Atmospheric Transport Schemes: Desirable Properties And A

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Chapter 8 Atmospheric Transport Schemes: Desirable Properties and a Semi-Lagrangian View on Finite-Volume Discretizations <*,*+ T5 X(7+#,@*16 <(73 :5 _33+#0" (1/ Y(-(0"(1/+(1 B5 N(#+ Abstract !"#$ 0"(),*+ "($ ,;&.&3/ )7+)&$*5 :.,*+ ( $"&+, #1,+&/70,#&1 ,& ,"* -($$ 0&1,#17#,9 *A7(,#&1$ #1 (,-&$)"*+#0 -&/*3$6 /*$#+(%3* )+&)*+,#*$ .&+ -($$ ,+(1$4 )&+, $0"*-*$ #1,*1/*/ .&+ -*,*&+&3&2#0(3 ())3#0(,#&1$ (+* /#$07$$*/ #1 $&-* /*,(#35 !"#$ #1037/*$ ( /#$07$$#&1 &1 ,"* 0&-)3#0(,#&1$ 0(7$*/ %9 ,"* 1&143#1*(+#,9 &. -&$, )+&%3*-$ &. #1,*+*$, ,"(, -('*$ #, "(+/ ,& /*!1* (007+(09 (1/ 0&18*+2*10* ($ ,"* `,+7,"Q #$ 1&, '1&;15 !"*+*(.,*+6 $&-* !1#,*48&37-* $0"*-*$ .+&- ,"* (,-&$)"*+#0 3#,*+(,7+* (+* +*8#*;*/ (1/ /#$07$$*/5 !& 0&-)3*-*1, ,"* 3(+2* *?#$,#12 3#,*+(,7+* &1 !1#,*48&37-* $0"*-*$6 ( 3*$$ .+*A7*1,39 /#$07$$*/ $*-#4X(2+(12#(1 /*+#8(,#&1 &. ,"* !1#,*48&37-* -*,"&/ #$ 2#8*1 ,"(, .&07$*$ &1 `+*-()4,9)*Q $0"*-*$ ;"*+* ,"* $)(0* (1/ ,#-* /#$0+*,#@(,#&1$ (+* 0&-%#1*/ +(,"*+ ,"(1 $*)(+(,*/5 : /#$07$4 $#&1 &1 ,"* 0"(33*12*$ #1 /*+#8#12 (007+(,* $0"*-*$ #1,*1/*/ .&+ 23&%(3 -&/*3$ (1/ 1&14,+(/#,#&1(3 $)"*+#0(3 2+#/$ #$ 2#8*1 ($ ;*335 <5T5 X(7+#,@*1 N(,#&1(3 D*1,*+ .&+ :,-&$)"*+#0 Y*$*(+0"{ 6 M[RL !(%3* U*$( B+#8*6 \&73/*+6 DH [LZLR6 _S:6 *4-(#3^ )*3m70(+5*/7 { !"* N(,#&1(3 D*1,*+ .&+ :,-&$)"*+#0 Y*$*(+0" #$ $)&1$&+*/ %9 ,"* N(,#&1(3 S0#*10* I&71/(,#&15 <5:5 _33+#0" _1#8*+$#,9 &. 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(#+ -($$ #1 (,-&$)"*+#0 -&/*3$ ($ ,"#$ "($ .71/(-*1,(3 #1"7*10* &1 "&; (33 &,"*+ $)*0#*$ (+* ,+*(,*/5 !"* /*1$#,9 &. ;*334 -#?*/ -&#$, (#+ l 6 0(1 %* $*)(+(,*/ #1,& ( /+9 (1/ ;*, )(+, l6 = 6% + 6# = l% + l# = l% + 4# l6 , T =[5K> ;"*+* 6% (1/ 6# (+* ,"* -($$*$ &. ,"* /+9 (#+ (1/ ;(,*+ 8()&+6 +*$)*0,#8*396 (1/ T #$ ( $-(33 8&37-*5 !"* /*1$#,9 &. /+9 (#+ (1/ ;(,*+ 8()&+ (+* /*1&,*/ l % (1/ l# 6 +*$)*0,#8*396 (1/ 4 # #$ ,"* $)*0#!0 "7-#/#,96 4# = 6# . 6% + 6# =[5Z> !& ( 8*+9 2&&/ ())+&?#-(,#&1 ,"* -($$ &. /+9 (#+ #$ ,"* -($$ &. ,"* /&-#1(1, ;*334 -#?*/ 2($*$^ N#,+&2*1 N K =0(5 h[5L5[v>6 H?92*1 H K =0(5 KL5ORv>6 :+2&1 :+ =0(5 L5OZv> (1/ D(+%&1 /#&?#/* DH K =(, )+*$*1, 0(5 L5LZ[v>5 !"*$* 2($*$ -('* 7) &8*+ OO5OO[v &. ,"* 8&37-* &. /+9 (#+ (1/ -(9 ,"*+*.&+* %* 0&1$#/*+*/ )*+-(1*1, =(34 ,"&72" (+2&1 (1/ 0(+%&14/#&?#/* (+* $3&;39 #10+*($#12>5 G1 (//#,#&16 $-(33 (-&71,$ &. ,+(0* 2($*$ (+* -#?*/ #1,& ,"* (#+ =;#," $&7+0*$ (1/ $#1'$ 8(+9#12 #1 $)(0* (1/ ,#-*>6 "&;*8*+6 ,"* 8(+#(,#&1 #1 ,"*$* `1&14)*+-(1*1,Q 2($*$ #$ 8*+9 $-(33 0&-)(+*/ ,& ,"* ,&,(3 -($$ &. (33 ,"* ,+(0* 2($*$5 !+*1%*+," (1/ S-#," =KLLR> *$,#-(,*/ ,"(, ,"* /+9 (#+ -($$ &. ,"* (,-&$)"*+* 0&++*$)&1/$ ,& ( $7+.(0* )+*$$7+* &. ())+&?#4 -(,*39 O[Z5LR "<( (1/ #, 8(+#*$ 3*$$ ,"(1 L5LM "<( %($*/ &1 0"(12*$ #1 (,-&$)"*+#0 0&-)&$#,#&15 S& ,"* 8(+#(,#&1 #1 ,"* /+9 (#+ -($$ %7/2*, #$ &1 ,"* &+/*+ &. L5LLMv5 S& ,& ( 8*+9 2&&/ ())+&?#-(,#&1 ,"* 0&1,#17#,9 *A7(,#&1 .&+ /+9 (#+ /&*$ 1&, "(8* (19 $&7+0* &+ $#1' ,*+-$6 (1/ ,"7$ +*(/$ , l% + ¢ · (l% v) = L, ,? =[5P> ;"*+* v #$ ,"* 8*3&0#,9 !*3/ (1/ `¢·Q #$ ,"* /#8*+2*10* &)*+(,&+5 !"* -($$ &. /+9 (#+ (00&71,$ .&+ ())+&?#-(,*39 OOv &. ,"* ,&,(3 -($$ &. (,-&$)"*+* (1/ ,"* +*-(#1#12 Mv #$ ())+&?#-(,*39 ,"* -($$ &. ;(,*+ 8()&+5 !"* 0&1,#17#,9 *A7(,#&1 .&+ "7-#/#,9 =;(,*+ 8()&+> #$ 2#8*1 %9 , (l6 4# ) + ¢ · (l6 4# v) = 14# l6 , ,? =[5R> ;"*+* 14# l6 +*)+*$*1,$ $&7+0*$ (1/ $#1'$ =#1 ,"#$ 0($* 0&1/*1$(,#&1 (1/ *8()&+(,#&1 )+&0*$$*$>5 U&#$,7+* 4 # 8(+#*$ $#21#!0(1,39 =+*3(,#8*39 $)*('#12> ;#," 8(37*$ 1*(+ @*+& .&+ 0&3/ /+9 (#+ (1/ ( .*; )*+0*1, #1 ;(+- -&#$, (#+5 !"* 0&1,#17,9 *A7(,#&1 .&+ -&#$, (#+ 0(1 %* &%,(#1*/ %9 (//#12 =[5P> (1/ =[5R>6 (1/ 7$#12 =[5K> ,& $#-)3#.95 !"* +*$73, #$ , l6 + ¢ · (l6 v) = 1l6 . =[5a> ,? !"#$ *A7(,#&1 #$ $#-#3(+ ,& ,"* *A7(,#&1 .&+ /+9 (#+ =[5P> *?0*), .&+ ,"* "7-#/#,9 .&+0#12 ,*+-$5 [ I#1#,*4E&37-* !+(1$)&+, S0"*-*$ MOM !"* )+&21&$,#0 8(+#(%3*$ 7$*/ .&+ ,+(0*+$ (+* 7$7(339 /*!1*/ #1 ,*+-$ &. -#?#12 +(,#&$5 G. -&#$, /*1$#,9 #$ )+&21&$*/6 ,"* -#?#12 +(,#&$ .&+ ,+(0*+$ (+* -&$, 0&18*4 1#*1,39 /*!1*/ #1 ,*+-$ &. ,"* $)*0#!0 0&10*1,+(,#&1 (*) 46 = 6(*) , 6% + 6# =[5h> ;"*+* 6(*) #$ ,"* -($$ &. 0&1$,#,7*1, (*)5 S& ,"* /*1$#,9 &. ,"* 0&1$,#,7*1, #$ l (*) = (*) (*) 46 l6 6 ;"*+* 46 #$ ,"* `-&#$,Q -#?#12 +(,#&5 T&;*8*+6 &1* -(9 (3$& $&38* ,"* (*) 0&1,#17#,9 *A7(,#&1 .&+ ,+(0*+$ #1 ,*+-$ &. ,"* `/+9Q -#?#12 +(,#& 4 % 6 /*!1*/ %9 (*) (*) 4% = 6% . 6% =[5[> :$ /#$07$$*/ #1 D&33#1$ *, (3 =KLLP> ,"* (/8(1,(2* &. 7$#12 =[5h> #$ ,"(, ,"* -($$ &. $)*0#*$ (*) #$ &%,(#1*/ %9 $#-)39 -73,#)39#12 ,"* -&#$, -#?#12 +(,#& ;#," ,"* (*) -&#$, (#+ /*1$#,9 4 6 l6 5 T&;*8*+6 ,"#$ ())+&(0" "($ ,"* /#$(/8(1,(2* &. #-)3#0#,39 (*) +*A7#+#12 ( 0"(12* #1 4 6 ;"*1*8*+ ,"* ;(,*+ 8()&+ 4 # 0"(12*$5 !"#$ /#$(/8(1,(2* /&*$ 1&, *?#$, #. =[5[> #$ 7$*/5 8.2.2 Consistency in the mass equations T*+*#1 ;* ;#33 +*$)*0,#8*39 7$* l (1/ 4 ,& /*1&,* (#+ /*1$#,9 (1/ -#?#12 +(,#& =;"#0" 0(1 %* *#,"*+ -&#$, &+ /+9> (1/ ;* ($$7-* 1& $&7+0*$ &+ $#1'$ =1& .&+0#12 ,*+-$>5 !"*1 ,"* ,;&4/#-*1$#&1(3 0&1,#17#,9 *A7(,#&1 .&+ (#+ /*1$#,9 l 0(1 %* ;+#,,*1 ($ ,l + ¢ · (l v) = L, ,? =[5O> (1/ $#-#3(+39 .&+ ( ,+(0*+ /*1$#,9 l 4 , (l 4) + ¢ · (l 4 v) = L, ,? =[5ML> ;"*+* v #$ ,"* 8*3&0#,9 8*0,&+5 N&,* ,"(, =[5O> (1/ =[5ML> #-)39 %4 = L, %? % , ≡ + v · ¢, %? ,? =[5MM> ;"#0" $,(,*$ ,"(, 4 #$ 0&1$*+8*/ (3&12 ,+(]*0,&+#*$V0"(+(0,*+#$,#0$ &. ,"* "&;5 N&,* ,"(, ,"* 0&1,#17#,9 *A7(,#&1$ =[5O> (1/ =[5ML> (+* 3#1'*/ #1 ,"* $*1$* ,"(, l ())*(+$ #1 %&," *A7(,#&1$5 T*10*6 17-*+#0(3 *++&+ #1,+&/70*/ #1 $#-73(,#12 ,"* *8&37,#&1 &. (#+ -($$ l -(9 %* +*"*0,*/ #1 ,"* )+&21&$*/ ,+(0* 2($ -#?#12 +(,#&$ ;"*1 0&18*+,#12 .+&- ,+(0*+ -($$ l 4 ,& -#?#12 +(,#& 45 MOK X(7+#,@*1 *, (35 !& $&38* (19 &. ,"* 0&1,#17#,9 *A7(,#&1$ 2#8*1 (%&8* ,"* "&; !*3/ v -7$, %* 2#8*15 !"* 0&1,#17#,9 *A7(,#&1 .&+ (#+ =[5O> #$ 0&7)3*/ ;#," ,"* -&-*1,7- *A7(4 ,#&1$ (1/ ,"*+-&/91(-#0 *A7(,#&1$5 T*10* ,"* ,"*+-&/91(-#0 8(+#(%3*$ (1/ &,"*+ )+&21&$,#0 8(+#(%3*$ .**/ %(0' &1 ,"* 8*3&0#,9 !*3/ ;"#0"6 #1 ,7+16 .**/$ %(0' &1 ,"* $&37,#&1 ,& ,"* 0&1,#17#,9 *A7(,#&15 G, .&33&;$ ,"(, ,"* 0&1,#17#,9 *A7(,#&1 .&+ (#+ 0(11&, %* $&38*/ #1 #$&3(,#&1 (1/ &1* -7$, &%*9 ,"* -(?#-7- (33&;(%3* ,#-*4$,*) +*$,+#0,#&1$ #-)&$*/ %9 ,"* .($,*$, ;(8*$ #1 ,"* $9$,*- Z =$** 0"(),*+ M (1/ a>5 !"* )($$#8* ,+(0*+ ,+(1$)&+, *A7(,#&1 =[5ML> &+ =[5MM> 0(1 %* $&38*/ #1 #$&3(,#&1 2#8*1 )+*$0+#%*/ ;#1/$ (1/ (#+ /*1$#,#*$6 (1/ #$ ,"*+*.&+* 1&, $7$0*),#%3* ,& ,"* $,+#0,*+ ,#-*4$,*) +*$,+#0,#&1$ #-)&$*/ %9 ,"* .($,*$, ;(8*$ #1 ,"* $9$,*- %7, `&139Q ,& ,"* 3*$$ +*$,+#0,#8* (/8*0,#8* 8*3&0#,#*$ P 5 T*10*6 #. .&+ $,(%#3#,9 +*3(,#8*39 $"&+, ,#-*4$,*)$ -7$, %* 7$*/ .&+ ,"* 0&1,#17#,9 *A7(,#&1 .&+ (#+6 &1* /&*$ 1&, 1*0*$$(+#39 1**/ ,& 7$* $"&+, ,#-*4$,*)$ .&+ ,"* ,+(0*+$ =(, 3*($, 1&, .&+ $,(%#3#,9 +*($&1$>5 !"(, #$6 &1* 0(1 $&38* ,"* ,+(0*+ ,+(1$)&+, *A7(,#&1$ ;#," ,#-*4$,*)$ 3&12*+ ,"(1 ;"(, (+* (33&;*/ .&+ $,(%#3#,9 #1 =[5O>5 !"#$ ,*0"1#A7* #$ +*.*++*/ ,& ($ ;5>K2J2*$)E6 ,"(, #$6 -73,#)3* 0903*$ &. /91(-#0$ =(#+ 0&1,#17#,9 *A7(,#&1> (+* )*+.&+-*/ ;#,"#1 &1* ,#-*4$,*) &. ,"* ,+(0*+$5 G1 / $& 0(+* -7$, %* ,('*1 ,& +*,(#1 ,"* 0&1$#$,*109 %*,;**1 ,+(04 *+$ (1/ (#+5 I&+ *?(-)3*6 #. 4 = M ,"*1 =[5ML> +*/70*$ ,& =[5O> (1/ (//#,#&1(3 0(+* -7$, %* ,('*1 ,& *1$7+* 0&1$#$,*109 %*,;**1 ,"*$* *A7(,#&1$ #1 ,"* /#$0+*,#@(,#&15 S)*0#!0 *?(-)3*$ (1/ /*,(#3$ &1 $7%40903#12 (+* 2#8*1 3(,*+ =$*0,#&1 [5h5K>5 I#+$,6 3*, 7$ 0&1$#/*+ #-)&+,(1, /*$#21 &%]*0,#8*$ .&+ ,+(0*+ ,+(1$)&+, $0"*-*$ #1,*1/*/ .&+ (,-&$)"*+#0 ())3#0(,#&1$5 8.3 Desirable properties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
, &. 8#*; )+&)*+,#*$ $70" ($ )(+(33*3 *.!0#*1096 2*&-*,+#0 "*?4 #%#3#,96 *,05 (+* (3$& 8*+9 #-)&+,(1, )+&)*+,#*$ &. ,"* !1(3 17-*+#0(3 (32&+#,"-5 C"(, .&33&;$ #$ ( 3#$, &. /*$#+(%3* )+&)*+,#*$ .&+ ,+(0*+ ,+(1$)&+, $0"*-*$ ,"(, (+* (33 =()(+, .+&- ,"* )+&)*+,#*$ +*3(,*/ ,& *.!0#*109> *$$*1,#(3 #12+*/#*1,$ &. $#-73(,#&1 8*+(0#,9^ 8.3.1 Accuracy (error norms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inear test cases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l 45 !"(, #$6 4 &+ l 4 #$ $*, *A7(3 ,& ,"* $(-* $)(,#(3 /#$,+#%7,#&1 (1/ ,"* -&/*3*+ R .&+ ( /#$07$$#&1 &1 0&1$*+8(,#&1 #1 ,"* 0&1,*?, &. ,"* .733 *A7(,#&1 $*, .&+ ,"* (,-&$)"*+* $** 0"(),*+ MM MOP X(7+#,@*1 *, (35 (a) (b) Fig. 8.1 F?(0, $&37,#&1$ .&+ ,"* =(> $&3#/ %&/9 (/8*0,#&1 &. ( 0&$#1* %*33 ,*$, 0($* (, ? = L =0*1,*+ &. )3&,> (1/ ? = PP" =.&+ ( `"&; +&,(,#&1 (123*Q &. PR◦ >6 (1/ =%> ,"* $,(,#0 8&+,*? ,*$, 0($* (, /(9 a5 #$ #-)3#0#,39 ($$7-#12 ,"(, l #$ &1* *8*+9;"*+* (1/ $#10* ,"* "&; #$ 1&14/#8*+2*1, l ;#33 +*-(#1 &1* ,"+&72"4&7, ,"* $#-73(,#&1 (, 3*($, #1 ,"* (1(39,#0(3 0($*5 T*10* ,"* -&/*3*+ #$ 1&, .&+0*/ ,& /#$,#127#$" %*,;**1 ,+(0*+ -($$ l 4 (1/ -#?#12 +(,#& 45 T&;*8*+6 .&+ ( /#8*+2*1,V0&18*+2*1, "&; &139 4 #$ 0&1$,(1, (3&12 )(+0*3 ,+(]*0,&+#*$ ;"*+*($ ,+(0*+ -($$ l 4 ;#33 #10+*($*V/*0+*($* #1 (+*($ &. 0&18*+2*10*V/#8*+2*10*5 I&+ ( .733*+ /#$07$$#&1 $** N(#+ (1/ X(7+#,@*1 =KLML>5 Translational. <+&%(%39 ,"* -&$, 0&--&139 7$*/ #/*(3#@*/ ,*$, 0($* #1 ,"* -*4 ,*&+&3&2#0(3 3#,*+(,7+* #$ ,"* $&3#/ %&/9 +&,(,#&1 &. ( 0&$#1* %*33 =I#27+* [5M(> =*5256 ,*$, 0($* &1* &. ,"* ;#/*39 7$*/ ,;&4/#-*1$#&1(3 ,*$, $7#,* &. C#33#(-$&1 *, (3 MOOK .&+ ,"* $"(33&;4;(,*+ *A7(,#&1$>5 !"* *?(0, $&37,#&1 #$ $#-)39 ,"* ,+(1$3(,#&1 &. ,"* #1#,#(3 0&1/#,#&1 (1/ $,(1/(+/ *++&+ 1&+-$ 0(1 %* 0&-)7,*/ (, *8*+9 ,#-*4$,*)5 !"#$ ,;&4/#-*1$#&1(3 ,*$, 0($* "($ %**1 *?,*1/*/ ,& ,"+** /#-*1$#&1$ #1 J(%3&1&;$'# *, (3 =KLML>5 :1&,"*+ ,"+** /#-*1$#&1(3 ,*$, 0($* &1 ,"* $)"*+* ;"*+* ,"* (1(39,#0 $&37,#&1 #$ '1&;1 ;($ )+&)&$*/ %9 f7%&8 *, (3 =MOOO>5 I&+ 0&18*+2*10* $,7/#*$ 7$*/ ,& ($$*$$ ,"* .&+-(3 &+/*+ &. (007+(09 &. ( $0"*-* ,"* ,+(1$3(,*/ /#$,+#%7,#&1 $"&73/ %* $7.!0#*1,39 $-&&,"5 I&+ *?(-)3*6 ,"* 0&$#1* %*33 /#$,+#%7,#&1 -(9 ())*(+ $-&&," %7, #, #$ &139 / M (, ,"* %($* &. ,"* %*335 D&14 $*A7*1,396 $0"*-*$ ,"(, (+* "#2"4&+/*+ (007+(,* #1 ,*+-$ &. ( !(93&+ S*+#*$ (1(39$#$ -(9 1&, $"&; ,"#$ "#2"4&+/*+ .&+-(3 0&18*+2*10* +(,* ;"*1 7$#12 ,"* 0&$#1* %*33 #1#,#(3 0&1/#,#&15 !& ($$*$$ `#/*(3Q 0&18*+2*10* +(,*$ #, #$ (/8#$*/ ,& 7$* / ' .7104 ,#&1$ $70" ($ W(7$$#(1 $7+.(0*$ =X*89 *, (36 KLLh>5 Deformational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s0 MOOK6 N(#+ (1/ U(0"*1"(7*+ KLLK>5 !"* *?(0, $&37,#&1 (, /(9 a #$ $"&;1 &1 I#25 [5M% a5 :$ 0(1 %**1 $**1 #1 ,"* I#24 a ,"* /#-*1$#&1(3#@(,#&1 &. ,"* 8&+,*? )+&%3*- 7$*/ "*+* .&33&;$ N(#+ (1/ J(%3&1&;$'# =KLL[> [ I#1#,*4E&37-* !+(1$)&+, S0"*-*$ MOR Fig. 8.2 F?(0, $&37,#&1$ .&+ ,"* -&8#12 8&+,*? ,*$, 0($* ;#," ( "&; &+#*1,(,#&1 &. 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C#$0&1$#1> (1/ )7%3#$"*/ #1 Y($0" *, (3 =KLLa>5 G, ,(+2*,$ ,"* -&/*3Q$ (%#3#,9 ,& $#-73(,* ,+(1$)&+, &. 0&1$*+8*/ ,+(0*+$ 0&1$#$,*1,39 (1/ ,"* -&/*3Q$ (%#3#,9 ,& -(#14 ,(#1 1&143#1*(+ +*3(,#&1$"#)$ %*,;**1 $#? /#..*+*1, 0&1$*+8*/ (1/ 1&140&1$*+8*/ ,+(0*+$5 G, 0(1 %* $"&;1 .+&- ,"* $*0&1/ 3(; &. ,"*+-&/91(-#0$ ,"(, ,;& )
,$ $*)(+(,*/ #1 $)(0* (1/ ,#-* 0&11*0,*/ %9 ( ,+(]*0,&+9 $"&73/ $(,#$.9 ( 1&143#1*(+ +*3(,#&1$"#) #1 ,*+-$ &. ,*-)*+(,7+* + 6 )&,*1,#(3 ,*-)*+(,7+* e (1/ )+*$$7+* @ =$** :))*1/#? &. Y($0" *, (3 KLLa>^ eM = ! eL +L " +M ! @L @M "(// @ , =[5MK> ;"*+* ,"* $7%$0+#), L (1/ M +*.*+ ,& ,"* ,;& )
,$6 (1/ ( (1/ / @ (+* ,"* 2($ 0&1$,(1, (1/ $)*0#!0 "*(, 0&1$,(1, (, 0&1$,(1, )+*$$7+*6 +*$)*0,#8*395 !"* ,*$, 0($* 0&1$#$,$ &. )+*/#0,#12 e 6 + (1/ @ (// @ $*)(+(,*39 (1/ ,"*1 0"*0' "&; ;*33 ,"*9 &%*9 =[5MK>5 !"* 3*8*3 &. (2+**-*1, %*,;**1 ,"*$* ,;& ;(9$ &. 0&-)7,#12 )&,*1,#(3 ,*-)*+(,7+* 9#*3/$ ( -*($7+* &. ,"* /*2+** &. 0&1$#$,*109 #1 ,"* -&/*35 S** Y($0" *, (3 =KLLa> .&+ /*,(#3$5 G, #$ )+&%(%39 #-)&$$#%3* ,& 0&1$,+70, ( $0"*-* ,"(, ;#33 *?(0,39 .73!33 ,"#$ 0&1$#$,*109 ,*$,6 "&;*8*+6 #, #$ /*$#+(%3* ,"(, $0"*-*$ $,+#8* ,& %* ($ 0&1$#$,*1, ($ )&$$#%3*5 8.3.2 Conservation of mass :$ /#$07$$*/ #1 $*0,#&1 [5K5M &1* &. ,"* -&$, .71/(-*1,(3 %7/2*,$ &. ,"* 23&%(3 (,-&$)"*+* #$ ,"(, .&+ ,"* -($$ &. /+9 (#+5 S#10* ,"* )"9$#0(3 8(+#(,#&1 #1 ,"* /+9 (#+ -($$ %7/2*, #$ &1 ,"* &+/*+ &. L5LLMv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h %*3&;> (1/ 0&1$#$,*109 %*,;**1 ,+(0*+ (1/ (#+ -($$ =$** $*0,#&1 [5Z5P %*3&;>5 !"*+*.&+* !1#,*4 8&37-* -*,"&/$ ,"(, (+* #1"*+*1,39 0&1$*+8(,#8*6 "(8* %*0&-* ( )&)73(+ 17-*+#0(3 -*,"&/ #1 03#-(,* (1/ 0"*-#$,+9 -&/*3#12 $#10* "%K,32 (/]7$,-*1,$ (+*6 #1 ,"*&+96 1&, 1*0*$$(+9O5 !"* 0&1,#17&7$ *A7(,#&1$ &. -&,#&1 0&1$*+8* (33 -&-*1,$ 1&, ]7$, -($$5 T&;4 *8*+6 !"7%7+1 =KLL[> (+27*/ =$** (3$& 0"(),*+ MM> #, -#2", 1&, %* /*$#+(%3* ,"(, ,"* (/8*0,#&1 $0"*-* (3$& )+*$*+8*$ "#2"*+4&+/*+ -&-*1,$5 8.3.3 Optimal diffusion and dispersion properties !"* 3#1*(+ /#..7$#&1 (1/ /#$)*+$#&1 )+&)*+,#*$ &. ( 3#1*(+#@*/ $0"*-* 0(1 %* ($$*$$*/ %9 )*+.&+-#12 ( 8&1 N*7-(11 $,(%#3#,9 (1(39$#$ =(3$& '1&;1 ($ ( I&7+#*+ $,(%#3#,9 (1(39$#$>5 G, #$ ( $,(1/(+/ (1(39,#0 (1(39$#$ ,*0"1#A7* (1/ #$ /*$0+#%*/ #1 -(19 ,*?,4 %&&'$ #1 ,"* 0&1,*?, &. 2+#/4)
, -*,"&/$ =$**6 *5256 B7++(1 MOOO6 T(3,#1*+ (1/ C#33#(-$ MO[L> (1/ #1 ,"* 0&1,*?, &. !1#,*48&37-* -*,"&/$ #1 X(7+#,@*1 =KLLh>5 !"* (1(39$#$ 0&1$#$,$ &. ($$*$$#12 (1(39,#0(339 "&; ( $#123* I&7+#*+ -&/* #$ /(-)*/ (1/ (00*3*+(,*/V/*0*3*+(,*/ %9 ,"* 17-*+#0(3 $0"*-* /7+#12 &1* ,#-*4$,*) ($$7-4 #12 ( 0&1$,(1, ;#1/ !*3/5 G1 &1* /#-*1$#&1 ,"* 8&1 N*7-(11 (1(39$#$ #$ )*+.&+-*/ %9 ($$7-#12 ( $&37,#&1 #1 ,"* .&+s ) (B) = s L K ) *?)(j" g B) , =[5MZ> ;"*+* "j #$ ,"* #-(2#1(+9 71#,6 s L ,"* #1#,#(3 (-)3#,7/*6 (1/ g = K / /G #$ ,"* ;(8*17-%*+ =G #$ ,"* ;(8*3*12,">6 (1/ ) #$ ,"* ,#-*43*8*3 #1/*?5 !"* /(-)#12 (1/ )"($* )+&)*+,#*$ &. ( $0"*-* (+* ($$*$$*/ %9 $7%$,#,7,#12 ,"* $&37,#&1 =[5MZ> #1,& ,"* .&+*0($, .&+-73( .&+ ,"* !1#,*48&37-* $0"*-* #1 A7*$,#&16 (1/ $7%$*A7*1,39 (14 (39@#12 ,"* 0&-)3*? (-)3#!0(,#&1 .(0,&+ K 5 !"* $,(%#3#,9 &. ( 17-*+#0(3 -*,"&/ #$ 2&8*+1*/ %9 ,"* -&/737$ &. ,"* 0&-)3*? (-)3#!0(,#&1 .(0,&+6 ,"(, #$6 ( )(+,#074 3(+ ;(8* ;#," ;(8*17-%*+ g #$ $,(%3* #. |K | ≤ M5 I&33&;#12 \(,*$ (1/ U0B&1(3/ =MO[K> ,"* /#$)*+$#&1 )+&)*+,#*$ &. ( $0"*-* #$ ($$*$$*/ %9 ;+#,#12 ,"* 0&-)3*? (-)3#!0(,#&1 .(0,&+ ($ K = |K | *?)(−j"t ∗ 6 ?) , =[5MP> O ;* ;+#,* `#1 ,"*&+9Q $#10* #. ( ,+(1$)&+, $0"*-* #$ 1&, $,+#0,39 -&1&,&1* 3&0(3 `(/ "&0Q (/]7$,-*1,$ -#2", %* 1*0*$$(+9 *8*1 .&+ !1#,*48&37-* -*,"&/$ [ I#1#,*4E&37-* !+(1$)&+, S0"*-*$ KLM ;"*+* t ∗ #$ ,"* 17-*+#0(3 .+*A7*1095 B*!1* ,"* +*3(,#8* .+*A7*109 ($ ( = t ∗ /t ;"*+* t #$ ,"* *?(0, .+*A7*109 2#8*1 %9 g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racer and air mass consistency !+(0*+ (1/ (#+ -($$ 0&1$#$,*109 #$ ( $,+#0,*+ 0&10*), ,"(1 $#-)3* -($$ 0&1$*+8(,#&1 &. ,"* #1/#8#/7(3 A7(1,#,#*$5 G, %($#0(339 $,(,*$ ,"(, ,"* /#$0+*,#@*/ ,+(0*+ ,+(1$)&+, $0"*-* $"&73/ +*/70* ,& ,"* /#$0+*,#@*/ 0&1,#17#,9 *A7(,#&1 .&+ (#+ ;"*1 4 = M ($ #$ ,"* 0($* .&+ ,"* 0&1,#17&7$ *A7(,#&1$^ =[5ML> +*/70*$ ,& =[5O> ;"*1 $*,,#12 4 = M5 !+(0*+4(#+ -($$ 0&1$#$,*109 0(16 .&+ *?(-)3*6 %* 8#&3(,*/ #. 7$#12 ( 17-*+#0(3 -*,"&/ .&+ ,+(0*+ ,+(1$)&+, ,"(, #$ /#..*+*1, .+&- ,"* $0"*-* 7$*/ .&+ )+*/#0,#12 ,"* *8&37,#&1 &. ,"* (#+ /*1$#,9 ML 5 !& (0"#*8* ( "#2" 3*8*3 &. 0&1$#$,*109 #, #$ 7$7(339 1*0*$$(+9 ,"(, ,"* $(-* 17-*+#0(3 (32&+#,"- #$ 7$*/ .&+ ,"* /91(-#0$ ($ ;*33 ($ .&+ ,+(0*+ ,+(1$)&+,5 I&+ -&+* /#$07$$#&1 $** U(0"*1"(7*+ *, (3 =KLLO>o X** *, (3 =KLLP>o Je&0'*3 *, (3 =KLLM>o f"(12 *, (3 =KLL[>5 8.3.5 Divergence preservation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l 4 %9 l .+&- =[5O>5 S& *8*1 #. ,"* 17-*+#0(3 $0"*-* #$ 71(%3* ,& )+*$*+8* ( 0&1$,(1, -($$ !*3/ l 6 #, #$ 7$7(339 )&$$#%3* ,& /*$#21 $0"*-*$ $& ,"(, ( 0&1$,(1, 4 !*3/ #$ +*0&8*+*/ ;"*1 /#8#/#12 l 4 %9 ,"* =)&,*1,#(339> 1&14/#8*+2*10* )+*$*+8#12 .&+*0($, &. l 5 8.3.6 Physical realizability (monotone, positive-definite, non-oscillatory, shape-preserving) G1 ,"* (%$*10* &. $&7+0*$ (1/ $#1'$ ,"* -#?#12 +(,#& &. ( X(2+(12#(1 )(+0*3 %*4 #12 ,+(1$)&+,*/ %9 ,"* "&; #$ #18(+#(1, =[5MM>5 G. ,"* 17-*+#0(3 $&37,#&1 .73!33$ ,"#$ )+&)*+,9 #, #$ 63)3?3)$2$?J MM @:-;-:#$)Eo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l 45 I&+ 0&18*+2*1, "&;$ l 4 0(1 )"9$#0(339 ,('* 8(37*$ &7,$#/* ,"* +(12* &. ,"* #1#,#(3 0&1/#,#&1 ;"*+*($ 4 $"&73/ 1&,5 S** N(#+ (1/ X(7+#,@*1 =KLML> .&+ ( /#$07$$#&1 (1/ $#-)3* #337$,+(,#&1 &. ,"* 3(,,*+ .&+ (1 #/*(3#@*/ "&; !*3/5 N&,* ,"(, $"()*4)+*$*+8(,#&1 0(1 %* *1.&+0*/ #1 !1#,*48&37-* $0"*-*$ %($*/ &1 =[5O> (1/ =[5ML> #. ,"*$* $0"*-*$ #-)39 $&-* /#$0+*,#@*/ 8*+$#&1 &. =[5MM>5 S0"*-*$ ,"(, +*,(#1 $70" ( )+&)*+,9 (+* ,*+-*/ 236@"?$>*- =S0"e(+ (1/ S-&3(+'#*;#0@6 MOOa>5 MM (,-&$)"*+#0 -&/*3*+$ ,*1/ ,& %* ( %#, 3&&$* ;#," ,"* ,*+- `-&1&,&1*Q (1/ 1&+-(339 ,"*9 /& 1&, +*.*+ ,& ,"* 0(+*.73 /*!1#,#&1 2#8*1 %9 T(+,*1 =MO[Z> [ I#1#,*4E&37-* !+(1$)&+, S0"*-*$ KLZ 8.3.7 Preservation of pre-existing functional relations between species (correlations) :$ /*$0+#%*/ #1 <37-% =KLLh>^ tY*3(,#&1$"#)$ %*,;**1 3&1243#8*/ $,+(,&$)"*+#0 ,+(04 *+$6 -(1#.*$,*/ #1 $#-#3(+ $)(,#(3 $,+70,7+*$ &1 $0(3*$ +(12#12 .+&- ( .*; ,& $*8*+(3 ,"&7$(1/ '#3&-*,*+$6 (+* /#$)3(9*/ -&$, $,+#'#1239 #. ,"* -#?#12 +(,#& &. &1* #$ )3&,4 ,*/ (2(#1$, (1&,"*+6 ;"*1 ,"* /(,( 0&33()$* &1,& +*-(+'(%39 0&-)(0, 07+8*$5u G1 &,"*+ ;&+/$6 /#..*+*1, 3&123#8*/ ,+(0* 0&1$,#,7*1,$ =$70" ($ 1#,+&7$ &?#/* . K W (1/ `,&,(3 &// 1#,+&2*1Q .W J > $**- ,& %* +*3(,*/ ,"+&72" +(,"*+ $#-)3* .710,#&1(3 +*4 3(,#&1$"#)$ #16 .&+ *?(-)3*6 ,"* )&3(+ $,+(,&$)"*+#0 8&+,*?5 S70" +*3(,#&1$"#)$ 0(1 (+#$* .+&- /#..*+*1, +*($&1$ =<37-% (1/ d&6 MOOK>6 "&;*8*+6 #, #$ ;*334'1&;1 ,"(, ,+(1$)&+, 0(1 *$,(%3#$" $70" +*3(,#&1$ =*525 !"7%7+1 (1/ U031,9+*6 MOOh>5 G1 &+/*+ ,& (007+(,*39 $#-73(,* $70" +*3(,#&1$"#)$ #1 17-*+#0(3 -&/*3$6 ,"* ,+(1$4 )&+, &)*+(,&+ $"&73/6 (, 3*($,6 %* (%3* ,& )+*$*+8* 3#1*(+ 0&++*3(,#&1$ =X#1 (1/ Y&&/6 MOOao !"7%7+1 (1/ U031,9+*6 MOOh>5 !"(, #$6 ,"* ,+(1$)&+, &)*+(,&+ $"&73/ -(#1,(#1 ,"* +*3(,#&1$"#) #1 =[5MR> ,"+&72"&7, ,"* $#-73(,#&1 4M = a (L) + a (M) 4K , =[5MR> ;"*+* a ($) 6 $ = L, M6 (+* 0&1$,(1,$6 (1/ 4 $ 6 $ = M, K6 (+* -#?#12 +(,#&$ &. ,;& 3#1*(+39 #1,*++*3(,*/ $)*0#*$5 : ,+(1$)&+, $0"*-* ;#33 )+*$*+8* 3#1*(+ )+*4*?#$,#12 .710,#&1(3 +*3(,#&1$ #. ,"* ,+(1$)&+, &)*+(,&+ T 6 ,"(, 7)/(,*$ 4 $ 6 $ = M, K6 #1 ,#-*6 #$ `$*-#43#1*(+Q $ # T (4M ) = T a (L) + a (M) 4K = a (L) T (M) + a (M) T (4K ) = a (L) + a (M) T (4K ), =[5Ma> =X#1 (1/ Y&&/6 MOOao !"7%7+1 (1/ U031,9+*6 MOOh>5 :$ 1&,*/ %9 !"7%7+1 (1/ U031,9+* =MOOh> ,"* $700*$$.73 )+*$*+8(,#&1 &. 3#1*(+ 0&++*3(,#&1$ %9 ( ,+(1$)&+, &)*+(,&+ /&*$ 1&, 1*0*$$(+#39 27+(1,** (1 (007+(,* $&37,#&1 $#10* $"()#12 ,;& ,+(0*+ !*3/$ ,"* $(-* ;(9 /&*$ 1&, 1*0*$$(+#39 #-)39 $"()#12 ,"*- ,"* +#2", ;(95 H1 ,"* &,"*+ "(1/6 #. ( -&/*3 $#21#!0(1,39 8#&3(,*$ ,"* )+*$*+8(,#&1 &. 3#1*(+ 0&++*3(,#&1$ %*,;**1 0"*-#0(3 0&1$,#,7*1,$6 ,"* -&/*3 #$ -&$, 3#'*39 1&, 2 ,& )+&8#/* ,+7,".73 $#-73(,#&1$ &. ,"* +*3(,#&1 %*,;**1 ,"&$* 0&1$,#,7*1,$5 S#10* #1,*++*3(,*/ ,+(0*+$ 0(1 (3$& %* +*3(,*/ 1&143#1*(+396 #, #$ (3$& &. #1,*+*$, ,& #18*$,#2(,* "&; ( ,+(1$)&+, &)*+(,&+ /#$,&+,$ $70" 1&143#1*(+ +*3(,#&15 I&+ *?(-)3*6 0&1$#/*+ ,;& ,+(0*+$ ,"(, (+* #1#,#(339 0&++*3(,*/ %9 ( .&7+,"4&+/*+ )&391&-#(3 4M = a (L) + a (M) (4K )P , =[5Mh> =!"7%7+1 (1/ U031,9+*6 MOOh> ;"*+* ,"* 0&1$,(1,$ a (L) (1/ a (M) $"&73/ %* 0"&4 $*1 $& ,"(, ,"* .710,#&1(3 +*3(,#&1 #$ *#,"*+ 0&18*? &+ 0&10(8* #1 ,"* +(12* &. ,"* #1#,#(3 0&1/#,#&1 8(37*$ &. 4 M (1/ 4K 5 F?0*), .&+ .7339 X(2+(12#(1 ,+(1$)&+, &)*+(4 ,&+$6 $0"*-*$ (+* 7$7(339 71(%3* ,& -(#1,(#1 1&143#1*(+ .710,#&1(3 +*3(,#&1$"#)$ (1/ ,"*#+ /*2+** &. 1&14)+*$*+8(,#&1 &. 0&++*3(,#&1$ *..*0,#8*39 ,+(1$3(,*$ #1,& 17-*+#4 0(3 -#?#12 &. ,"* 0&1$,#,7*1,$5 G1#,#(3#@#12 ,;& ,+(0*+$ ,"(, (+*6 .&+ *?(-)3*6 +*3(,*/ ,"+&72" =[5Mh> (1/ 3*,,#12 ,"* ,+(0*+$ %* ,+(1$)&+,*/ %9 ( 0"(33*12#12 "&; ,"(, /*4 KLP X(7+#,@*1 *, (35 8*3&)$ .*(,7+*$ ,"(, 0&33()$* ,& ,"* 1*(+ 2+#/4$0(3*6 )+&8#/*$ )"9$#0(3 #1$#2", #1,& ,"* 17-*+#0(3 -#?#12 ,"(, ,"* ,+(1$)&+, &)*+(,&+ #1,+&/70*$ =!"7%7+1 (1/ U031,9+*6 MOOh>5 N& )+(0,#0(3 F73*+#(1 (1/ $*-#4X(2+(12#(1 $0"*-* 0(1 )+*$*+8* =[5Mh> (1/ ;#33 ,"*+*.&+* )+&/70* $0(,,*+ )
,$ ,"(, /*8#(,* .+&- ,"* )+*4*?#$,#12 .710,#&1(3 +*3(,#&1$"#) =[5Mh>5 C"*1 $0(,,*+ )
,$ /*8#(,* .+&- ,"* )+*4*?#$,#12 .710,#&1(3 +*4 3(,#&1 07+8* ,"* ,+(1$)&+, &)*+(,&+ #$ #1,+&/70#12 17-*+#0(3 -#?#125 !"* 17-*+#0(3 -#?#12 0(1 *#,"*+ %* $)7+#&7$ &+ +*$*-%3* `+*(3Q -#?#125 G. ,"* $0(,,*+ 8(37*$ (+* &1 ,"* 0&10(8* $#/* &. ,"* )+*4*?#$,#12 .710,#&1(3 +*3(,#&16 ,"* 17-*+#0(3 -#?#12 #$ $#-#3(+ ,& `+*(3 -#?#12Q ,"(, #$ &%$*+8*/ #1 ,"* (,-&$)"*+* =$** I#25 [5R>5 U#?#12 #1 ,"* (,-&$)"*+* &007+$6 .&+ *?(-)3*6 ;"*1 ,"* )&3(+ $,+(,&$)"*+#0 8&+,*? %+*('$ 7) =*525 C(72" *, (36 MOOh>5 G. $0(,,*+ 8(37*$ ())*(+ &7,$#/* ,"* `0&18*? "733Q =*#,"*+ %9 )+&/70#12 $0(,,*+ )
,$ &1 ,"* 0&18*? $#/* &. ,"* )+*4*?#$,#12 .710,#&1(3 +*3(,#&14 $"#) 07+8* (1/V&+ &7,$#/* ,"* +(12* &. ,"* #1#,#(3 0&1/#,#&1 .&+ 4 $ 6 $ = M, K>6 ,"* -&/*3 )+&/70*$ 17-*+#0(3 71-#?#12 ;"#0" #$ 713#'* `+*(3 -#?#12Q5 !"7%7+1 (1/ U031,9+* =MOOh> )+&8*/ ,"(, #1 &+/*+ ,& 27(+(1,** &139 `+*(3Q 17-*+#0(3 -#?#126 ,"* ,+(1$)&+, &)*+(,&+ $"&73/ %* `$*-#43#1*(+Q (1/ -&1&,&1* (00&+/#12 ,& T(+,*1 =MO[Z>5 _14 .&+,71(,*39 &139 !+$,4&+/*+ $0"*-*$ ;#33 -**, ,"*$* +*A7#+*-*1,$5 S#10* !+$,4&+/*+ $0"*-*$ (+* ,&& /#..7$#8* .&+ -&$, (,-&$)"*+#0 ())3#0(,#&1$6 &1* -7$, (00*), $&-* 3*8*3 &. 71-#?#125 I&+ ( -&+* 0&-)3*,* /#$07$$#&1 &. ,"#$ ,&)#0 ,"* +*(/*+ #$ +*.*++*/ ,& !"7%7+1 (1/ U031,9+* =MOOh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`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c6"Bd K AK Ac6$)d K Ac6$)d M AM Ac6"Bd M Fig. 8.5 S0"*-(,#0 8#*; &. ,"* *..*0, &. -#?#12 &1 ( $0(,,*+ )3&, ;"*+* -#?#12 +(,#& .&+ ,+(0*+ M6 4M 6 #$ )3&,,*/ (2(#1$, ,+(0*+ K6 4K 5 !"* ,;& ,+(0*+$ (+* #1#,#(339 1&143#1*(+39 0&++*3(,*/6 ,"(, #$6 ,"* $0(,,*+ )
,$ (4M , 4K ) (+* &1 ,"* )+*4*?#$,#12 .710,#&1(3 +*3(,#&1 07+8* =$&3#/ ,"#0' 3#1* 07+8*>5 !"* (6$)) (6"B) (6$)) (6"B) ] (1/ [4K , 4K ]6 +*$)*0,#8*395 <(+,#(3 -#?#12 #1#,#(3 +(12*$ .&+ ,"* ,;& ,+(0*+$ (+* [4M , 4M &. ,;& (#+ -($$*$ =,;& !33*/ 0#+03*$ 4 $0(,,*+ )
,$> ;#33 ,*1/ ,& -&8* ,"* ,;& $0(,,*+ )
,$ ,&;(+/$ *(0" &,"*+ (3&12 ,"* $,+(#2", =+*/> 3#1* =(3$& +*.*++*/ ,& ($ ( `-#?#12 3#1*Q>5 T*10* `+*(3 -#?#12Q &007++#12 #1 ,"* (,-&$)"*+* ;#33 ,*1/ ,& -&8* )
,$ &1 ,"* $0(,,*+ )3&, ,& ,"* 0&10(8* $#/* &. ,"* )+*4*?#$,#12 .710,#&1(3 +*3(,#&1 07+8* =(3$& +*.*++*/ ,& ($ ,"* `0&18*? "733Q 4 $"(/*/ (+*(>5 8.3.8 Robustness !"* 17-*+#0(3 -*,"&/ $"&73/ +*-(#1 $,(%3* (1/ +*,(#1 $#-73(,#&1 8*+(0#,9 ,"+&72"4 &7, ,"* #1,*2+(,#&15 Y&%7$,1*$$ 0(1 %* ($$*$$*/ %9 ,*$,#12 ,"* (32&+#,"- .&+ -(19 /#..*+*1, "&; !*3/$6 ,*-)&+(3 (1/ $)(,#(3 +*$&37,#&1$5 8.3.9 Parallel computational efficiency <*+.&+-(10* #-)+&8*-*1,$ (+* 3(+2*39 /7* ,& #10+*($*/ )(+(33*3#$- +(,"*+ ,"(1 #-)+&8*/ -#0+&)+&0*$$&+ 03&0' .+*A7*1095 T*10* ,"* 17-*+#0(3 (32&+#,"- $"&73/ %* (-*1(%3* .&+ *?*07,#&1 &1 -($$#8*39 )(+(33*3 0&-)7,#12 )3(,.&+-$5 : ;(9 ,& (0"#*8* ,"#$ #$ ,& 7$* 3&0(3 -*,"&/$ ;#," -#1#-(3 23&%(3 /*)*1/*10* =.&+ -&+* /#$4 07$$#&1 $** 0"(),*+ Ma>5 KLa X(7+#,@*1 *, (35 G, #$ ;&+," 1&,#12 ,"(, (3,"&72" 0&-)7,#12 )&;*+ "($ #10+*($*/ /+(-(,#0(339 #1 ,"* 3($, KL 9*(+$ &+ $&6 ,"*$* *?,+( 0&-)7,(,#&1(3 +*$&7+0*$ "(8* 3(+2*39 %**1 7$*/ ,& $(,#$.9 /*-(1/$ .&+ "#2"*+ +*$&37,#&16 -&+* (/8(10*/ )"9$#0(3 )(+(-*,*+#@(,#&1$ (1/ 0&7)3#12 ,"* (,-&$)"*+#0 0&-)&1*1, ,& &0*(16 3(1/6 (1/ #0* 0&-)&1*1,$ =#5*5 0&7)3*/ -&/*3$>5 T*10* #, #$ $,#33 /*$#+(%3* ,& /*8*3&) *.!0#*1, /91(-#0(3 0&+* (32&4 +#,"-$6 #1 )(+,#073(+6 $0"*-*$ .&+ *.!0#*1, ,+(0*+ ,+(1$)&+, =$** )(+(2+()" &1 A5*?$K ?:"2-: - *8*1 ,"&72" 0&-)7,#12 )&;*+ #$ #10+*($#125 8.3.10 Multi-tracer efficiency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o B7'&;#0@ (1/ \(7-2(+/1*+6 KLLL> (1/V&+ ,+(1$)&+, ,+(0*+$ ;#," 3&12*+ ,#-*4$,*)$ ,"(1 7$*/ .&+ ,"* 0&1,#17#,9 *A7(,#&1 .&+ (#+ #1 ,"* /91(-#0(3 0&+* =(3$& +*.*++*/ ,& ($ `$7)*+40903#12Q &. ,+(0*+$ ;#," +*$)*0, ,& (#+ &+6 -&+* 0&--&1396 `$7%4 0903#12Q &. (#+ ;#," +*$)*0, ,& ,+(0*+$o $** $*0,#&1 [5h5K>5 8.3.11 Geometric flexibility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o .&+ (1 &8*+8#*; $** S-&3(+'#*;#0@ =KLLa>5 8.4 Problem formulation: Discrete schemes I#1#,*48&37-* -*,"&/$ (+* 17-*+#0(3 -*,"&/$ ;"*+* *(0" )+&21&$,#0 8(+#(%3* #$ $,&+*/ ($ (1 (8*+(2* A7(1,#,9 &8*+ ( 0*+,(#1 !1#,*39 3(+2* 0&1,+&3 8&37-* =(3$& [ I#1#,*4E&37-* !+(1$)&+, S0"*-*$ KLh +*.*++*/ ,& ($ 0*334#1,*2+(,*/ -*,"&/$>5 !"#$ 0"�* /#..*+$ .+&- -*,"&/$ ,"(, (+* %($*/ &1 2+#/4)
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o (5'5(5 X(2+(12#(1 .&+-5 G1 X(2+(12#(1 .&+- ,"*+* #$ 1& "7? &. -($$ ,"+&72" ,"* `;(33$Q &. ,"* 8&37-*5 \&," &. ,"*$* .&+-$ &. ,"* !1#,*48&37-* /#$0+*,#@(,#&1 &. ,"* 0&1,#17#,9 *A7(,#&1 (+* )+*$*1,*/ 1*?, (.,*+ ,"* #1,+&/70,#&1 &. $&-* 1&,(,#&15 I&+ $#-)3#04 #,9 ;* 0&1$#/*+ ,"* ,;&4/#-*1$#&1(3 )+&%3*- #1 D(+,*$#(1 2*&-*,+9 (1/ /*.*+ ,"* /#$07$$#&1 &. ,"* *?,*1$#&1 ,& $)"*+#0(3 2*&-*,+9 (1/ ,"+**4/#-*1$#&1$ ,& $*0,#&1 [5R5Z5Z (1/ [5a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sC = M 6 &C & &C s (B, J) %&, s = l &+ l 4, =[5M[> ;"*+* s (B, J) #$ ,"* *?(0, $&37,#&15 G1 ,#-* ;* /#$0+*,#@* #1 ,*+-$ (, *A7#/#$,(1, ,#-*43*8*3$6 #5*5 $7)*+$0+#), ) +*.*+$ ,& ,"* A7(1,#,9 (, ,#-* ? = ) 6 ? ;"*+* 6 ? #$ ,"* ,#-*4$,*)5 S& ,"* $,(,* &. ( ,+(0*+ #1 0*33 & C (, ,#-*43*8*3 ) #$ /*1&,*/ s )C 5 MZ ,"* F73*+#(1 (1/ X(2+(12#(1 .&+-$ (+* 3#-#,$ &. ,"* -&+* 2*1*+(3 (+%#,+(+9 X(2+(12#(14F73*+#(1 =:XF> .&+- =T#+, *, (36 MOhP> KL[ X(7+#,@*1 *, (35 8.4.1 (Semi-)Lagrangian schemes D&1$#/*+ (1 (+%#,+(+9 X(2+(12#(1 (+*( &(?)5 \9 /*!1#,#&1 ,"* (+*( &(?) -&8*$ ;#," ,"* "&; ;#,"&7, (19 "7? &. -($$ ,"+&72" #,$ $#/*$ (1/ "*10* #, (3;(9$ 0&1,(#1$ ,"* $(-* -(,*+#(3 )(+,#03*$5 S#10* ,"*+* #$ 1& "7? &. -($$ ,"+&72" ,"* %&71/(+#*$ &. &(?)6 ,"* -($$ #1 ,"* (+*( #$ 0&1$*+8*/5 G1 -(,"*-(,#0(3 ,*+-$ ,"#$ 0(1 %* ;+#,,*1 ($ % %? & &(?) s %& = L, s = l &+ l 4. =[5MO> FA7(,#&1 =[5MO> #$ +*.*++*/ ,& ($ ,"* X(2+(12#(1 !1#,*48&37-* .&+- &. ,"* 0&1,#17#,9 *A7(,#&15 : ,*-)&+(3 /#$0+*,#@(,#&1 &. =[5MO> +*(/$ & &(?+6 ?) s %& = & &(?) s %&. =[5KL> G. ,"* $(-* X(2+(12#(1 0*33 &(?) #$ ,+(0'*/ ,"+&72"&7, ,"* $#-73(,#&1 ,"* +*$73,#12 $0"*-* #$ +*.*++*/ ,& ($ .7339 X(2+(12#(15 !"* 0"(33*12* #1 $70" $0"*-*$ #$ ,"(, .&+ 1&14,+#8#(3 "&;$ ,"* (+*($ A7#0'39 /*.&+- #1,& ,"#1 !3(-*1,$ $& ,"(, ,"* +*$&37,#&1 #$ 1& 3&12*+ 71#.&+- =$** I#27+* K #1 0"(),*+ h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a(>5 C* ($$7-* ,"(, 6 #$ 0"&$*1 $70" ,"(, (33 ,"* /*.&+-*/ (+*($ " C (+* $#-)39 0&11*0,*/5 N&,* ,"(, ,"*+* *?#$,$ ( &1*4,&4&1* 0&++*$)&1/*10* %*,;**1 & C (1/ "C $70" ,"(, ,"* "C Q$ $)(1 1 ;#,"&7, 2()$ &+ &8*+3()$ %*,;**1 ,"*. ' C=M "C = 1 , (1/ "C ∩ "! = LV ∀ C (= !. =[5KM> :$$7-* ,"(, ,"* *8&37,#&1 &. ,"* X(2+(12#(1 2+#/ #$ '1&;1 (1(39,#0(339 $& ;* '1&; ,"* 0"(+(0,*+#$,#0$ &+ ,+(]*0,&+#*$ .&+ *(0" "7#/ )(+0*3 (, (33 ,#-*$5 !"* 0&-)7,(,#&1 MP 1&,* ,"(, &1* -#2", *A7(339 ;*33 0&1$#/*+ /&;1$,+*(- $0"*-*$ ;"*+* &1* 0&1$#/*+$ F73*+#(1 =+*273(+> 2+#/ 0*33$ (, ,#-*43*8*3 ) (1/ 3*, ,"*- %* ,+(1$)&+,*/ ;#," ,"* "&; .&+ &1* ,#-*4$,*)5 [ I#1#,*4E&37-* !+(1$)&+, S0"*-*$ KLO (a) (b) Fig. 8.6 : 2+()"#0(3 #337$,+(,#&1 &. ,"* 7)$,+*(- $*-#4X(2+(12#(1 1&-*103(,7+*5 =(> !"* $,(,#0 F73*+#(1 0*33 &C =3#2", $"(/#12> (1/ ,"* 0&++*$)&1/#12 7)$,+*(- X(2+(12#(1 (+*( "C =/(+' $"(/#12> ,"(, *1/$ 7) (, &C (.,*+ &1* ,#-*4$,*)5 I&+ #337$,+(,#&1 ,"* ,+(]*0,&+#*$ &. ,"* 8*+,#0*$ =!33*/ 0#+03*$> &. &C (+* /*)#0,*/ ;#," (++&;$5 !"* 0&++*$)&1/#12 7)$,+*(- 8*+,#0*$ =/*)(+,7+* )
,$> (+* $"&;1 ;#," &)*1 0#+03*$5 =%> !"* 1&,(,#&1 7$*/ ,& /*!1* &8*+3() (+*($ %*,;**1 F73*+#(1 0*33 &! (1/ 7)$,+*(- X(2+(12#(1 (+*( "C #$ "C! = &! ∩ "C =/(+' $"(/*/ (+*(> &. "7#/ )(+0*3 ,+(]*0,&+#*$ #$ ;*33 /*8*3&)*/ #1 ,"* $*-#4X(2+(12#(1 3#,*+(,7+* =*5256 S,(1#.&+," (1/ Dj&,s* MOOM6 S,(1#.&+," *, (3 KLLZ6 T&+,(3 KLLK> (1/ #1 ,"* #1,*+*$, &. %+*8#,9 #, #$ 1&, /#$07$$*/ .7+,"*+ #1 ,"#$ 0"(),*+6 (3,"&72" (007+(,* ,+(]*0,&+#*$ (+* 8#,(3 .&+ ,"* (007+(09 &. (19 X(2+(12#(1 -*,"&/5 C#," ,"* 1&,(,#&1 #1,+&/70*/ (%&8* ,"* .&+*0($, *A7(,#&1 =[5KL> 0(1 %* ;+#,,*1 ($ s )+M 6 &C = s )C 6 "C . C =[5KK> ;"*+* s )C #$ ,"* (8*+(2* ,+(0*+ /*1$#,9 &8*+ ,"* 7)$,+*(- (+*( " C s )C = M 6 "C & "C s ) (B, J) %&, =[5KZ> (1/ s )+M #$ ,"* 0*33 (8*+(2*/ 8(37* &. s &8*+ ,"* +*273(+ (+*( & C (, ,#-*43*8*3 ) + M5 C !"* .710,#&1 s ) (B, J) #$ ,"* 0&1,#17&7$ /#$,+#%7,#&1 &. s (, ,#-*43*8*3 )5 H%8#&7$396 $#10* ,"* )+&21&$,#0 8(+#(%3*$ (+* 0*33 (8*+(2*$ s ;* /& 1&, '1&; ,"* 8(+#(,#&1 &. s (, ,"* $7%42+#/ $0(3* (1/ s ) (B, J) -7$, %* +*0&1$,+70,*/ .+&- ,"* )+&21&$,#0 0*33 (8*+(2*$MR 5 !"#$ )+&0*/7+* #$ +*.*++*/ ,& ($ $7%42+#/4$0(3* +*0&1$,+70,#&15 G1 !1#,*48&37-* $0"*-*$ ,"* +*0&1$,+70,#&1 #$ 7$7(339 3&0(3 +(,"*+ ,"(1 23&%(35 S& *(0" 0*33 C ;#33 "(8* (1 ($$&0#(,*/ $7%42+#/4$0(3* +*0&1$,+70,#&1 .710,#&1 s C (B, J) +(,"*+ ,"(1 &1* 23&%(3 +*0&1$,+70,#&1 .710,#&1 &8*+ (33 0*33$ $70" ($ ,"* $)"*+#0(3 "(+-&1#0 .710,#&1$ 7$*/ #1 $)*0,+(3 ,+(1$.&+- -&/*3$5 T*10* ,"* 23&%(3 +*0&1$,+70,#&1 .710,#&1 #$ ( 0&33*0,#&1 &. 3&0(3 +*0&1$,+70,#&1 .710,#&1$ s (B, J) = . - X&C sC (B, J), =[5KP> C=M MR 713*$$ 8(+#(%3*$ $70" ($ 2+(/#*1,$ (+* (3$& 0(++#*/ ($ )+&21&$,#0 8(+#(%3*$ =*525 c(%* *, (36 KLLM> KML X(7+#,@*1 *, (35 ;"*+* X&C #$ ,"* #1/#0(,&+ .710,#&1 X&C =   M, (B, J) ∈ &C ,  L, (B, J) ∈ / &C . =[5KR> D&--&139 7$*/ -*,"&/$ .&+ 0&-)7,#12 s C (B, J) .+&- s C (+* /#$07$$*/ #1 $*0,#&1 [5R5K5 I#+$,6 ;* 1&,* ,"(, s (B, J) #$ 1&, 1*0*$$(+9 0&1,#17&7$ &+ /#..*+*1,#(%3* (0+&$$ 0*33 %&71/(+#*$5 S& #. ,"* 7)$,+*(- (+*( " C 0&8*+$ $*8*+(3 F73*+#(1 0*33$ =*5256 I#25 [5a>6 ,"* #1,*2+(3 &1 ,"* +#2",4"(1/ $#/* &. =[5KZ> -7$, %* %+&'*1 7) #1,& &8*+3() (+*($ %*,;**1 F73*+#(1 0*33$ (1/ " C 5 !"* /#$0+*,#@*/ $*-#4X(2+(12#(1 !1#,*48&37-* 0&1,#17#,9 *A7(,#&1 =[5KK> ,"*1 +*(/$ s )+M 6 &C = C GC - & !=M "C! s!) (B, J) %&. =[5Ka> !"* 17-%*+ &. 1&14*-),9 &8*+3() (+*($ %*,;**1 ,"* 7)$,+*(- 0*33 =/*)(+,7+* 0*33> "C (1/ ,"* F73*+#(1 2+#/ 0*33$ #$ /*1&,*/ G C 5 N&,* ,"(, GC /*)*1/$ &1 ,"* "&; (1/ ,#-*4$,*) $#@*6 (1/ .&+ ,#-*48(+9#12 "&;$ #, #$ 1&, 1*0*$$(+#39 0&1$,(1,5 !"* (+*( " C! #$ ,"* 1&14*-),9 &8*+3() (+*( %*,;**1 ,"* 7)$,+*(- 0*33 " C (1/ ,"* F73*+#(1 2+#/ 0*33 &! =$** I#25 [5a%> "C! = "C ∩ &! , "C! (= Lo V ! = M, . . . , GC , (1/ M ≤ GC ≤ ., =[5Kh> ;"*+* . #$ ,"* 17-%*+ &. 0*33$ #1 ,"* /&-(#15 !;& 0&1/#,#&1$ -7$, %* .73!33*/ ,& 2*, 0&1$*+8(,#&1 &. -($$ #1 X(2+(12#(1 !1#,*4 8&37-* $0"*-*$^ I#+$,396 ,"* 7)$,+*(- 0*33$ " C -7$, %* $#-)39 0&11*0,*/ /&-(#1$ (1/ ,"*9 -7$, $)(1 1 ;#,"&7, 2()$ &+ &8*+3()$ =*A7(,#&1 [5KM>5 S*0&1/396 ,"* +*4 0&1$,+70,#&1 .710,#&1 #1 0*33 C6 s C (B, J)6 -7$, %* 0&1$*+8(,#8* #1 ,"* $*1$* ,"(, ,"* #1,*2+(3 &. sC (B, J) &8*+ &C -7$, 9#*3/ ,"* 0*334(8*+(2* 8(37* ,"(, #$ 7$*/ ($ )+&24 1&$,#0 8(+#(%3*6 & M sC (B, J) %& = s C . =[5K[> 6 &C &C FA7(,#&1 =[5Ka> #$ ,"* %($#0 !1#,*48&37-* .&+- &. ,"* 0&1,#17#,9 *A7(,#&1 ;"*1 7$#12 (1 7)$,+*(- !1#,*48&37-* $*-#4X(2+(12#(1 ())+&(0"5 H%8#&7$39 ;* /& 1&, '1&; ,"* *?(0, X(2+(12#(1 ,+(]*0,&+9 &. *8*+9 )(+0*3 #1 ,"* /&-(#1 $& $&-* ()4 )+&?#-(,#&1 ,& " C #$ 1*0*$$(+9 .&+ ,"* /*+#8(,#&1 &. (19 )+(0,#0(3 $0"*-*5 !"#$ #$ /#$07$$*/ #1 $*0,#&1 [5R5M5 G1 ,"* /#$07$$#&1 (%&8* s 2*1*+#0(339 +*.*+$ ,& %&," l (1/ l 45 G1 ,"* +*0&1$,+704 ,#&1 &. l 4 &1* -(9 0"&$* ,& +*0&1$,+70, l (1/ 4 $*)(+(,*39 (1/ 0&-%#1* ,"*- ,& )+&8#/* ( +*0&1$,+70,#&1 .&+ ,"* )+&/70, l 45 !"*+* (+* $*8*+(3 +*($&1$ .&+ 0"&&$#12 ,"#$ ())+&(0"5 I#+$,6 #, #$ 4 (1/ 1&, l 4 ,"(, #$ 0&1$*+8*/ (3&12 )(+0*3 ,+(]*0,&+#*$ =$** *A7(,#&1 [5MM> (1/ 4 $"&73/ ,"*+*.&+* &%*9 -&1&,&1#0#,9 +*A7#+*-*1,$5 T*10* &1* 0(1 (+27* ,"(, -&1&,&1* +*0&1$,+70,#&1 .710,#&1 !3,*+$ =/#$07$$*/ #1 $*0,#&1 [5R5K> [ I#1#,*4E&37-* !+(1$)&+, S0"*-*$ KMM $"&73/ %* ())3#*/ ,& 4 (1/ 1&, l 45 S*0&1/6 ,"* 0&1$#$,*1, 0&7)3#12 &. ,+(0*+$ (1/ (#+ /*1$#,9 *A7(,#&1$ #1 0*334#1,*2+(,*/ $*-#4X(2+(12#(1 $0"*-*$ ($ ;*33 ($ *1$7+#12 -&1&,&1* .&+*0($,$ &. 46 #$ )*+"()$ *($#*+ ;"*1 0"&&$#12 ,"#$ ())+&(0" =N(#+ (1/ X(7+#,@*16 KLML>5 !"* +*0&1$,+70,#&1$ .&+ l (1/ 4 0(1 %* 0&-%#1*/ ,& )+&8#/* ( +*0&1$,+70,#&1 .&+ l 4 %9 $#-)39 -73,#)39#12 ,"* +*0&1$,+70,#&1 .710,#&1$ .&+ l (1/ 4 ($ /&1* #1 B7'&;#0@ (1/ \(7-2(+/1*+ =KLLL>5 T&;*8*+6 #1 / $& -($$40&++*0,#&1 ,*+-$ -(9 %* 1**/*/ ,& $(,#$.9 =[5K[> .&+ "#2"*+4&+/*+ +*0&1$,+70,#&1$5 !"* /&;1$#/* &. ,"#$ ())+&(0" #$ ,"(, #.6 .&+ *?(-)3*6 ,"* +*0&1$,+70,#&1 .710,#&1 .&+ l (1/ 4 (+* )&394 1&-#(3$ &. $," (1/ L," &+/*+ ,"* )+&/70, ;#33 %* )&391&-#(3$ &. ($ + L)," &+/*+ ;"#0" -(9 %* 0&-)7,(,#&1(339 #1,*1$#8* ,& #1,*2+(,*5 H1* -(9 $#-)3#.9 %9 +*-&8#12 $&-* ,*+-$ .+&- ,"* )+&/70, ($ /&1* #1 N(#+ (1/ X(7+#,@*1 =KLML>5 !"* 3(,,*+ (3$& .(0#3#4 ,(,*$ +*1/*+#12 $0"*-*$ -&1&,&1* #1 45 G1 F73*+#(1 $0"*-*$6 /#$07$$*/ 1*?,6 ,+(0*+ -#?#12 +(,#& (1/ (#+ /*1$#,9 (+* 7$7(339 +*0&1$,+70,*/ $*)(+(,*39 .&+ $7%40903#12 =$** $*0,#&1 [5h5K> 8.4.2 Eulerian scheme D&1,+(+9 ,& ,"* X(2+(12#(1 /*+#8(,#&1$ #1 ,"* )+*8#&7$ $*0,#&16 ,"* *A7(,#&1$ &. -&,#&1 (+* ,9)#0(339 /*+#8*/ #1 F73*+#(1 .&+-5 G1 ,"* 0&1,*?, &. ,"* !1#,*48&37-* .&+- &. ,"* 0&1,#17#,9 *A7(,#&1 ,"* F73*+#(1 ())+&(0" '**)$ ,+(0' &. ,"* "7? &. -($$ ,"+&72" ,"* F73*+#(1 0*33 ;(33$ +(,"*+ ,"(1 ,+(0'#12 ,"* -($$ #1 ( 0*33 -&8#12 ;#," ,"* "&;5 : -&+* .&+-(3 /*+#8(,#&1 #$ 2#8*1 %*3&;5 I#+$,6 #1,*2+(,* =[5O> &+ =[5ML> #1 $)(0* &8*+ ( 2+#/ 0*33 & C & &C ,s %& + ,? & &C ¢ · (s v) %& = L, ;"*+* s = l , l 4. =[5KO> H1 #1,*2+(,#12 ,"* !+$, ,*+- &1 ,"* 3*.,4"(1/ $#/* &. =[5KO> ,& 2*, ,"* (+*( (8*+(2* (1/ ())39#12 ,"* /#8*+2*10* ,"*&+*- ,& ,"* $*0&1/ ,*+- ;* 2*, % (s 6 &C ) + %? C + , &C (s v) · n %0 = L, =[5ZL> ;"*+* , &C #$ ,"* %&71/(+9 &. & C (1/ n ,"* &7,;(+/ 1&+-(3 8*0,&+ ,& , & C 5 !"* $*0&1/4,*+- &1 ,"* 3*.,4"(1/ $#/* &. =[5KO> +*)+*$*1,$ ,"* #1$,(1,(1*&7$ "7? &. -($$ ,"+&72" ,"* %&71/(+#*$ &. & C 5 !*-)&+(3 #1,*2+(,#&1 &. =[5ZL> &8*+ &1* ,#-*4$,*) 9#*3/$ , ) s )+M C 6 &C = s C 6 &C − & ()+M) 6 ? + )6 ? , &C (s v) · n %0 %?, =[5ZM> (.,*+ +*4(++(12#12 ,*+-$5 !"* $*0&1/ ,*+- &1 ,"* +#2",4"(1/ $#/* &. =[5ZM> #$ ,"* "7? &. -($$ ,"+&72" ,"* ;(33$ &. & C /7+#12 &1* ,#-*4$,*)5 : 2+()"#0(3 #337$,+(,#&1 &. ,"* "7?*$ #$ 2#8*1 #1 I#27+* [5h (1/ /#$07$$*/ #1 ,"* 1*?, )(+(2+()"5 KMK X(7+#,@*1 *, (35 (a) (b) :' ( 'o!" ( 'o!% (c) :' (d) :' :' ( 'o!# ( 'o!$ :'ïM Fig. 8.7 : 2+()"#0(3 #337$,+(,#&1 &. ,"* `"7?4(+*($Q ($$&0#(,*/ ;#," F73*+#(1 0*33 &C =(+*( #1 ,"* 7))*+ +#2", 0&+1*+ &. *(0" )3&, %&71/*/ %9 ,"#0' 3#1*$>5 I&+ *(0" 8*+,*? &. 0*33 &C =!33*/ 0#+03*$> ,"* 7)$,+*(- ,+(]*0,&+#*$ (+* $"&;1 =07+8*/ (++&;$ /*)(+,#12 .+&- &)*1 0#+03*$>5 !"* $"(/*/ (+*($ $"&; ,"* "7?4(+*($ .&+ ,"* =(> *($, "Co =M 6 =%> 1&+," "Co =K 6 =0> ;*$, "Co =Z (1/ =/> $&7," "Co =P .(0*6 +*$)*0,#8*396 7$#12 $,(1/(+/ 0&-)($$ &+#*1,(,#&15 !"*$* (+*($ (+* $;*), ,"+&72" *(0" .(0* /7+#12 &1* ,#-*4$,*)5 S** ,*?, .&+ /*,(#3$5 X*, o /*1&,* ,"* .(0* 17-%*+ (1/ . < ,"* 17-%*+ &. .(0*$ &. ,"* 0*33$5 I&+ $#-)3#04 #,9 ;* ($$7-* ( A7(/+#3(,*+(3 -*$" . < = P6 "&;*8*+6 ,"* -*,"&/ 0(1 (00&--&/(,* (19 '#1/ &. -*$" =.&+ *?(-)3*6 .&+ ( ,+#(1273(+ (1/ "*?(2&1(3 -*$" . < ;&73/ %* Z (1/ a6 +*$)*0,#8*39>5 : 2+()"#0(3 #337$,+(,#&1 &. ,"* "7?*$ ,"+&72" ,"* 0*33 ;(33$ .&+ F73*+#(1 0*33 C (+* $"&;1 &1 I#25 [5h5 :$ ;#33 %*0&-* 03*(+6 ,"* I#27+* (3$& $"&;$ ,"* 7)$,+*(- X(2+(12#(1 0*33 (3,"&72" #, #$ 1&, *?)3#0#,39 1**/*/ .&+ "7? 0&-)7,(4 ,#&1$5 !"* $#/*$ &. ,"* F73*+#(1 0&1,+&3 8&37-* (+* 17-%*+*/ 0&71,*+403&0';#$* $& ,"(, $#/*$ o = M, K, Z, P 0&++*$)&1/ ,& ,"* *($,6 1&+,"6 ;*$, (1/ $&7," ;(33$6 +*$)*04 ,#8*39 =7$#12 $,(1/(+/ 0&-)($$ 1&,(,#&1>5 !"* "7? &. -($$ ,"+&72" ,"* $#/* o = M 0&++*$)&1/$ ,& ,"* -($$ &8*+ ,"* $"(/*/ (+*( &1 I#25 [5h( ,"(, #$ `$;*),Q ,"+&72" ,"* ;(33 /7+#12 &1* ,#-* $,*)5 !"* $"(/*/ (+*(6 +*.*++*/ ,& ($ " Co =M 6 #$ %&71/*/ %9 ,"* .(0* o = M6 ,"* ,;& 7)$,+*(- ,+(]*0,&+#*$ .&+ ,"* *1/ )
,$ &. .(0* o = M6 (1/ ,"* 7)$,+*(- ,+(1$3(,#&1 &. ,"* $#/* o = M5 C* ;#33 +*.*+ ,& " Co =M ($ ,"* `"7?4(+*(Q .&+ .(0* o = M5 S#-#3(+396 ,"* "7?*$ ,"+&72" ,"* +*-(#1#12 0*33 $#/*$ (+* #337$,+(,*/ #1 I#25 [5h%4/5 _$#12 ,"* 1&,(,#&1 #1,+&/70*/ (%&8* =[5ZM> 0(1 %* ;+#,,*1 ($ .< s )+M 6 &C = s )C 6 &C − - SCo , C o =M ;"*+* SCo #$ ,"* "7? &. -($$ ,"+&72" .(0* o /7+#12 &1* ,#-*4$,*) =[5ZK> [ I#1#,*4E&37-* !+(1$)&+, S0"*-*$ KMZ SCo = ;oC & "oC s ) (B, J) %&. =[5ZZ> !"* `"&;4/#+*0,#&1Q .710,#&1 ; oC #$ 7$*/ ,& #1/#0(,* #1"&; (1/ &7,"&; ;oC = $21(v · n), =[5ZP> ;"*+* $21(·) #$ ,"* $#214.710,#&15 T*10* ; oC #$ M .&+ &7,"&; (1/ 4M .&+ #1"&; Ma 5 G1 I#27+* [5h ,"* "&;4/#+*0,#&1 .710,#&1 ; oC #$ M .&+ o = M, K (1/ 4M .&+ o = Z, P5 N&,* ,"(, ,"* "7? &. -($$ ,"+&72" &1* 0*33 ;(33 #$ #/*1,#0(36 %7, ;#," &))&$#,* $#216 ,& ,"* "7? &. -($$ ,"+&72" ,"* 1*#2"%&+#12 0*33 ,"(, #, $"(+*$ ( .(0* ;#,"5 I&+ ,"* *?(-)3* &1 I#27+* [5h/ o =K , =[5ZR> "Co =P = "C−M ;"*+* ,"* 0*33 3&0(,*/ #--*/#(,*39 ,& ,"* $&7," &. ,"* F73*+#(1 0*33 & C #$ &C−M 5 S& #1 ( )+(0,#0(3 #-)3*-*1,(,#&1 &. ( $0"*-* %($*/ &1 =[5ZK> &139 ,;& "7?*$ )*+ 0*33 (+* 0&-)7,*/ #. . < = P5 :3,"&72" ,"* $0"*-* &7,3#1*/ (%&8* #$ ,*+-*/ `F73*+#(1Q #, #$ 1&, F73*+#(1 #1 ,"* 03($$#0(3 $*1$* ;"*+* ,"* $)(0* (1/ ,#-* /#-*1$#&1$ (+* $*)(+(,*/5 G1 &,"*+ ;&+/$6 ,"* $0"*-* &7,3#1*/ (%&8* 0&73/ (3$& %* ,*+-*/ "7?4.&+- $*-#4X(2+(12#(1 $#10* "7?4(+*($ ,"(, -&8* ;#," ,"* "&; (+* ,+(0'*/ =`+*-()4,9)*Q $0"*-*>5 G, #$ F73*+#(1 #1 ,"* $*1$* ,"(, ;* 0&1$#/*+ ,"* "7? &. -($$ ,"+&72" ,"* =$,(,#&1(+9 &+ F73*+#(1> 0*33 ;(33$5 C"*1 $*)(+(,#12 ,"* ,*-)&+(3 (1/ $)(,#(3 /#-*1$#&1$6 ($ /&1* #1 03($$#0(3 F73*+#(1 $0"*-*$6 ,"*+* (+* 1& ,+(]*0,&+9 0(3073(,#&1$ (1/ "7?*$ (+* 0&-)7,*/ 7$#12 3&0(3 #1.&+-(,#&1 (1/ )(+,#(3 /*+#8(,#8*$ (3&12 ,"* 0&&+/#1(,* /#+*0,#&1$ (, $)*0#!0 ,#-*$5 !"* ,*-)&+(3 /#$0+*,#@(,#&1 #$ 7$7(339 %($*/ &1 Y712(4d7,,( -*,"&/$ =$** 0"(),*+ a>5 H1* -(9 (+27* ,"(, ,"* 03($$#0(3 F73*+#(1 $0"*-*$ (+* (1 ())+&?#-(,#&1 &. ,"* 2*1*+(3 F73*+#(14X(2+(12#(1 0&10*), )+*$*1,*/ #1 ,"#$ 0"(),*+ ;"*+* ,+7* =(3&12 ,"* ,+(]*0,&+#*$> "7?*$ (+* ())+&?#-(,*/ ;#," )(+,#(3 "7?*$ =#5*56 ,"* )(+,#03* )(," 8*0,&+ 0(1 %* /*0&-)&$*/ #1,& 8*0,&+ 0&-)&1*1,$ (3&12 ,"* 0&&+/#1(,* (?*$>5 8.4.3 Equivalence between the Lagrangian and Eulerian discretizations G, #$ #1,*+*$,#12 ,& 1&,* ,"* *A7#8(3*10* %*,;**1 ,"* X(2+(12#(1 !1#,*48&37-* 0&1,#4 17#,9 *A7(,#&1 =[5Ka> (1/ ,"* F73*+#(1 8*+$#&1 =[5ZK>^ G. ,('#12 ,"* $7- &. ,"* "7?4 (+*($ "oC ;#," ;*#2", M .&+ &7,"&; (1/ ;*#2", 4M .&+ #1"&; ($ ;*33 ($ & C ;#," ;*#2", M =(33 (+*($ #18&38*/ &1 ,"* +#2",4"(1/ $#/* &. =[5ZK>>6 ,"* 7)$,+*(- X(2+(12#(1 (+*( "C +*$73,$ =$** *?(-)3* &1 I#25 [5h>5 !"(, #$6 ,"* +#2",4"(1/ $#/* &. =[5ZK> ;+#,,*1 #1 ,*+-$ &. (+*($ #$ Ma .&+ $#-)3#0#,9 ;* /& 1&, 0&1$#/*+ ,"* $#,7(,#&1 ;"*+* ;oC #$ -73,#48(37*/ (3&12 ( )(+,#073(+ .(0*5 I&+ -&+* /*,(#3$ &1 $70" ( $#,7(,#&1 $** T(++#$ *, (3 =KLML> KMP X(7+#,@*1 *, (35 .< 6 &C − - (;oC 6 "oC ) = 6 "C . =[5Za> o =M S& ,"* X(2+(12#(1 (1/ F73*+#(1 $0"*-*$ (+* #/*1,#0(36 ($ *?)*0,*/6 $#10* 1& ())+&?4 #-(,#&1$ "(8* %**1 -(/* $& .(+ =*8*1 #. ())+&?#-(,#&1$ (+* -(/* (1/ ,"* +*$73,#12 $0"*-*$ (+* ())3#*/ ,& ,"* F73*+ *A7(,#&1$6 F73*+#(1 (1/ $*-#4X(2+(12#(1 $0"*-*$ -(9 )+&/70* 8*+9 $#-#3(+ +*$73,$ ($ $"&;1 #1 X*$3#* (1/ B#*,(0"-(9*+6 MOOh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o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iscrete schemes: Approximations !"* X(2+(12#(1 (1/ F73*+#(1 !1#,*48&37-* $0"*-*$6 2#8*1 #1 =[5Ka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a) KMR (b) @4(?#$ 94(?#$ ?4(?#$ Geometric error Reconstruction error Fig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s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pproximation to areas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a X(7+#,@*1 *, (35 (a) (b) (c) (d) Fig. 8.9 W+()"#0(3 #337$,+(,#&1 &. ())+&?#-(,#&1$ ,& ,"* 7)$,+*(- X(2+(12#(1 0*33 "C (5'5(5 ,"* /*)(+,7+* 0*335 :$$7-* ,"* /*)(+,7+* )
,$ 0&++*$)&1/#12 ,& ,"* 8*+,#0*$ &. ,"* F73*+#(1 2+#/ 0*33 (+* '1&;1 =&)*1 0#+03*$>5 =(> F?(0, /*)(+,7+* 0*33 =$"(/*/ (+*(> ;#," $#/*$ /*)#0,*/ ;#," ,"#0' 3#1*$5 =%> S#/*$ &. ,"* /*)(+,7+* 0*33 ())+&?#-(,*/ ;#," $,+(#2", 3#1*$ %9 0&11*0,#12 ,"* /*)(+,7+* )
,$5 =0> B*)(+,7+* 0*33 ())+&?#-(,#&1 7$*/ #1 N(#+ (1/ U(0"*1"(7*+ =KLLK> ;"*+* ,"* *($, (1/ ;*$, $#/*$ (+* $,+(#2", 3#1*$ )(+(33*3 ,& ,"* F73*+#(1 3&12#,7/*$ =J4(?#$ &1 ,"* )3&,> (1/ ,"* 1&+," (1/ $&7," $#/*$ (+* ())+&?#-(,*/ ;#," `$,*) .710,#&1$Q5 =/> !"* X(2+(12#(1 0*33 7$*/ #1 ,"* 0($0(/* $0"*-*$ ,"(, (+* %($*/ &1 #1,*+$*0,#&1$ =0+&$$*$> %*,;**1 ,"* X(2+(12#(1 3(,#,7/*$ =/($"*/V$&3#/ 07+8*/ 3#1*$> (1/ ,"* F73*+#(1 3&12#,7/*$5 !"* `$,*)Q #1 ,"* $,*) .710,#&1$ 7$*/ #1 ,"* 0($0(/* $0"*-*$ (3;(9$ 0 #/*$ ;#," ,"* F73*+#(1 3&12#,7/*$ =B4#$&3#1*$ &1 ,"* I#27+*>5 8.5.1.1 Lagrangian area approximations S5**J ?D3K%$6-);$3)"* G"E:")E$") ":-" "@@:3B$6"?$3); <+&%(%39 ,"* -&$, +#2&+&7$ ())+&?#-(,#&1 ,& ,"* *?(0, 7)$,+*(- 0*33 " C 6 I#27+* [5O(6 #$ ,& .&33&; ,"* ,+(]*0,&+#*$ &. ,"* 8*+,#0*$ &. & C 7)$,+*(- (1/ ,"*1 0&11*0, ,"* 7)4 $,+*(- 8*+,#0*$ ;#," $,+(#2", 3#1*$ =I#25 [5O%>o =Y(1#0#s0 MOOK6 X(7+#,@*1 *, (3 KLML>5 [ I#1#,*4E&37-* !+(1$)&+, S0"*-*$ KMh :33 &,"*+ ())+&?#-(,#&1$ #18&38* ())+&?#-(,#12 " C ;#," 3#1* $*2-*1,$ )(+(33*3 ,& ,"* 0&&+/#1(,*$ (?#$ ;"#0"6 #1 2*1*+(36 $#-)3#!*$ ,"* &8*+3()4(+*( #1,*2+(,#&1 (32&4 +#,"-5 S&-* *?(-)3*$ (+* 2#8*1 &1 I#25 [5O5 I&+ -&+* /*,(#3$ &1 X(2+(12#(1 0*33 ())+&?#-(,#&1$ .&+ &+,"&2&1(3 -*$"*$ $** ,"* 0&-)+*"*1$#8* +*8#*; %9 X(7+#,@*1 *, (3 =KLLa> (1/ U(0"*1"(7*+ *, (3 =KLLO>5 S*3DK;@*$? G"E:")E$") ":-" "@@:3B$6"?$3); U&+* +*0*1,39 ,"* !1#,*48&37-* 0($0(/* Mh ())+&(0" ;($ $722*$,*/ %9 N(#+ *, (3 =KLLK> (1/ f*++&7'(, *, (3 =KLLK> ;"#0" 7$*$ ( 0&-%#1(,#&1 &. F73*+#(1 (1/ X(4 2+(12#(1 &)*+(,&+$6 ,"(, #$6 ,"* &1*4/#-*1$#&1(3 &)*+(,&+$ (+* $700*$$#8*39 ())3#*/ (3&12 ( 0&&+/#1(,* 3#1* (1/ ( X(2+(12#(1 3#1*6 +*$)*0,#8*395 :1 *?(-)3* #$ 2#8*1 #1 I#25 [5O/ ;"*+* ,"* !+$, &1*4/#-*1$#&1(3 &)*+(,&+ #$ ())3#*/ (3&12 ,"* F73*+#(1 3&12#,7/#1(3 /#+*0,#&1 (1/ ,"* $*0&1/ #$ ())3#*/ (3&12 ,"* /*.&+-*/ X(2+(12#(1 3(,4 #,7/* =07+8*/ $&3#/V/($" 3#1*$ &1 I#27+* [5O/>5 S& +(,"*+ ,"(1 %*#12 ( !?*/ /#+*0,#&1 %($*/ $)3#,,#12 -*,"&/ #, #$ "&;4%($*/ =.&+ ( +*8#*; $** U(0"*1"(7*+ *, (3 KLLO>5 !"* 7)$,+*(- X(2+(12#(1 0*33 .&+ ,"* 0($0(/* $0"*-* #$ #337$,+(,*/ &1 I#27+* [5O/5 !"* -(#1 /#..*+*10* %*,;**1 ,"* .7339 ,;&4/#-*1$#&1(3 (+*( ())+&?#-(,#&1 7$*/ #1 N(#+ (1/ U(0"*1"(7*+ =KLLK>6 $"&;1 &1 I#27+* [5O06 (1/ ,"* 0($0(/* $0"*-* (+*( ())+&?#-(,#&16 #$ ,"* 3&0(,#&1 &. ,"* `]7-)Q #1 ,"* 1&+," (1/ $&7," $#/*$ &. ,"* /*4 )(+,7+* 0*335 S#10* ,"* !+$, 0($0(/* `$;**)Q #$ (3&12 F73*+#(1 3&12#,7/*$ ,"* ]7-) #1 ,"* 1&+," (1/ $&7," $#/*$ 0 #/* ;#," (1 F73*+#(1 3&12#,7/*5 G1 ,"* N(#+ (1/ U(0"*1"(7*+ =KLLK> ,"* ]7-) #$ 3&0(,*/ -#/;(9 %*,;**1 ,"* *($, (1/ ;*$, 0*33 $#/*$5 :))+&?#-(,#12 ,"* X(2+(12#(1 0*33 ;#," 3#1*4$*2-*1,$ )(+(33*3 ,& ,"* 0&&+/#1(,* (?#$6 *#,"*+ ;#," .7339 ,;&4/#-*1$#&1(3 &+ 0($0(/* -*,"&/$6 #$ (,,+(0,#8* .&+ &+,"&24 &1(3 2+#/$ $70" ($ ( D(+,*$#(1 +*0,(1273(+ -*$" =*5256 f*++&7'(, *, (3 KLLK> (1/ ( +*273(+ 3(,#,7/*43&12#,7/* 2+#/ &1 ,"* $)"*+* =*5256 N(#+ (1/ U(0"*1"(7*+ KLLK6 N(#+ *, (3 KLLK6 f*++&7'(, *, (3 KLLP>5 G, #$ 3*$$ &%8#&7$ "&; ,& *?,*1/ $70" ())+&(0"*$ ,& 1&14&+,"&2&1(3 2+#/$ $70" ($ ,+#(1273(+ &+ "*?(2&1(3 2+#/$ $#10* ,"* 0*33 $#/*$ (+* 1& 3&12*+ &+,"&2&1(35 8.5.1.2 Eulerian flux area approximations !"* ())+&?#-(,#&1 ,& "7?4(+*($ #1 F73*+#(1 $0"*-*$ 0(1 %* /#8#/*/ #1,& ,;& 0(,*4 2&+#*$^ I7339 ,;&4/#-*1$#&1(3 ())+&?#-(,#&1$ ,& ,"* "7?4(+*($ (1/ /#-*1$#&1(339 $)3#, (+*( ())+&?#-(,#&1$5 C* +*-#1/ ,"* +*(/*+ ,"(, &139 -*,"&/$ ,"(, "(8* %**1 *?,*1/*/ ,& 23&%(3 $)"*+#0(3 /&-(#1$ (+* /#$07$$*/ "*+*5 C* (+* ,"*+*%9 *?037/#12 -(19 ,+(1$)&+, $0"*-*$ )7%3#$"*/ #1 ,"* -*,*&+&3&2#0(3 3#,*+(,7+*5 Mh ,"* 1&140&1$*+8(,#8* 0($0(/* #1,*+)&3(,#&1 -*,"&/ #1 D(+,*$#(1 2*&-*,+9 ;($ #1,+&/70*/ %9 <7+$*+ (1/ X*$3#* =MOOM> KM[ X(7+#,@*1 *, (35 1":"**-*3E:"6e<*5Be":-"; (a) (b) (c) ( 'o!" :' ( 'o!% (d) (f) (e) ( 'o!# (' ( 'o!$ N5"%:$*"?-:"*e<*5Be":-"; (g) (h) (i) ( 'o!" :' ( 'o!% (l) (k) (j) ( 'o!# (' ( 'o!$ Fig. 8.10 : $0"*-(,#0 #337$,+(,#&1 &. /#..*+*1, "7? ())+&?#-(,#&1$6 )(+(33*3&2+(- =(6%6/6*> (1/ A7(/+#3(,*+(3 =26"6]6'> "7?4(+*($6 (1/ ,"* *A7#8(3*1, 7)$,+*(- X(2+(12#(1 (+*($ =.63>5 !"* *A7#8(3*1, 7)$,+*(- (+*($ (+* 0&-)7,*/ %9 ,('#12 ,"* $7- &. (33 (+*($ #18&38*/ #1 ,"* .&+*0($, =(6%606/6*> &+ =26"6#6]6'> ;#," ())+&)+#(,* $#21$ =$** *A7(,#&1 =[5Za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h> $722*$,*/ ,& ())+&?#-(,* ,"* "7?4(+*($ .+&- ( .(0*4 0*1,*+*/ ;#1/ 8*3&0#,95 S& ,"* ,;& 8*+,#0*$ &. ,"* .(0* ;&73/ "(8* #/*1,#0(3 7)4 [ I#1#,*4E&37-* !+(1$)&+, S0"*-*$ KMO $,+*(- /#$)3(0*-*1,$ %($*/ &1 ,"* $(-* .(0*40*1,*+*/ 8*3&0#,9 8*0,&+5 !"* ,+(]*0,&4 +#*$ (+* ,"*+*.&+* )(+(33*3 (1/ "(8* ,"* *?(0, $(-* 3*12,"5 T*10* ,"* "7?4(+*($ " oC 6 o = M, K, Z, P6 (+* )(+(33*3&2+(-$ =I#25 [5ML(6%6/6*6 +*$)*0,#8*39>5 !"* .(0, ,"(, ,"* 7)$,+*(- (+*( #$ ( )(+(33*3&2+(- -(9 $#-)3#.9 ,"* )+(0,#0(3 #1,*2+(,#&1 &. &8*+3() (+*($ (, ,"* *?)*1$* &. $&-* )&,*1,#(3 3&$$ &. (007+(09 #. ,"* "&; #$ "#2"39 /*.&+4 -(,#&1(35 !"#$ #$ #337$,+(,*/ %9 0&-)7,#12 ,"* *..*0,#8* 7)$,+*(- X(2+(12#(1 (+*( .&+ ,"* U#7+( =KLLh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s06 MOOKo X#)$0&-% (1/ Y#123*+6 KLLRo c*"6 KLLh>5 !"#$ *?,+( /*2+** &. .+**/&- (33&;$ ,"* "7?4(+*($ ,& /*.&+- #1,& (+%#,+(+9 A7(/+#3(,*+(3$5 !"* *A7#8(3*1, 7)$,+*(- (+*( 1&; *A7(3$ ,"* X(2+(12#(1 (+*( +*4 $73,#12 .+&- 0&11*0,#12 ,"* 7)$,+*(- )
,$ ;#," $,+(#2", 3#1*$5 !"#$ 0(1 %* $"&;1 ($ (%&8* %9 ,('#12 ,"* $7- &. ,"* (+*($ #18&38*/ #1 ,"* .&+*0($, *A7(,#&1 =[5ZK>6 I#25 [5ML26"6]6'6#6 ;#," ())+&)+#(,* ;*#2",$ =$#21$>5 :$ .&+ ,"* F73*+#(14X(2+(12#(1 *A7#8(3*10* #1 ,"* 0&1,#17&7$ 0($*6 /#$07$$*/ #1 $*0,#&1 [5P5Z6 ,"#$ ())+&?#-(,* "7?4 .&+- $0"*-* #$ *?(0,39 *A7#8(3*1, ,& ,"* ())+&?#-(,* X(2+(12#(1 $0"*-* /#$07$$*/ (%&8* ;"*+* ,"* /*)(+,7+* )
,$ (+* 0&11*0,*/ ;#," $,+(#2", 3#1*$ =I#25 [5ML3 (1/ I#25 [5O%6 +*$)*0,#8*39>5 G-)+&8#12 ,"* *..*0,#8* ())+&?#-(,#&1 ,& ,"* 7)$,+*(- (+*( .7+,"*+ ;&73/ #14 8&38* ,"* #1,+&/70,#&1 &. -&+* )(+0*3$ ,"(, (+* ,+(0'*/ =($ $722*$,*/ %9 X(7+#,@*1 *, (3 KLML> &+ $&-* ())+&?#-(,#&1 ,& ,"* $#/*$ ;#," 07+8*/ 3#1*$5 : 07+$&+9 $,7/9 (//+*$$#12 ,"* )&,*1,#(3 %*1*!,$ &. ())+&?#-(,#12 ,"* 7)$,+*(- (+*($ ;#," "#2"*+4 KKL X(7+#,@*1 *, (35 Fig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a> (1/ X*&1(+/ *, (3 =MOOa>5 G1 ,"* )+*$*1, /#$07$$#&1 &1 *..*0,#8* 7)$,+*(- (+*($6 ,"#$ &)*+(,&+ $)3#,,#12 ()4 )+&(0" ;($ (1(39@*/ %9 X(7+#,@*1 =KLLh> (1/ U(0"*1"(7*+ *, (3 =KLLO>5 C"*1 7$4 #12 /#-*1$#&1(339 $)3#, ())+&(0"*$ ,"* *..*0,#8* 7)$,+*(- (+*( #$ ())+&?#-(,*/ ;#," ( 0&-%#1(,#&1 &. +*0,(123*$ (3#21*/ ;#," ,"* 2+#/ 3#1*$ (1/ ;#," /#..*+*1, ;*#2",$ =$** U(0"*1"(7*+ *, (3 KLLO>5 H1*4/#-*1$#&1(3 &)*+(,&+$ 0(11&, +*)+*$*1, (+*($ $'*; ,& ,"* .(0* #1 A7*$,#&15 :$ (1 *?(-)3* &. (1 &)*+(,&+ $)3#,,#12 ())+&(0" ,"* *.4 .*0,#8* /*)(+,7+* (+*( .&+ ,"* X#1 (1/ Y&&/ =MOOa> $0"*-* #$ 2#8*1 &1 I#25 [5MK .&+ [ I#1#,*4E&37-* !+(1$)&+, S0"*-*$ KKM (' Fig. 8.12 : 2+()"#0(3 #337$,+(,#&1 &. ,"* *..*0,#8* /*)(+,7+* (+*( "C .&+ ,"* X#1 (1/ Y&&/ =MOOa> $0"*-* 7$#12 (1 (1(39,#0 ;#1/ !*3/ ;"#0" #$ /*.&+-(,#&1(36 +&,(,#&1(3 (1/ /#8*+2*1,5 !"* *?(0, /*)(+,7+* 0*33 #$ $"&;1 ;#," ,"#0' %3(0' 3#1*$ =(1/ &)*1 0#+03*$ ($ 8*+,#0*$>5 X#2", $"(/#12 $"&;$ ,"* )(+,$ &. ,"* /*)(+,7+* (+*( ;"*+* -($$ #$ ;*#2",*/ ;#," MVK (1/ /(+' $"(/*/ (+*($ (+* ;*#2",*/ ;#," M5 S** U(0"*1"(7*+ *, (3 =KLLO> .&+ /*,(#3$5 ( "&; ,"(, "($ ( ,+(1$3(,#&1(36 /*.&+-(,#&1 (1/ +&,(,#&1(3 0&-)&1*1, =$** U(0"*14 "(7*+ *, (3 KLLO .&+ /*,(#3$>5 G1 /#-*1$#&1(3 $)3#, $0"*-*$ &1* 0(1 &%,(#1 )+*$*+8(,#&1 &. ( 0&1$,(1, /*1$#,9 !*3/ #1 ( 1&14/#8*+2*1, "&; !*3/5 !"#$ )+&)*+,9 #$ "(+/*+ ,& &%,(#1 ;#," .7339 ,;&4/#-*1$#&1(3 $*-#4X(2+(12#(1 $0"*-*$ %7, #, #$ )&$$#%3* ;#," 0($0(/* $*-#4 X(2+(12#(1 $0"*-*$ =!"7%7+1 *, (36 KLML>5 8.5.1.3 Comment on area approximations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ig. 8.13 =(> D7%*/4$)"*+* 2+#/ $"&;1 ;#," 3#2", $"(/*/ 3#1*$ (1/ )(1*3 */2*$ ;#," %3(0' 3#1*$5 =%> !"* 7)$,+*(-V/*)(+,7+* 2+#/ =/($"*/ 3#1*$> $"&;1 &1 ( 3&0(3 =21&-&1#0> )+&]*0,#&1 .&+ &1* &. ,"* 07%*/4$)"*+* )(1*3$ 7$#12 ,"* $&3#/4%&/9 (/8*0,#&1 "&; !*3/ =,#-*4$,*) #$ $& ,"(, &1* +*8&37,#&1 #$ 0&-)3*,*/ #1 hK ,#-*4$,*)$5 !"* $&3#/ 3#1*$ $"&; ,"* F73*+#(1 $,(,#0 2+#/5 !"* $'*;*/ /*)(+,7+* 0*33$ (+* 0*33$ *1,*+#12 .+&- 1*#2"%&+#12 )(1*3$ /7+#12 ,"* ,#-*4$,*)5 !"* )(+,$ &. ,"* /*)(+,7+* 0*33$ &7,$#/* ,"* )(1*3 "(8* %**1 `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"7?4(+*( ())+&?#-(,#&1 (1/ 8(+#(%3* $,(22*+#12 (+* `#1,*+,;#1*/Q (1/ ,"* 0"�*$ #-)(0, 1&, &139 ,"* (007+(09 &. ,"* ,+(1$)&+, &)*+(,&+ %7, (3$& ;(8* )+&)(2(,#&1 )+&)*+,#*$ #1 .733 -&/*3$ ($ ;*33 ($ &,"*+ )+&)*+,#*$ $70" ($ ,"* 1**/ .&+ !3,*+#12 *,05 =$**6 *5256 0"(),*+ MZ (1/ MP>5 : *?"(7$,#8* /#$07$$#&1 &. &),#-(3 8(+#(%3* $,(22*+#12 (1/ "7?4(+*( ())+&?#-(,#&1 #$ %*9&1/ ,"* $0&)* &. ,"#$ 0"(),*+5 8.5.2 Sub-grid-scale reconstruction G1 ,"* )+*8#&7$ $*0,#&1$ ,"* 2*&-*,+#0(3 ())+&?#-(,#&1 ,& ,"* 7)$,+*(- (+*($ (1/ "7?4(+*($ "(8* %**1 /#$07$$*/5 N*?, 0&-*$ ,"* (0,7(3 #1,*2+(,#&1 &. s (B, J) &8*+ ,"*$* (+*($6 .&+ ;"#0" ( $7%42+#/4$0(3* +*0&1$,+70,#&1 &. ,"* ,+(0*+ !*3/ #$ 1**/*/5 C* $,(+, %9 /#$07$$#12 +*0&1$,+70,#&1 -*,"&/$ #1 &1* $)(,#(3 /#-*1$#&1 (1/ ,"*1 [ I#1#,*4E&37-* !+(1$)&+, S0"*-*$ KKZ %+#*"9 /#$07$$ ,;& /#-*1$#&1(3 *?,*1$#&1$ %*.&+* 0&8*+#12 ,"* #1,*2+(,#&1 &. s (B, J) &8*+ &8*+3() (+*($5 8.5.2.1 One-dimensional reconstruction functions !"* $7%42+#/4$0(3* +*0&1$,+70,#&1 #$ 8#,(3 .&+ ,"* &8*+(33 (007+(09 (1/ *.!0#*109 &. ( $0"*-*6 (1/ ( ,"&+&72" /#$07$$#&1 #$ %*9&1/ ,"* $0&)* &. ,"#$ 0"(),*+5 C* ;#336 "&;*8*+6 /#$07$$ $&-* &. ,"* -&$, ;#/*39 7$*/ -*,"&/$5 G1 )+#10#)3* (19 .710,#&1 0&73/ %* 7$*/ .&+ +*0&1$,+70,#&1$6 "&;*8*+6 ,"* 0"�* &. +*0&1$,+70,#&1 .710,#&1 "($ 0&1$*A7*10*$ .&+ (19 !1#,*48&37-* $0"*-*5 T*+* (+* $&-* /*$#+(%3* )+&)*+,#*$ .&+ +*0&1$,+70,#&1 .710,#&1$ ,"(, $"&73/ %* 0&1$#/*+*/^ • G32"*$?J' X&0(3#,9 #$ 2*1*+(339 /*$#+(%3* ,& -(?#-#@* )(+(33*3 *.!0#*109o ,"(, #$6 ,"* $,*10#3 =&+ "(3&> 7$*/ .&+ ,"* +*0&1$,+70,#&1 #1 (19 0*33 $"&73/ 7$* &139 ( 3#-#,*/ 17-%*+ &. 1*#2"%&+#12 2+#/ 0*33$5 !"* 0*33$ 7$*/ #1 ,"* +*0&1$,+70,#&1 &. ( 2#8*1 0*33 (+* +*.*++*/ ,& ($ ,"* $,*10#3 &. ,"(, 0*335 • X)?-E:">$*$?J' !"* +*0&1$,+70,#&1 .710,#&1 -7$, 3(,*+ %* #1,*2+(,*/ &8*+ &8*+3() (+*($ (1/ #, #$ 0&18*1#*1, ,& 7$* .710,#&1$ ,"(, 0(1 %* #1,*2+(,*/ *?(0,395 G. )&394 1&-#(3$ (+* 7$*/6 )&391&-#(3$ &. $700*$$#8*39 "#2"*+ /*2+** ;#33 3*(/ ,& -&+* 0&-)7,(,#&1(339 *?)*1$#8* $0"*-*$5 • /3);-:#"?$3)' I&+ X(2+(12#(1 !1#,*48&37-* $0"*-*$ -($$40&1$*+8(,#&1 &. ,"* !1(3 (32&+#,"- +*A7#+*$ ,"* +*0&1$,+70,#&1 .710,#&1 ,& $(,#$.9 ,"* $&40(33*/ 0*334 (8*+(2*/ )+&)*+,9o 1(-*396 #1,*2+(,#&1 &. ,"* +*0&1$,+70,#&1 &8*+ ,"* 0*33 =[5K[> 9#*3/$ ,"* '1&;1 0*334(8*+(2* =.&+ *(0" )+&21&$,#0 8(+#(%3*>5 !"#$ +*A7#+*-*1, #$ 1&, $,+#0,39 1*0*$$(+9 .&+ F73*+#(1 "7?4.&+- $0"*-*$ %7,6 #1 2*1*+(36 3*(/$ ,& -&+* (007+(,* +*0&1$,+70,#&1$ =S'(-(+&0' KLLOo )*+$&1(3 0&--71#0(,#&1>5 •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`-#?#12Q ,"* =7$7(339 3&;4 &+/*+> -&1&,&1* "7? ;#," ,"* =7$7(339 "#2"*+4&+/*+> 1&14-&1&,&1* "7? =f(3*4 $('6 MOhO>5 G1 ,"* 3#,*+(,7+* ,"* " @3;?-:$3:$ !3,*+#12 #$ &.,*1 +*.*++*/ ,& ($ 3#-#,4 #125 :1 *?0*33*1, +*8#*; &1 3#-#,#12 #$ 2#8*1 #1 B7++(1 =MOOO>6 (1/ ;* -('* 1& *..&+, ,& ,+9 (1/ +*)+&/70* #, "*+*5 D*+,(#1 +*0&1$,+70,#&1$ 0(1 (3$& %* 7$*/ ,"(, (+* #1"*+*1,39 1&14&$0#33(,&+9 %9 /*$#216 $70" ($ ,"* 03($$ &. =C>FNH $0"*-*$ ==C*#2",*/> F$$*1,#(339 N&14H$0#33(,&+9 $0"*-*$>6 ;"#0" 2*1*+(339 /& 1&, +*4 A7#+* !3,*+#12 &+ 3#-#,#125 • MB"2?)-;;' : +*0&1$,+70,#&1 (32&+#,"- #$ +*.*++*/ ,& ($ @4*?(0, #. #, *?(0,39 +*)+&4 /70*$ ( 23&%(3 )&391&-#(3 &. /*2+** @ =\(+," (1/ I+*/*+#0'$&16 MOOL>5 W*1*+(339 $)*('#126 $,+#0, *?(0,1*$$ 0&1$,+(#1,$ ;#33 3*(/ ,& (1 #10+*($* #1 (007+(09 &. ,"* +*0&1$,+70,#&1 .710,#&15 KKP X(7+#,@*1 *, (35 <&391&-#(3 +*0&1$,+70,#&1 .710,#&1$6 -*1,#&1*/ ( 0&7)3* &. ,#-*$ (%&8*6 (+* ( )&)73(+ 0"�* #1 ,"* 3#,*+(,7+* (1/ (33 )+&)*+,#*$ /#$07$$*/ (%&8* 0(1 %* 0&18*4 1#*1,39 /*(3, ;#," 7$#12 $70" ( %($#$5 S&-* ;&+' "($ %**1 /&1* &1 1&1)&391&-#(34 %($*/ +*0&1$,+70,#&1 .710,#&1$ =*525 N&+-(1 (1/ N(#+6 KLL[o l#(& *, (36 KLLK>6 "&;*8*+6 ;* ;#33 .&07$ &1 ,"* .&+-*+ "*+*5 : 0&-)(+#$&1 &. 8(+#&7$ +*0&1$,+704 ,#&1 .710,#&1$ #1 ,"* 0&1,*?, &. 0&1$*+8(,#8* 0($0(/* #1,*+)&3(,#&1 ;($ ,(0'3*/ %9 N&+-(1 *, (3 =KLLO>5 (-23);?:52?$3) @:3>*-6 <3:65*"?$3) c3)- %$6-);$3)d !"* &1*4/#-*1$#&1(3 +*0&1$,+70,#&1 )+&%3*- .&+ ( !1#,*48&37-* $0"*-* 7,#3#@#12 ( )&391&-#(3 %($#$ 0(1 %* $,(,*/ ($ .&33&;$^ W#8*1 /#$0+*,* 0*334(8*+(2*/ 8(37*$ s C ($) &8*+ 0*33$ &C ="*+* &C +*.*+$ ,& ( MB 0*33>6 /*,*+-#1* 0&*.!0#*1,$ 2 C 6 $ = M, . . . , @6 $& ,"(, (L) (M) (K) (@) sC (B) = 2C + 2C B + 2C BK + . . . + 2C B @ , =[5Zh> #$ (1 ())+&?#-(,#&1 ,& ,"* 71/*+39#12 !*3/ s #1 0*33 & C 5 :$ -*1,#&1*/ )+*8#&7$396 #, #$ /*$#+(%3* ,"(, ,"* +*0&1$,+70,#&1 $(,#$!*$ ,"* 0*334(8*+(2*/ )+&)*+,96 & &C sC (B) %B = s C 6 BC , =[5Z[> ;"*+* 6 BC #$ ,"* ;#/," &. 0*33 &C 5 G1 ,"* 0&1,*?, &. $*-#4X(2+(12#(1 (/8*0,#&1 $0"*-*$6 ,"#$ )+&)*+,9 #$ (3$& +*.*++*/ ,& ($ ,"* -($$40&1$*+8(,#&1 )+&)*+,95 G1 ,"* 0*334#1,*2+(,*/ 0&1,#17#,9 *A7(,#&1 =[5M[> s +*.*+$ ,& *#,"*+ 0*334(8*+(2*/ (#+ /*1$#,9 l &+ ,+(0*+ /*1$#,9 l 46 "&;*8*+6 #1 ,"* 0&1,*?, &. +*0&1$,+70,#&1$ #, 0(1 %* /*$#+(%3* ,& +*0&1$,+70,#&1 l (1/ 4 $*)(+(,*39 =($ -*1,#&1*/ #1 $*0,#&1 [5P5M>5 G1 )(+,#073(+ ;"*1 *1.&+0#12 $"()*4)+*$*+8(,#&1 #, -(9 %* 0&18*1#*1, ,& ())39 ,"* !3,*+$V3#-#,*+$ ,& 4 (1/ 1&, l 4 =*525 N(#+ (1/ X(7+#,@*16 KLML>5 T*10*6 .&+ ,"* /#$4 07$$#&1 &1 +*0&1$,+70,#&1$ s 0(1 *#,"*+ +*.*+ ,& l 6 l 4 &+ 45 +,- 1$-2-D$;- /3);?")? A-?,3% c1/3Ad <*+"()$ ,"* $#-)3*$, $7%42+#/4$0(3* +*)+*$*1,(,#&1 #$ ,"* $&40(33*/ )#*0*;#$* 0&14 $,(1, -*,"&/ =6 ;"#0" $#-)39 7$*$ sC (B) = s C . =[5ZO> !"#$ ())+&(0" #$ (,,+#%7,*/ ,& W&/71&8 =MORO> (1/ ,+#8#(339 $(,#$!*$ =[5Z[>6 /&*$ 1&, 1**/ ( "(3&6 (1/ #$ (3$& #1"*+*1,39 -&1&,&1* $#10* #, 0(11&, 3*(/ ,& 1*; *?4 ,+*-(5 !"#$ ())+&(0" #$ (3$& .&+-(339 !+$,4&+/*+ (007+(,* (1/ "#2"39 /#..7$#8* ;"*1 7$*/ ;#," (19 $0"*-* &8*+ $-&&," "&;$ (1/ /#$,+#%7,#&1$5 :$ ( 0&1$*A7*10*6 ,"#$ 0"�* &. +*0&1$,+70,#&1 #$ 0&1$#/*+*/ ,&& /#..7$#8* .&+ (,-&$)"*+#0 ,+(1$)&+, )+&%4 [ I#1#,*4E&37-* !+(1$)&+, S0"*-*$ KKR 3*-$ =713*$$ ,"* "&; #$ `+&72"Q>6 (1/ $& ;* -7$, ,7+1 &7+ (,,*1,#&1 ,& "#2"*+4&+/*+ +*0&1$,+70,#&1$5 F$E,-:K3:%-: :-23);?:52?$3); N&,* ,"(, %9 ())+&)+#(,*39 $"#.,#12 ,"* )&391&-#(3 =[5Zh>6 ;* 0(1 (3;(9$ -() & C &1,& ,"* 1&+-(3#@*/ #1,*+8(3 B ∈ [−6 B C /K, 6 BC /K] ;#," 0*1,*+)
, B = L5 \9 / $&6 ,"* -(," %*"#1/ ,"* +*0&1$,+70,#&1 #$ /+(-(,#0(339 $#-)3#!*/6 (1/ $& ;* ;#33 "*+*(.,*+ ($$7-* ,"(, ;* (+* ;&+'#12 &8*+ ,"#$ /&-(#15 I7+,"*+6 ;* ;#33 ($$7-* ,"(, ,"* 2+#/ #$ 5)$<3:6 $& ,"(, 6 B L = 6 B .&+ (33 L5 Y*0&1$,+70,#&1$ %($*/ &1 1&14 71#.&+- 2+#/$ (+* 2*1*+(339 ( $,+(#2",.&+;(+/ *?,*1$#&1 &. ,"* 71#.&+- 0($*5 <*+"()$ ,"* -&$, #1,7#,#8* -*,"&/ .&+ /*,*+-#1#12 ,"* 0&*.!0#*1,$ &. =[5Zh> #$ ,& 7$* ( !(93&+ $*+#*$ *?)(1$#&1 (%&7, ,"* 0*1,*+ &. ,"* 0*33 =B = L>6 . sC (B) = sC . ! , sC + B=L ,B ".. . . . B=L M B+ K ! , K sC , BK ".. . . . ! @ ".. , sC . M BK + . . . + . B@ @| , B @ . B=L B=L 0 / + O (6 B) @+M . =[5PL> \9 )(#+#12 ,*+-$ &. *A7(3 &+/*+6 ;* &%,(#1 ,"* ($$&0#(,#&1 ". ! M , $ sC .. (L) ($) 2C = sC (L), 2C = . $| , B$ . . =[5PM> B=L S#10* ;* /& 1&, '1&; ,"* *?(0, 8(37* &. s C &+ #,$ /*+#8(,#8*$6 ;* -7$, ())+&?#4 -(,* ,"*$* 8(37*$ 7$#126 .&+ *?(-)3*6 #1,*+)&3(,*/ )&391&-#(3$ ,"+&72" '1&;1 0*334 (8*+(2*/ 8(37*$5 N&,* ,"(, &1* -7$, %* 0(+*.73 #1 0"&&$#12 ,"* 0&++*0, ())+&?#-(,#&1$ ,& ,"*$* /*+#8(,#8*$ ,& )+*$*+8* "#2"4&+/*+ (007+(095 S)*0#!0(3396 .&+ =[5PL> ,& %* .&+-(339 O [(6 B) @ ] (007+(,*6 *(0" &. ,"* /*+#8(,#8*$ , ) sC /, B) -7$, %* ())+&?#-(,*/ ,& &+/*+ O [(6 B) @−) ]6 (1/ sC (L) -7$, %* ())+&?#-(,*/ ,& &+/*+ O [(6 B) @ ]5 !"* +(,#&1(3* %*"#1/ ,"#$ 03(#- #$ ($ .&33&;$^ C"*1 *8(37(,#12 ,"* +*0&1$,+70,#&1 =[5PL>6 *(0" &. ,"* /*+#8(,#8*$ , ) sC /, B) #$ -73,#)3#*/ %9 B @ 6 ;"#0" -7$, $(,#$.9 |B| @ ≤ (6 B) @ 5 T*10*6 #. , ) sC /, B) #$ ())+&?#-(,*/ ,& O [(6 B) )−@ ] ,"*1 *(0" ,*+- #1 ,"* $*+#*$ =[5PL> #$ ())+&?#-(,*/ ,& O [(6 B) @ ]5 T&;*8*+6 $#10* s C (L) #$ 1&, -73,#)3#*/ %9 (19 )&;*+ &. B6 #, -7$, %* ())+&?#-(,*/ ,& .733 &+/*+4&.4(007+(095 S$)$?-K%$<<-:-)2- "@@:3B$6"?$3); H1 (8*+(2#12 ,"* !(93&+ $*+#*$ =[5PL> &8*+ ( 0*33 & C 6 ;* &%,(#1 ". ! & 0 / M , K sC .. M 6 B/K sC (B)%B = sC (L) + . (6 B)K + O (6 B)P . K . 6 B −6 B/K MK , B B=L =[5PK> KKa X(7+#,@*1 *, (35 !"* 3*.,4"(1/4$#/* &. ,"#$ *?)+*$$#&1 #$ $#-)39 ,"* 0*33 (8*+(2* s C 6 ;"#0" #$ '1&;1 #1 ( !1#,*48&37-* 0&1,*?,5 !"* !+$, ,*+- &1 ,"* +#2",4"(1/4$#/* #$ ,"* 8(37* &. s C (B) *8(37(,*/ (, ,"* 0*3340*1,*+)
, (1/ 0 #, #$ .&33&;*/ %9 "#2"*+4&+/*+ ,*+-$5 T*10*6 ;* / 0(1 0&1037/* ,"(, s C #$ ( O (6 B)K ())+&?#-(,#&1 ,& ,"* 8(37* &. s C (B) *8(37(,*/ (, ,"* 0*1,*+)
,5 !"#$ +*$73, #-)3#*$ ,"(, #. ;* 7,#3#@* !1#,*4/#..*+*10* ())+&?#-(,#&1$ ,& ())+&?#-(,* &. (19 &+/*+ (, B = L6 $70" ())+&?#-(,#&1$ ;#33 &139 %* 0 / /*+#8(,#8*$ 8(3#/ 7) ,& O (6 B)K #1 ( !1#,*48&37-* 0&1,*?,5 !"* $#-)3*$, !1#,*4/#..*+*10* ())+&?#-(,#&1 #$ ,"* )#*0*;#$*43#1*(+ -*,"&/ =6 2#8*1 %9 ! ". , sC .. sC (B) = s C + =[5PZ> . B, ,B . B=L =8(1 X**+6 MOhh> ;"*+* , s C /, B #$ (, 3*($, ( !+$,4&+/*+4(007+(,* ())+&?#-(,#&1 ,& ,"* /*+#8(,#8* (, B = L5 S&-* 0"�*$ #1037/* (1 7);#1/ /#$0+*,#@(,#&16 ! ". s − s C−M , sC .. + O(6 B), = C =[5PP> . ,B . 6B B=L &+ ( 0*1,*+*/ /#$0+*,#@(,#&1 ! ". , sC .. . ,B . = B=L / 0 s C+M − s C−M + O (6 B)K . K6 B =[5PR> F#,"*+ 0"�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
, 8(37* s C (L)6 ". ! 0 / (6 B)K , K sC .. =[5Pa> sC (L) = s C − . + O (6 B)P , K . KP ,B B=L ;"#0" #$ ( .&7+,"4&+/*+4(007+(,* ())+&?#-(,#&1 ,&/ ,"* )
,;#$* 8(37* &. s C (L)6 ($ 0 3&12 ($ , K sC /, BK #$ ())+&?#-(,*/ ,& (, 3*($, O (6 B)K 5 D&-%#1#12 ,"#$ ())+&?#4 -(,#&1 ;#," =[5PL>6 ;* &%,(#1 ( ,"#+/4&+/*+ =)(+(%&3#0> +*0&1$,+70,#&1 [ I#1#,*4E&37-* !+(1$)&+, S0"*-*$ | K | 2 R 0.9 (a) 5 0.9 5 KKh 0.6 (b) L (6x) L (6x) 0.3 8 0. 0.6 3 4 0.8 3 0.3 2 0 0.2 0. 6 0.6 1.0 1.0 4 PCoM PLM 0.4 0.4 0.4 0.6 &! u0 6t / 6x 0.8 1 2 1.2 1.4 0 0.2 0.4 0.6 &! u0 6t / 6x 0.8 1 Fig. 8.14 !"* $,(%#3#,9 )+&)*+,#*$ =$** $*0,#&1 [5Z5Z (1/V&+ X(7+#,@*1 KLLh> &. ( &1*4/#-*1$#&1(3 !1#,*48&37-* $0"*-* %($*/ &1 (1/ 6 +*$)*0,#8*395 N&,* ,"(, #1 &1* /#-*1$#&1 (33 !1#,*48&37-* $0"*-*$ /#$07$$*/ #1 ,"#$ 0"(),*+ (+* #/*1,#0(3 ;"*1 7$#12 ,"* $(-* +*0&1$,+70,#&1 -*,"&/5 =(> SA7(+*/ -&/737$ &. ,"* (-)3#!0(,#&1 .(0,&+ =|K |K > ($ ( .710,#&1 &. D&7+(1, 17-%*+ =B4(?#$> (1/ ;(8*3*12," G =J4(?#$>5 T*10* =(> $"&;$ "&; -70" *(0" ;(8*3*12," #$ /(-)*/ #1 &1* ,#-*4$,*) ($ ( .710,#&1 &. D&7+(1, 17-%*+5 I&+ ( !?*/ D&7+(1, 17-%*+ & ,"* /(-)#12 /*0+*($*$ -&1&,&1#0(339 ($ ( .710,#&1 &. ;(8*3*12," G (1/ 3#-G→' |K |K = M5 I&+ D&7+(1, 17-%*+ L &+ M ,"* $0"*-* #$ *?(0, (1/ "*10* |K |K = M5 =%> S(-* ($ =(> %7, .&+ ,"* +*3(,#8* )"($* $)**/ =(>6 ,"(, #$6 "&; -70" *(0" I&7+#*+ -&/* #$ (00*3*+(,*/ &+ /*0*3*+(,*/ ($ ( .710,#&1 &. D&7+(1, 17-%*+5 ".. . . . ".. . . . ! " / 0 (6 B)K BK − + O (6 B)Z , MK B=L B=L =[5Ph> ;"*1 0&-%#1*/ ;#," $#-)3* !1#,*4/#..*+*10* ())+&?#-(,#&1$ &. ,"* .&+- =[5PR> (1/ ! K ".. 0 / s − Ks C + s C+M , sC . =[5P[> = C−M + O (6 B)K . . K K . ,B (6 B) ! , sC sC (B) = s C + ,B M B+ K ! , K sC , BK B=L G1 .(0,6 #, 0(1 %* A7#0'39 8*+#!*/ ,"(, =[5Ph> (3$& $(,#$!*$ ,"* 0*334(8*+(2*/ )+&)*+,9 =[5Z[>5 !"#$ -*,"&/ "($ ,"* "#2"*$, .&+-(3 &+/*+ &. (007+(09 ,"(, 0(1 %* &%,(#1*/ %9 ,+*(,#12 !1#,*48&37-* -*,"&/$ #1 ( !1#,*4/#..*+*10* 0&1,*?,5 !"#$ 0"�* &. +*4 0&1$,+70,#&1 ;($ 7$*/ %9 X()+#$* (1/ <3(1,* =MOOR>5 S$)$?-K#3*56- "@@:3B$6"?$3); !& &%,(#1 ())+&?#-(,#&1$ "#2"*+ ,"(1 ,"#+/4&+/*+ #1 (007+(096 ;* -7$, !+$, ,('* ( $,*) %(0' (1/ 71/*+$,(1/ "&; !1#,*48&37-* -*,"&/$ (+* .&+-73(,*/5 I#+$,6 +*0(33 ,"(, !1#,*48&37-* -*,"&/$ 7$* 0*334(8*+(2*/ 8(37*$6 ;"#0" #-)3#*$ ,"(, ,"* 71/*+4 39#12 $0(3(+ !*3/ #$ 1&, '1&;1 )
,4%94)
,5 G1$,*(/6 #, #$ 0*334(8*+(2*/ 8(37*$ ,"(, (+* '1&;1 *?(0,39 & M 6 B/K sC = sC (B)%B. =[5PO> 6 B −6 B/K KK[ X(7+#,@*1 *, (35 T*10*6 #1 ,"* 0&1,*?, &. !1#,*48&37-* -*,"&/$6 "#2"4&+/*+ $7%42+#/4$0(3* +*0&14 $,+70,#&1$ 0(11&, %* #1,*+)&3(,*/ ,"+&72" $)*0#!0 )
,$ =($ ;#," !1#,*4/#..*+*10* -*,"&/$>6 %7, -7$, #1$,*(/ $(,#$.9 0*+,(#1 0*334(8*+(2*/ )+&)*+,#*$5 !& %7#3/ ( +*0&1$,+70,#&1 ,"(, 7,#3#@*$ 0*334(8*+(2*$6 &1* 2*1*+(339 /*!1*$ ( 25K 65*"?$#- 6";; <5)2?$3) Q (B) 8#( Q (B) = & B BC− L−M/K sC (B)% r B, r =[5RL> ;"*+* BC− L−M/K /*1&,*$ ,"* 3*.,4$#/* */2* &. 0*33 & C− L 5 N&;6 &%$*+8* Q (BC− L−M/K ) = L, # $ Q (BC− L+M/K ) = 6 B s C− L , # $ Q (BC− L+Z/K ) = 6 B s C− L + s C− L+M , 55 5 H8*+ $70" ( $*, &. 0&1$*07,#8* 0*33$ &1* 0(1 ,"*1 /*!1* (1 #1,*+)&3(,#12 )&391&4 -#(3 &. /*2+** 6 ,"(, ())+&?#-(,*$ ,"* *?(0, 07-73(,#8* -($$ .710,#&1 Q (B)5 C* /*1&,* ,"#$ ())+&?#-(,#&1 %9 Qr (B)5 I#1(3396 ;* &%$*+8* ,"(, #1 (00&+/(10* ;#," ,"* .71/(-*1,(3 ,"*&+*- &. D(30737$6 /#..*+*1,#(,#12 =[5RL> 2#8*$ %Q (B) = sC (B). %B =[5RM> r \9/ *8(37(,#12 0 ,"* !+$, /*+#8(,#8* &. Q (B) (, ( 2#8*1 )
,6 ;* (0,7(339 &%,(#1 ( O (6 B)6−M ())+&?#-(,#&1 ,& ,"* 71/*+39#12 !*3/ s C (B) .+&- #,$ 0*334(8*+(2*$5 !"#$ -*,"&/ 0(1 ,"*1 %* 7$*/ ,& +*0&1$,+70, s C (B) (, (19 )
, (1/6 %9 ,('#12 (//#4 ,#&1(3 /*+#8(,#8*$ &. Qr (B)6 #,$ 0&++*$)&1/#12 /*+#8(,#8*$5 :3,*+1(,#8*396 &1* 0(1 &%,(#1 (1 #/*1,#0(3 +*0&1$,+70,#&1 %9 *1.&+0#12 ,"* 0*334 (8*+(2*/ 0&1$,+(#1, &1 (1 #1,*+)&3(,#12 )&391&-#(3 #1 1*#2"%&+#12 0*33$ =f*++&7'(, 1C (B) &. /*2+** @ ,"(, *?(0,39 $(,#$!*$ ,"* -($$4 *, (36 KLLK>5 !"(, #$6 ( )&391&-#(3 s 0&1$*+8(,#&1 0&1$,+(#1, 1&, &139 #1 0*33 C %7, (3$& #1 @ (/](0*1, 0*33$^ & B # L+M/K @$ @$ # 1C (B) %B = s L 6 B, L = C − .. C + , =[5RK> s K K B L−M/K .&+ @ *8*1 (1/ & B L+M/K B L−M/K 1C (B) %B = s L 6 B, s ! " ! " @+M @−M L = C− .. C + , K K =[5RZ> .&+ @ &//5 r (B) = s 1C (B)>6 (3,"&72" F#,"*+ -*,"&/ ;#33 9#*3/ (1 #/*1,#0(3 +*0&1$,+70,#&1 = Q ,"* 3(,,*+ #$ -&+* (/(),(%3* ,& ,;& /#-*1$#&1$ (1/ %*9&1/5 [ I#1#,*4E&37-* !+(1$)&+, S0"*-*$ KKO G. ;* 7,#3#@* ,"* (.&+*-*1,#&1*/ )+&0*/7+* &8*+ ( Z40*33 $,*10#3 =0&1$#$,#12 &. 0*33$ C − M6 C (1/ C + M>6 ;* ;#33 *?(0,39 &%,(#1 =[5PR>6 =[5Pa> (1/ =[5P[>5 T&;*8*+6 %*9&1/ Z 0*33$6 ,"* !1#,*4/#..*+*10* (1/ !1#,*48&37-* +*0&1$,+70,#&1$ ;#33 /#..*+ $7%$,(1,#(3395 I&+ #1$,(10*6 &8*+ ( 0*1,*+*/ R40*33 $,*10#36 ;* &%,(#1 ())+&?#-(,#&1$ 3 2 Os C−K − MMas C−M + KMZPs C − MMas C+M + Os C+K + O (6 B)a , sC (L) = MOKL ! ".. 2 3 Rs C−K − ZPs C−M + ZPs C+M − Rs C+K , s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sC (B) = 2C + 2C B + 2C BK . =[5RP> S#10* ;* "(8* ,"+** /*2+**$ &. .+**/&- #1 ,"#$ )&391&-#(36 ;* 0(1 0"&&$* ,& *14 .&+0* sC (−6 B/K) = sCG (1/ sC (6 B/K) = sC( 6 ;"*+* sCG (1/ sC( (+* +*0&1$,+70,*/ 8(37*$ (, ,"* 3*.,4 (1/ +#2",4 */2*$6 +*$)*0,#8*395 !"*$* (+* )7+)&$*39 0"&$*1 ,& %* 0&1$#$,*1, %*,;**1 1*#2"%&+#12 0*33$6 ;"#0" 2#8*$ 7$ ,"* /*$#+*/ 0&1,#17#,9 +*$,+#04 ,#&15 C#," &7+ +*-(#1#12 /*2+** &. .+**/&- ;* *1.&+0* ,"* 0*334(8*+(2*/ 0&1/#,#&1 =[5Z[>5 !"#$ $0"*-* #$ ,"* ;*334'1&;1 )#*0*;#$*4)(+(%&3#0 -*,"&/ =< &. D&3*33( (1/ C&&/;(+/ =MO[P>5 !& &%,(#1 */2*)
, 8(37*$ s CG (1/ sC( 6 < ($) G = s ( 5 G1 ,*+-$ &. ,"* 0&*.!0#*1,$ 2 6 =(3$& $** f*++&7'(, *, (3 =KLLK>> (1/ s C+M C C ,"#$ +*0&1$,+70,#&1 0(1 %* ;+#,,*1 ($ KZL X(7+#,@*1 *, (35 (a) Picewise-linear reconstruction exact PLM PLM+filter Density 4 3 2 1 0.1 0.3 0.5 0.7 x-axis (b) Picewise-parabolic reconstruction exact PPM PPM+filter Density 4 3 2 1 0.1 0.3 0.5 0.7 x-axis Fig. 8.15 Y*0&1$,+70,#&1$ .&+ ,"* #++*273(+ $#21(3 &. S-&3(+'#*;#0@ (1/ W+(%&;$'# =MOOL6 %37* 3#1*> 7$#12 ,"* =(> )#*0*;#$*43#1*(+ -*,"&/ = (1/ =%> ,"* )#*0*;#$* )(+(%&3#0 -*,"&/ =< ;#," +*0&1$,+70,#&1 .710,#&1 !3,*+ =2+*9> (1/ ;#,"&7, =+*/>5 !"* !3,*+ .&+ ;#," ,"*,(wM (1/ ,"* <5 (K) 2 (L) 2C = s C − (6 B)K C , MK , 5 M 4 5 M K4 (M) s − s C−M − s − s C−K , 2C = 6 B Z C+M MK C+K , 4 5 M4 5 M (K) −Rs C + Z s C+M + s C−M − s C+K + s C−K . 2C = K K (6 B)K [ I#1#,*4E&37-* !+(1$)&+, S0"*-*$ | K | 2 R 5 0.9 (a) 0.9 4 L (6x) L (6x) 5 KZM 0.9 0.6 3 4 3 (b) PPM-s PPM2 PPM3 PPM4 PPM5 0.8 0.8 0.6 0.3 1.2 2 0 0.2 0.4 0.6 &! u0 6t / 6x 0.8 1 2 1.4 0 0.2 0.4 0.6 &! u0 6t / 6x 0.8 1 Fig. 8.16 S(-* ($ I#25 [5MP %7, .&+ < ($ ;*33 ($ <5 <6 ,"(, #$6 7$#12 =[5RK> ;#," @ = K ,& /*,*+-#1* ,"* $7%42+#/40*33 +*0&1$,+70,#&1 .710,#&15 ($) :33/ &. ,"*0 0&*.!0#*1,$ 2 C #1 ,"#$ 0($* ())+&?#-(,* s C (B) (1/ #,$ /*+#8(,#8*$ ,& O (6 B)P 5 !"* ())+&?#-(,#&1 ,& s CG #1 =[5RR> (1/ s C( (+* .&7+,"4&+/*+ (007+(,* .&+ 71#.&+2+#/$5 H%8#&7$396 &1* 0&73/ (3$& /*+#8* $*0&1/6 ,"#+/6 .&7+,"6 !.," &+ $#?,"4&+/*+ (007+(,* *$,#-(,#&1$ %9 !,,#12 ( 3#1*(+6 )(+(%&3#06 07%#06 A7(+,#0 (1/ A7#1,#0 )&394 1&-#(3 $& ,"(, =[5RL> &+ =[5RK4[5RZ> #$ $(,#$!*/ #1 K6 Z6 P6 R (1/ a (/](0*1, 0*33$6 +*$)*0,#8*395 C* ;#33 +*.*+ ,& < ;"*+*($ < ;"*+*($ <5 H1* 0&73/ 1(,7+(339 ($' ,"* A7*$,#&1 ;"9 $"&73/ &1* 1&, 7$* ,"* "#2"*$,4&+/*+ )&391&-#(3 ,"(, 0(1 %* ())+&?#-(,*/ ;#," ( 2#8*1 "(3& =$,*1$#3>k I&+ *?(-)3*6 ,"* 07%#0 )&391&-#(3 7$*/ ,& *$,#-(,* ,"* */2* 8(37* #1 <5 Y*0&1$,+70,#&1$ %($*/ &1 )&391&-#(3$ &. /*2+** "#2"*+ ,"(1 K "(8* %**1 )+&4 )&$*/ #1 ,"* 3#,*+(,7+* %7, "(8* 1&, %**1 ;#/*39 (/&),*/ #1 ,+(1$)&+, $0"*-*$ ($ &. ,"* ,#-* &. ;+#,#125 f*++&7'(, *, (3 =KLLK> #1,+&/70*/ ( $9--*,+#0 )#*0*;#$*407%#0 -*,"&/ =6 (3&12 ;#," (/8(10*/ !3,*+#12 ,*0"1#A7*$ =f*++&7'(, *, (36 KLLR>6 KZK X(7+#,@*1 *, (35 | K | R 5 (a) 0.9 4 L (6x) L (6x) 5 2 0.9 0.6 3 4 3 (b) PQM3 PQM4 PQM5 PQM6 0.8 0.8 0.6 0.3 1.2 2 0 0.2 0.4 0.6 &! u0 6t / 6x 0.8 1 2 1.4 0 0.2 0.4 0.6 &! u0 6t / 6x 0.8 1 Fig. 8.17 S(-* ($ I#25 [5Ma %7, .&+ )+&)&$*/ ( $9--*,+#0 )#*0*;#$*4A7(+,#0 -*,"&/ = %($*/ &1 )&391&-#(3$ &. /*2+** P5 :$ .&+ ,"* <5 T&;*8*+6 *8*1 .&+ ,"* -&$, (007+(,* #$ -&/*$, 0&-)(+*/ ,& < (1/ P = 0(1 "(8* K (1/ Z *?4 ,+*-( ;#,"#1 ( 2+#/ 0*33 -('#12 #, "(+/*+ ,& !3,*+ $70" )&391&-#(3$ 0&-)(+*/ ,& ( =+*3(,#8*39 3&;4&+/*+> )(+(%&3(5 1$-2-D$;- 45"%:"?$2 ;@*$)-;' :1 #1,*+*$,#12 8(+#(1, &1 ,"* +*0&1$,+70,#&1 -*,"&/$ /#$07$$*/ $& .(+6 (1/ (3$& %($*/ &1 )(+(%&3($6 #$ ,"* )#*0*;#$* A7(/+(,#0 $)3#1* -*,"&/ =f*++&7'(, *, (3 KLLa> (1/ "#2"*+4&+/*+ *?,*1$#&1$ $70" ($ ,"&$* )+*$*1,*/ #1 f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f*++&7'(, *, (3 =KLLK> ,"* )#*0*;#$* $)3#1* +*0&1$,+70,#&1 -*,"&/ #$ $7)*+#&+ ,& <5 !"* )+#0* ,& )(96 #1 ,*+-$ &. ( )(+(33*3 0&-)7,(,#&1(3 *18#+&1-*1,6 #$ ,"(, $)3#1*$ (+* #1"*+*1,39 23&%(3 $#10* ,"* #18*+$#&1 &. ( 23&%(3 ,+#4/#(2&1(3 -(,+#? #$ 1*0*$$(+95 M;;-)?$"**J )3)K3;2$**"?3:J cM.Wd :-23);?:52?$3);' F$$*1,#(339 1&14&$0#33(,&+9 +*0&1$,+70,#&1$ ;*+* &+#2#1(339 /*8*3&)*/ %9 T(+,*1 *, (3 =MO[h> .&+ $"&0' "9/+&/91(-#0$ )+&%3*-$5 !"#$ ())+&(0" #$ )(+,#073(+39 #14 [ I#1#,*4E&37-* !+(1$)&+, S0"*-*$ KZZ ,*+*$,#12 $#10* #, 3*(/$ ,& ( +*0&1$,+70,#&1 ,"(, #$ =71/*+ -&$, 0#+07-$,(10*$> -&1&4 ,&1#0 (1/ )&$#,#8*5 !"* FNH $0"*-* ;&+'$ %9 ())39#12 *#,"*+ ( !1#,*4/#..*+*10* &+ !1#,*48&37-* ())+&(0" =($ /#$07$$*/ *(+3#*+> &1 ( 8(+#*,9 &. $,*10#3$5 !"* +*0&14 $,+70,#&1 ,"(, $(,#$!*$ $&-* 3*($,4&$0#33(,&+9 )+&)*+,96 (-&12 (33 )&$$#%3* $,*10#3$6 #$ ,"*1 0"&$*1 ,& 2#8* ,"* `,+7*Q +*0&1$,+70,#&15 !"* -(#1 /+(;%(0' &. ,"#$ ())+&(0" #$ ,"(, #, +*A7#+*$ ( 3(+2* $,*10#3 #1 &+/*+ ,& &%,(#1 ,"* $(-* &+/*+ &. (007+(09 ($ `8(1#33(Q !1#,*4/#..*+*10* &+ !1#,*48&37-* -*,"&/$5 G-";?K;45":-; !"* 3*($,4$A7(+*$ ,*0"1#A7* #$ &1* &. ,"* .*; ())+&(0"*$ (8(#3(%3* .&+ &%,(#1#12 ())+&?#-(,* +*0&1$,+70,#&1$ &1 71$,+70,7+*/ 2+#/$5 _1/*+ ,"#$ -*,"&/6 ;* #1,+&4 /70* $&-* A7(1,#!0(,#&1 &. ,"* -#$!, %*,;**1 ,"* +*0&1$,+70,#&1 =[5Zh> (1/ ,"* '1&;1 0*334(8*+(2*$6 7$7(339 2#8*1 %9 ,"* $A7(+* &. ,"* /#..*+*10* %*,;**1 ,"* 0*334 (8*+(2*$ &. ,"* +*0&1$,+70,#&1 (1/ ,"* '1&;1 0*334(8*+(2*$5 !"* -#$!, #$ ,"*1 -#14 #-#@*/ &8*+ (33 )&$$#%3* +*0&1$,+70,#&1$ #1 &+/*+ ,& 2#8* ,"* t%*$, )&$$#%3*u +*0&14 $,+70,#&15 :1 *?(-)3* &. ( D(+,*$#(1 !1#,*48&37-* $0"*-* &1 (1 71$,+70,7+*/ 2+#/ %($*/ &1 ,"* 3*($,4$A7(+*$ ,*0"1#A7* 0(1 %* .&71/ #1 \(+," (1/ I+*/*+#0'$&1 =MOOL>5 W)-K%$6-);$3)"* :-23);?:52?$3) *$6$?-:;f!*?-:; :$ /#$07$$*/ #1 $*0,#&1 [5Z5a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hh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t07, &..u &+ "(,,*1 $-&&,"6 )"9$#4 0(3 -(?#-( (1/ -#1#-(5 S*8*+(3 ())+&(0"*$ "(8* %**1 )+&)&$*/ ,& +*,(#1 )"9$#0(3 *?,+*-( ;"#3* !3,*+#12 &7, $)7+#&7$ 2+#/4$0(3* 1&#$* =$**6 .&+ *?(-)3*6 8(1 :3%(/( *, (3 =MO[K>6 f*++&7'(, *, (3 =KLLR>6 X#7 *, (3 =KLLh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wo-dimensional reconstruction functions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s C &8*+ 0*33$ &C 6 /*,*+-#1* 0&*.!0#*1,$ 2 ($, L) 6 $ + L ≤ @ =$ (1/ L (+* L &+ )&$#,#8* #1,*2*+$>6 $& ,"(, sC (B, J) = - $+ L≤@ ($, L) $ L 2C BJ , =[5Ra> #$ (1 ())+&?#-(,#&1 ,& ,"* 71/*+39#12 !*3/ s #1 0*33 & C 5 !"* 0*334(8*+(2*/ )+&)*+,9 #1 ,;& /#-*1$#&1$ ,"*1 +*(/$6 & &C sC (B, J) %& = s C 6 &C . =[5Rh> :2(#1 ;* 0(1 0"&&$* ,& $"#., ,"* +*0&1$,+70,#&1 $& ,"(,6 .&+ $#-)3#0#,96 & C "($ ( 0*1,+&#/ 3&0(,*/ (, (B, J) = (L, L)5 1$-2-D$;- /3);?")? A-?,3% c?D3 %$6-);$3);d !"* *?,*1$#&1 &. ,"* [ I#1#,*4E&37-* !+(1$)&+, S0"*-*$ KZR !"#$ $0"*-* $7..*+$ .+&- ,"* $(-* /*!0#*10#*$ ($ /#$07$$*/ #1 ,"* &1*4/#-*1$#&1(3 0($*6 (1/ $& #$ 1&, /#$07$$*/ .7+,"*+ "*+*5 1$-2-D$;- G$)-": A-?,3% c?D3 %$6-);$3);d !"* ,;&4/#-*1$#&1(3 )#*0*;#$* 3#1*(+ +*0&1$,+70,#&1 0(1 %* ;+#,,*1 ($ (M,L) sC (B, J) = s C + 2C (M,L) :19 0"�* &. 2 C (L,M) (1/ 2C (L,M) B + 2C J. =[5RO> ;#33 9#*3/ ( -($$40&1$*+8(,#8* +*0&1$,+70,#&1 (1/6 ($ (M,L) (L,M) (1/ 2C 0&++*$)&1/ ,& ,"* 0&-)&1*1,$ &. ;#," ,"* &1*4/#-*1$#&1(3 5 F$E,K3:%-: :-23);?:52?$3); c?D3 %$6-);$3);d !+7* ,"#+/4&+/*+ (1/ "#2"*+ $0"*-*$ +*A7#+* $&-* -*,"&/ &. #10&+)&+(,#12 0+&$$4 /*+#8(,#8*$ #1 &+/*+ ,& %* .&+-(339 ,"#+/4&+/*+ (007+(,*5 I&+ *?(-)3*6 ( ,+7* ,"#+/4 &+/*+ )(+(%&3#0 +*0&1$,+70,#&1 0&73/ -('* 7$* &. ( +*0&1$,+70,#&1 &. ,"* .&+(M,L) (L,M) sC (B, J) = s C + 2C B + 2C J =[5aL> ! ! " " K K (6 B) (6 J) (K,L) (M,M) (L,K) +2C BK − JK − + 2C BJ + 2C , MK MK (M,L) (L,M) (K,L) (L,K) (M,M) (+* &%,(#1*/ %9 (2(#1 ())+&?#-(,#12 ,"* ;"*+* 2C 6 2C 6 2C 6 2C 6 (1/ 2C /*+#8(,#8*$ &. s 5 N&,* ,"(, 2 (L.L) /&*$ 1&, *A7(3 ,"* 0*33 (8*+(2* s %7, #1037/*$ -&+* ,*+-$ ,& *1$7+* ,"* -($$40&1$*+8(,#&1 )+&)*+,95 F?,*1$#&1$ &. ,"#$ .&+- (+* /*$0+#%*/ #1 N(#+ (1/ U(0"*1"(7*+ =KLLK> (1/ _33+#0" *, (3 =KLML>5 G, "($ %**1 $"&;1 ,"(, ,"* 3&$$ &. (007+(09 (,,+#%7,*/ ,& 1*23*0,#12 ,"* 0+&$$4/*+#8(,#8* ,*+(M,M) 0(1 %* 3(+2*6 %7, #$ 3*$$ $#21#!0(1, &1 2+#/$ &. 3&; +*$&37,#&1 =X(7+#,@*1 *, (36 2C KLML>5 KZa X(7+#,@*1 *, (35 1$-2-D$;- 1":">3*$2 A-?,3% c?D3 %$6-);$3);d : +#2&+&7$ *?,*1$#&1 &. ,"* < ;&73/ +*A7#+* ,"* .7339 ,;& /#-*1$#&1(3 %#)(+(%&3#0 )&391&-#(3$ ,& %* 0&14 ,#17&7$ (0+&$$ 0*334%&+/*+$ (, $*3*0,*/ )
,$ (1/V&+ #1 $&-* (8*+(2* $*1$*5 H1* $70" *?,*1$#&1 ;($ /*8*3&)*/ %9 Y(1#0#s0 =MOOK>6 ;"& 0"&$* s (B, J) = qK (J) BK + qM (J) B + qL (J), =[5aM> qL (J) = qLK JK + qLM J + qLL , qM (J) = qMK JK + qMM J + qML , qK (J) = qKK JK + qKM J + qKL . =[5aK> ;"*+* =[5aZ> =[5aP> !"#$ +*0&1$,+70,#&1 "($ 1#1* /*2+**$ &. .+**/&-6 ;"#0" (+* +*$,+#0,*/ %9 $(,#$.9#12 =#> ,"* 0*334(8*+(2*/ 0&1$,+(#1, =[5Rh> #1 0*33 '6 =##> *A7(3 (8*+(2* 8(37*$ (3&12 *(0" &. ,"* .&7+ */2*$ &. ,"* A7(/+#3(,*+(3 0*33$6 (1/ =###> 0&1,#17#,9 (, ,"* 0&+1*+ )
,$ &. *(0" 0*335 !"*$* +*$,+#0,#&1$ 3*(/ ,& O 0&1$,+(#1,$6 (1/ "*10* /*!1* ( 71#A7* +*0&1$,+70,#&15 N&,* ,"(, ,"* +*0&1$,+70,#&1 #$ 1&, 23&%(339 0&1,#17&7$5 C* +*.*+ ,& Y(1#0#s0 =MOOK> .&+ .7+,"*+ /*,(#3$ &1 ,"#$ (32&+#,"-5 MB?-);$3); ?3 $::-E5*": E:$%; :33 &. ,"* -*,"&/$ /*$0+#%*/ (%&8* (+* ,#*/ ,& A7(/+#3(,*+(3 =&+,"&2&1(3> -*$"*$ (1/ ,"* *?,*1$#&1 ,& ,+#(1273(+6 "*?(2&1(3 (1/ &,"*+ 2+#/$ ;"*+* ,"* 0*33$ /& 1&, "(8* *?(0,39 .&7+ 8*+,#0*$6 #$ 1&, &%8#&7$5 !"* (7,"&+$ (+* 1&, (;(+* &. (19 +#2&+&7$ *?,*1$#&1$ &. < 2C = s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h %* &),#-(35 G1 $70" 0($*$ ( 3*($,4$A7(+*$ ())+&(0" -#2", %* ( -&+* 1(,7+(3 0"�* ,& (8&#/ %#($*$ #1,+&/70*/ %9 *?037/#12 $&-* (/](0*1, 0*33$ (1/ 1&, &,"*+$5 C"*1 7$#12 ( 3*($, $A7(+*$ -*,"&/ &1* -(9 0"&$* ,& &),#-#@* ,"* ())+&?#-(,#&1 ,& ,"* 0&*.!0#*1,$ 1&, ]7$, ,& -($$40&1$*+8(,#&1 #1 (/](0*1, 0*33$ %7, (3$& ,& @4*?(0,1*$$ .&+ *?(-)3* =$**6 *5256 \(+," (1/ I+*/*+#0'$&16 MOOL>5 +D3K%$6-);$3)"* *$6$?-:; f !*?-:; Y*0&1$,+70,#&1 .710,#&1 !3,*+#12 #1 ,;&4/#-*1$#&1$ #$ $#21#!0(1,39 -&+* 0&-)3#4 0(,*/ ,"(1 #1 &1* /#-*1$#&16 $#-)39 %*0(7$* ( ,;&4/#-*1$#&1(3 )&391&-#(3 &. /*4 2+** K =( )(+(%&3#0 +*0&1$,+70,#&1> 0(1 )&$$*$$ (1 *?,+*-( ;#,"#1 ( 0*336 (3&12 ( 0*33 %&71/(+96 &+ (, ( 0&+1*+4)
, =(33 &. ;"#0" -7$, %* 0"*0'*/>5 T*10*6 !3,*+4 #12 0&-*$ #1 ,;& "(8&+$^ B#-*1$#&1(339 $)3#, !3,*+#12 (1/ .7339 ,;&4/#-*1$#&1(3 ())+&(0"*$5 !"* /#-*1$#&1(3 $)3#, ())+&(0" $#-)39 ())3#*$ ,"* &1*4/#-*1$#&1(3 !3,*+$ )+*4 $*1,*/ #1 $*0,#&1 [5R5K5M #1 *(0" 0&&+/#1(,* /#+*0,#&15 T&;*8*+6 %9 / $& ,"* *1,#+* +*0&1$,+70,#&1 s C #$ 1&, 27(+(1,**/ ,& %* -&1&,&1#0 ;#,"#1 & C 5 S)*0#!0(3396 ,"*+* (+* 1& 27(+(1,**$ &. -&1&,&1#0#,9 (, 0*33 0&+1*+4)
,$ ;"*+* ,"* +*0&1$,+704 ,#&1 #1 *(0" 0&&+/#1(,* /#+*0,#&1 #$ (//#,#8* =X(7+#,@*1 *, (36 KLLa>5 S,+#0, -&1&,&1#0#,9 (, (33 )
,$ ;#,"#1 ( 0*33 0(1 %* 27(+(1,**/ 7$#12 ,"* .7339 ,;&4/#-*1$#&1(3 ())+&(0" &. \(+," (1/ J*$)*+$*1 =MO[O>6 ;"#0" 0(1 (3$& %* ())3#*/ ,& 71$,+70,7+*/ 2+#/$5 !"#$ !3,*+ 27(+(1,**$ $,+#0, -&1&,&1#0#,9 &. 3#1*(+ +*0&1$,+704 ,#&1$ %9 !+$, /*,*+-#1#12 ;"*+* ( 2#8*1 3#1*(+ +*0&1$,+70,#&1 "($ *?,+*-( =,"#$ #$ ,9)#0(339 ,"* 0*33 0&+1*+4)
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f(3*$(' =MOhO>5 :$ #1 &1* /#-*1$#&1 #, $**'$ ,& !1/ ,"* &),#-(3 t%3*1/#12u &. ( -&1&,&1* "7? (1/ ( "#2"4&+/*+ 1&14-&1&,&1* "7?5 ID! #$ /*4 $0+#%*/ #1 /*,(#3 #1 B7++(1 =MOOO> (1/ "*10* 1&, +*)*(,*/ "*+*5 8.5.3 Practical integration over areas I&+ ,"* ())+&?#-(,#&1 &. ,"* &8*+3() #1,*2+(3$ #1 =[5Ka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s! (B, J) = - 2! B$ J L , =[5aa> $+ L≤K ($, L) ;"*+* 2! (+* +*0&1$,+70,#&1 0&*.!0#*1,$6 0&-)7,* ,"* #1,*2+(3 & "C! s! (B, J)%&. =[5ah> 8.5.3.1 Direct integration using Gaussian quadrature I&+ ,"* /#+*0, #1,*2+(,#&1 7$#12 W(7$$#(1 A7(/+(,7+* #, #$ &.,*1 0&18*1#*1, ,& %+*(' 7) "C! #1,& ,+#(123*$ ;"#0" #$ ,"* 0($* ;* ;#33 /#$07$$ "*+*5 S&6 .&+ $#-)3#0#,96 $7)4 )&$* ,"* &8*+3() (+*( #$ (3+*(/9 (1 (+%#,+(+9 ,+#(123* M[ ;#," 8*+,#0*$ 3&0(,*/ (, B C!,, 6 JC!,, 6 , = M, K, Z6 (1/ 17-%*+*/ 0&71,*+403&0';#$*5 F?(0, #1,*2+(,#&1 &. ,"* )&394 1&-#(3 =[5aa> 0(1 %* (0"#*8*/ 7$#12 W(7$$#(1 A7(/+(,7+* ;"#0" ())+&?#-(,*$ ,"* #1,*2+(3 #1 ,*+-$ &. ( ;*#2",*/ $7- &. .710,#&1(3 *8(37(,#&1$ (, A7(/+(,7+* )
,$5 !"* A7(/+(,7+* )
,$ (+* (") (") (>) (>) (BC! , JC! ) = (BC! , JC! ) = (2) (2) (BC! , JC! ) = 4 5 M a PBC!,M + BC!,K + BC!,Z , PJC!,M + JC!,K + JC!,Z , 4 5 M a BC!,M + PBC!,K + BC!,Z , JC!,M + PJC!,K + JC!,Z , 4 5 M a BC!,M + BC!,K + PBC!,Z , JC!,M + JC!,K + PJC!,Z . =[5a[> =[5aO> =[5hL> =B7'&;#0@ (1/ \(7-2(+/1*+6 KLLL> (1/ ,"* #1,*2+(3 &. s ! (B, J) &8*+ ,"* &8*+3() ,+#(123* "C! #$ 2#8*1 %9 & "C! s! (B, J) %& = 3 6 "C! 2 (") (") (>) (>) (2) (2) s! (BC! , JC! ) + s! (BC! , JC! ) + s! (BC! , JC! ) , Z =[5hM> ;"*+* 6 "C! #$ ,"* (+*( &. "C! 6 "C! = M[ 0 M/ (BC!,K − BC!,M)(JC!,Z − JC!,M) − (JC!,K − JC!,M)(BC!,Z − BC!,M ) . K 1&,* ,"(, (19 (+*( ;#," $,+(#2", 3#1* $#/*$ 0(1 %* %+&'*1 7) #1,& ,+#(123*$ =[5hK> [ I#1#,*4E&37-* !+(1$)&+, S0"*-*$ KZO N&,* ,"(, ,"* A7(/+(,7+* )
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onverting area-integrals into line-integrals !"#$ ())+&(0" ;($ &+#2#1(339 #1,+&/70*/ %9 B7'&;#0@ =MO[P> (1/ B7'&;#0@ (1/ d&/#$ =MO[h> #1 17-*+#0(3 $0"*-*$^ I&+ ,"* $#-)39 0&11*0,*/ &8*+3() (+*( " C! =1&, 1*0*$$(+#39 ( ,+#(123* %7, (19 )&392&1> ,"* .&33&;#12 #1,*2+(3 *A7(,#&1 "&3/$6 && "C! s! (B, J) %& = + , "C! [1 %B + N %J], =[5hZ> ;"*+* , "C! #$ ,"* %&71/(+9 &. " C! 5 !"* )&,*1,#(3$ 1 = 1(B, J) (1/ N = N(B, J) (+* 0"&$*1 $70" ,"(, ,"*9 $(,#$.9 − ,1 ,N + = s! (B, J). ,J ,B !"* #1,*2+(3 &. ,"* )&391&-#(3 +*0&1$,+70,#&1 .710,#&1 s ! (B, J) #1 =[5aa> 0(1 %* ;+#,,*1 ($ & ($, L) ($, L) s! (B, J) %& = - 2! DC! , =[5hP> "C! ($, L) ;"*+* 2! %9 $+ L≤K ($, L) (+* ,"* +*0&1$,+70,#&1 .710,#&1 0&*.!0#*1,$ (1/ D C! (+* ;*#2",$ 2#8*1 KPL X(7+#,@*1 *, (35 54 5 M ., 4 BC!,, + BC!,,−M JC!,, − JC!,,−M , K ,=M (L,L) = (M,L) = (L,M) =− (K,L) = (L,K) =− DC! DC! DC! DC! DC! (M,M) DC! =[5hR> 54 5 M ., 4 K BC!,, + BC!,, BC!,,−M + BKC!,,−M JC!,, − JC!,,−M , a ,=M 54 5 M ., 4 K - JC!,, + JC!,, JC!,,−M + JKC!,,−M BC!,, − BC!,,−M , a ,=M 54 54 5 M ., 4 - BC!,, + BC!,,−M BKC!,, + BKC!,,−M JC!,, − JC!,,−M , MK ,=M =[5ha> =[5hh> =[5h[> 54 54 5 M ., 4 JC!,, + JC!,,−M JKC!,, + JKC!,,−M BC!,, − BC!,,−M , =[5hO> MK ,=M 6, 4 5 M ., = JC!,, Z BKC!,, + K BC!,, BC!,,−M + BKC!,,−M + KP ,=M 4 K 5 K JC!,,−M BC!,, + K BC!,, BC!,,−M + Z BC!,,−M 7 (JC!,, − JC!,,−M) , =[5[L> (1/ (BC!,, , JC!,, ), , = M, . . . , ., =[5[M> (+* ,"* 0&&+/#1(,*$ .&+ ,"* $#/*$ &. ,"* &8*+3() (+*( " C! 17-%*+*/ 0&71,*+403&0';#$*5 S& ., = Z .&+ ,+#(1273(+ &8*+3() (+*($6 . , = P .&+ A7(/+#3(,*+(3 " C! Q$ *,05 N&,* ,"(, =BC!,,−M 6 JC!,,−M > (1/ =BC!,, 6 JC!,, > (+* 0&1,#27&7$ )
,$ =/*!1#12 ( 3#1* $*2-*1,> (1/ ,"* #1/*? , #$ 0903#0 $& ,"(, , = L *A7(3$ , = . , 5 ($, L) !"* ;*#2",$ DC! 2#8*1 #1 *A7(,#&1$ =[5hR4[5[L> "(8* %**1 /*+#8*/ %9 7$#12 =[5hZ> ;#," ,"* .&33&;#12 )(#+$ (1 ($, L) , N($, L) ) ! " 1(L,L) = L, N(L,L) = B , " ! K 1(M,L) = L, N(M,L) = BK , " ! JK (L,M) (L,M) =− , N =L , 1 K " ! (K,L) BZ (K,L) = L, N = Z , 1 " ! JZ (L,K) (L,K) =− , N =L , 1 Z " ! BK J (M,M) (M,M) = L, N = K . 1 [ I#1#,*4E&37-* !+(1$)&+, S0"*-*$ KPM !"* 0"�* &. 1 (1/ N #$ 1&, 71#A7* (1/ 0(1 &.,*1 %* 0"&$*1 .&+ 0&18*1#*10*5 T*+* ;* "(8*1 0"&$*1 1 (1/ N ($ #1 \&0'-(1 =MO[O>5 N&,* ,"(, ,"* #1,*2+(,#&1 &. ,"* )&391&-#(3$ #$ *?(0,5 _$#12 ,"* 3#1*4#1,*2+(3 ())+&(0" ,"* !1(3 /#$0+*,#@*/ ,+(1$)&+, $0"*-* #1 X(4 2+(12#(1 (1/ F73*+#(1 .&+- 0(1 %* ;+#,,*1 ($ 8 9 & s )+M 6 &C = C GC - !=M "C! s! (B, J) %& = GC - - !=M $+ L (1/ P  GoC  P s )+M 6 &C = s )C 6 &C + -  - SC!o  = s )C 6 &C + C o =M !=M o =M  8  GoC !=M 9  ($, L) ($, L) - 2! DC! (o )  , $+ L +*$)*0,#8*396 ;"*+* ,"* #1/#8#/7(3 &8*+3() "7?*$ (+* ;+#,,*1 ($ SC!o = ;oC! & "oC! s! (B, J) %&. =[5[P> I&+ *(0" &8*+3() (+*( ,"* $#214.710,#&1 ; oC! #$ +M .&+ #1"&; (1/ −M .&+ &7,"&;5 !"* $7%$0+#), ! #1 ;o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xtension to spherical geometry F?,*1/#12 ,"* (.&+*-*1,#&1*/ ())+&(0"*$ ,& $)"*+#0(3 2*&-*,+9 2*1*+(339 0&-4 )&71/$ ,"* 0&-)3*?#,9 &. ,"* )+&%3*-6 $#10* *?,+( 0(+* -7$, %* ,('*1 ;"*1 -*,+#0 ,*+-$ (+* )+*$*1,5 S& #1$,*(/ &. "(8#12 #1,*+)&3(,* ( )&391&-#(3 ( -&+* 0&-)3#0(,*/ .710,#&1 -7$, %* #1,*2+(,*/ & & "C! E(_ , ` ) s! (_ , ` )% _ % ` , =[5[R> ;"*+* (_ , ` ) #$ ,"* 0&&+/#1(,* .&+ ,"* 0&-)7,(,#&1(3 $)(0* 0"&$*1 .&+ ,"* #1,*2+(4 ,#&1MO (1/ E(_ , ` ) #$ ,"* -*,+#0 ,*+-5 I&+ *?(-)3*6 #. &1* 0"&&$*$ 2*&2+()"#0 0&4 &+/#1(,*$ (_ , ` ) = (h , e )6 ;"*+* h #$ ,"* 3&12#,7/* (1/ e #$ 3(,#,7/*6 (1/ ,"*1 ,"* MO .&+ $#-)3#0#,9 ;* &139 0&1$#/*+ ,;&4/#-*1$#&1(3 0&-)7,(,#&1(3 $)(0*$ (3,"&72" &1* -(9 (3$& 7$* ,"+**4/#-*1$#&1(3 D(+,*$#(1 0&&+/#1(,*$ .&+ "&+#@&1,(3 )+&%3*-$ &1 ,"* $)"*+* KPK X(7+#,@*1 *, (35 -*,+#0 ,*+- #$ E = (K 0&$(e ) ;"*+* ( #$ ,"* +(/#7$ &. ,"* F(+,"5 S& #1$,*(/ &. "(84 #12 ,& #1,*2+(,* ( )&391&-#(3 ( -70" -&+* 0&-)3#0(,*/ .710,#&1 -7$, #1,*2+(,*/5 G1 2*1*+(36 *?(0, #1,*2+(,#&1 #$ 1& 3&12*+ )&$$#%3* ($ ;($ ,"* 0($* #1 D(+,*$#(1 2*4 &-*,+95 !"*+* (+*6 "&;*8*+6 $&-* $)*0#(3 0($*$ ;"*+* /#+*0, #1,*2+(,#&1 #$ )&$$#%3* =/#$07$$*/ %*3&;>5 !"* 0"�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e = "h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
,$ ;#," ( $,+(#2", 3#1* #1 ,"(, 0&-)7,(,#&1(3 $)(0* ;#33 -#++&+ ( 2+*(,40#+03* (+0 &1 ,"* $)"*+*5 :1&,"*+ %*1*!0#(3 )+&)*+,9 &. ,"#$ 0&-)7,(,#&1(3 $)(0* #$ ,"(, *?(0, #1,*2+(,#&1 &. =[5[R> #$ )&$$#%3* (3&12 0&&+/#1(,* 3#1*$ #1 ,"* 21&-&1#0 0&&+/#1(,* $9$,*- ;"*1 ())39#12 ,"* W(7$$4W+**1Q$ ,"*&+*- =_33+#0" *, (36 KLLO>5 I&+ 3#1*$ 1&, )(+(33*3 ,& ,"* 0&&+/#1(,* 3#1*$ ,"* )&,*1,#(3$ ,"(, 1**/ ,& %* #1,*2+(,*/ #1 ,"* 3#1*4 #1,*2+(3$ 0(1 %* *8(37(,*/V())+&?#-(,*/ 7$#12 &1*4/#-*1$#&1(3 W(7$$#(1 A7(/+(4 ,7+* =X(7+#,@*1 *, (36 KLML>5 :2(#16 /#+*0, #1,*2+(,#&1 ;#33 (3;(9$ %* #1*?(0, /7* ,& ,"* 21&-&1#0 -*,+#0 ,*+-$5 8.6 Extension to three dimensions !"* /#$07$$#&1 $& .(+ "($ %**1 3#-#,*/ ,& ,;& $)(,#(3 /#-*1$#&1$ (1/ ;* ;#33 &139 %+#*"9 /#$07$$ ,"+**4/#-*1$#&1(3 $0"*-*$6 ($ ( -&+* ,"&+&72" /#$07$$#&1 ;&73/ 1**/ (, 3*($, ( 0"(),*+ &1 #,$ &;15 !"*+* (+* %($#0(339 ,"+** ;(9$ &. *?,*1/#12 $0"*-*$ ,& ,"+** /#-*1$#&1$ ;"#0" ;* ;#33 /#$07$$ $*)(+(,*39 %*3&;5 [ I#1#,*4E&37-* !+(1$)&+, S0"*-*$ KPZ pt o p Topography ps Fig. 8.18 : 2+()"#0(3 #337$,+(,#&1 ,"* "&(,#12 X(2+(12#(1 0&&+/#1(,*5 !"* 8*+,#0(3 0&&+/#1(,* #$ )+*$$7+*5 @; (1/ @?3@ #$ ,"* )+*$$7+* (, ,"* $7+.(0* (1/ -&/*3 ,&)6 +*$)*0,#8*395 !"* /($"*/ 3#1*$ #$ ,"* +*.*+*10* F73*+#(1 2+#/ (1/ $&3#/ 3#1*$ (+* X(2+(12#(1 $7+.(0*$ +*$73,#12 .+&- 3*,,#12 ,"* F73*+#(1 3*8*3$ *8&38* #1 ,#-*6 (1/ +*A7#+* ( )*+#&/#0 +*-())#125 8.6.1 Floating Lagrangian vertical coordinate !"* "&(,#12 X(2+(12#(1 0&&+/#1(,* ;($ #1,+&/70*/ #1 ( ,"*&+*,#0(3 0&1,*?, %9 S,(++ =MOPR> (1/ !+$, ())3#*/ #1 /#$0+*,#@*/ -&/*3$ &8*+ "(3. ( 0*1,7+9 3(,*+ =*525 X#16 KLLPo X(7+#,@*1 *, (36 KLL[o N(#+ *, (36 KLLO>5 G1$,*(/ &. 7$#12 8*+,#0(3 0&&+/#1(,*$ %($*/ &1 "*#2", &+ )+*$$7+*6 ( 8*+,#0(3 0&&+/#1(,* c ,"(, #$ 0&1$,(1, (3&12 ,"+**4 /#-*1$#&1(3 )(+0*3 ,+(]*0,&+#*$ #$ 7$*/ %c = L, %? =[5[a> =$** I#25 [5M[>5 !"* %*1*!, &. 7$#12 $70" ( 8*+,#0(3 0&&+/#1(,* #$ ,"(, ,"* 8*+,#0(3 (/4 8*0,#&1 ,*+-$ #1 ,"* *A7(,#&1$ &. -&,#&1 (+* *3#-#1(,*/ (1/ &139 ,;&4/#-*1$#&1(3 ,+(1$)&+,V(/8*0,#&1 &)*+(,&+$ (+* 1*0*$$(+95 !"* /&;1$#/*6 ($ ;#," (19 &,"*+ X(4 2+(12#(1 ())+&(0"6 #$ ,"(, ,"* 8*+,#0(3 0&&+/#1(,* /*.&+- ($ ,"* "&; *8&38*$5 G1 &+/*+ ,& (8&#/ &8*+39 /*.&+-*/ 8*+,#0(3 0&&+/#1(,*$ ( +*-())#12 &. ,"* )+&21&$4 ,#0 8(+#(%3*$ #1 ,"* 8*+,#0(3 ,& $&-* +*.*+*10* 8*+,#0(3 0&&+/#1(,* #$ 1*0*$$(+95 !"#$ -(9 %* ( $&7+0* &. 8*+,#0(3 /#..7$#&1 #1 ,"* -&/*35 N&,* ,"(, #$*1,+&)#0 8*+,#0(3 0&4 &+/#1(,*$ (+* ( $7%$*, &. "&(,#12 X(2+(12#(1 8*+,#0(3 0&&+/#1(,*$ ($ ,"*9 (+* (3$& -(,*+#(3 $7+.(0*$ .&+ (/#(%(,#0 "&;5 KPP X(7+#,@*1 *, (35 8.6.2 Operator splitting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a[6 X#1 (1/ Y&&/ MOOa>5 !"* ,+(/#,#&1(3 F73*4 +#(1 ,9)* &)*+(,&+ $)3#,,#12 ())+&(0" -(9 %* +*.*++*/ ,& ($ ( !?*/ /#+*0,#&1 %($*/ $)3#,,#12 -*,"&/ ($ &))&$*/ ,& ,"* "&;4%($*/ $)3#,,#12 ())+&(0" /#$07$$*/ %*3&;5 U&+* +*0*1,39 ,"* !1#,*48&37-* 0($0(/* ())+&(0" ;($ $722*$,*/ %9 N(#+ *, (3 =KLLK> (1/ f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igorous three-dimensional approach I7339 ,"+**4/#-*1$#&1(3 $0"*-*$ %($*/ &1 ,"* $)(0*4,#-* !1#,*48&37-* ())+&(0" /#$07$$*/ #1 ,"#$ 0"(),*+ (+* +(,"*+ 0&-)3*?5 G1$,*(/ &. "(8#12 ,& /*(3 ;#," &8*+3() (+*($ =($ /#$07$$*/ #1 ,"#$ 0"(),*+> &1* "($ ,& 0&-)7,* &8*+3() 8&37-*$ ;"#0" #$ $#21#!0(1,39 0&-)3#0(,#12 ,"* )+&%3*-5 F?(-)3*$ &. .7339 ,"+**4/#-*1$#&1(3 +*-()4 )#12 (32&+#,"-$ (+* 2#8*1 #16 *5256 W(+#-*33( *, (3 =KLLh> .&+ D(+,*$#(1 2*&-*,+95 !"* (7,"&+$ (+* 1&, (;(+* &. (19 .7339 ,"+**4/#-*1$#&1(3 !1#,*48&37-* +*-())#12 $0"*-*$ &1 ,"* $)"*+*5 8.7 Time-integration and tracer transport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ifferent schemes air and tracers G. /#..*+*1, $0"*-*$ (+* 7$*/ .&+ (#+ (1/ ,+(0*+$ 0&1$#$,*109 0(11&, %* (0"#*8*/ &,"*+ ,"(1 ;#," !?*+$ ,"(, *1.&+0* 0&1$#$,*109 #1 ( `(/ "&0Q (1/ $&-*;"(, (+%#,+(+9 -(11*+5 S** $*0,#&1 [5Z5P (1/ +*.*+*10*$ ,"*+*#15 8.7.2 Different time-steps for air and tracers (sub-cycling, super-cycling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 $& #, #$ #-)&+,(1, ,& +*,(#1 ,"* 0&1$#$,*109 /#$07$$*/ #1 $*0,#&1 [5K5K6 ,"(, #$6 .&+ ( 0&1$,(1, -#?#12 +(,#& =4 = M> ,"* ,+(0*+ ,+(1$)&+, *A7(,#&1 $"&73/ 9#*3/ ,"* $(-* $&37,#&1 ($ ,"* 0&14 ,#17#,9 *A7(,#&1 .&+ (#+ =(5'5(5 `.+**4$,+*(- )+*$*+8#12Q>5 : $0"*-* )&$$*$$#12 ,"* `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l ) 6 ;"*+* ;* "(8* ($$7-*/ ,"(, ,"* 0*33 ;#/," #$ &1* =2+*9 (+*( &1 I#25 [5MO(>5 C* ,"*1 #1,*2+(,* ,"* .733 /91(-#0(3 $9$,*- &. *A7(,#&1$ =0&1,#17#,9 *A7(,#&1 .&+ (#+6 -&-*1,7- *A7(,#&1$ (1/ ,"*+-&/91(-#0 *A7(,#&1> .&+;(+/ #1 ,#-* ,& ? = () + M/C;@*$?)6 ?5 !"* "7? &. -($$ #1,& ,"* 0*33 /7+#12 ,"#$ .&+;(+/ #1,*2+(,#&1 0&++*$)&1/$ ,& ,"* +*/ (+*( `$;*),Q ,"+&72" ,"* 3*., 0*33 ;(336 &1 I#25 [5MO( =3*., 0&37-1> (1/ "*10* ,"* (#+ -($$ #1 ,"* 0*33 #10+*($*$ %9 ,"* +*/ (+*( #1 0*33 C =I#25 [5MO( +#2", 0&37-1>5 !"#$ )+&0*/7+* #$ +*)*(,*/ ,"+**6 &+ =C;@*$? − M>6 -&+* ,#-*4$,*)$ /7+#12 ;"#0" ,"* %37*6 9*33&; (1/ 2+**1 (+*($ (+* `$;*),Q ,"+&72" ,"* 3*., 0*33 ;(33 (1/ (//#12 ,& ,"* ,&,(3 (#+ -($$ #1 ,"* 0*33 =I#25 [5MO%606/6 +*$)*0,#8*39>5 KPa X(7+#,@*1 *, (35 l (a) l 1zMVP 1 l l 716, l (b) l 1zMVP l l1zKVP 1zMVP 7 6, l (c) 1zKVP l l l1zZVP 71zKVP6, l (d) l1zZVP l l1zM 71zZVP6, (e) l l1zM 1zZVP =7pppppppppppppppppppppppppppppppppppppppp> z 71zKVP z 71zMVPz 71 6, Fig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h &. 4) &8*+ ,"* %+&;1 (+*( #1 I#25 [5MO* /*1&,*/ +4 ) ,5 !"*1 ,"* !1(3 .&+*0($, .&+ ,"* ,+(0*+ #$ 2#8*1 %9 ,"* )+&/70, %*,;**1 ,"* %(0'2+&71/ "&; &. -($$ (1/ (1 *$,#-(,* &. ,"* -#?#12 +(,#& &8*+ ,"* 3&12 ,#-*4$,*) 9 8 (l 4))+M = (l 4)) + +4) , C;@*$? - 6 l )+$/C;@*$? , =[5[h> $=M ;"*+* b l )+$/C;@*$? #$ ,"* "7? &. (#+ -($$ #1,& ,"* 0*33 /7+#12 &1* $7%40903*/ ,#-*4 $,*) 6 ?/C;@*$?5 G. 4 = M ,"*1 =[5[h> +*/70*$ ,& ,"* *A7(,#&1 .&+ (#+ -($$ (1/ 0&1$*4 A7*1,39 ,"* $0"*-* #$ .+**4$,+*(- )+*$*+8#125 N&,* ,"(, 7)/(,#12 ,"* ,+(0*+$ &1 ,"* $"&+, ,#-*4$,*) ;#33 1&, 9#*3/ ,"* $(-* +*$73,5 8.7.3 Semi-implicit time-stepping for air and explicit for tracers G. $*-#4#-)3#0#, ,#-*4$,*))#12 #$ 7$*/ =$** 0"(),*+ a> ,"*1 ,"* )+&21&$,#0 *A7(,#&1 .&+ (#+ /*1$#,9 0(1 %* ;+#,,*1 ($ )+M l )+M = l-B@ + 6 ? :- < 4 )+M r )+M 5 ! , l −! K =[5[[> )+M =*525 X(7+#,@*1 *, (36 KLLa> ;"*+* l -B@ #$ ,"* *?)3#0#, )+*/#0,#&16 l :- < #$ ( 0&1$,(1, +*.*+*10* /*1$#,96 ! #$ ,"* /#8*+2*10* (1/ !r #$ ,"* /#8*+2*10* *?,+()&3(,*/ ,& ,#-*4 3*8*3 ) + M5 !"* ,*+-$ &1 ,"* +#2",4"(1/ $#/* &. =[5[[> #18&38#12 ! (+* +*.*++*/ ,& ($ ,"* $*-#4#-)3#0#, 0&++*0,#&1 ,*+-$ (1/ +*)+*$*1, ,"* #-)3#0#, 0&7)3#12 ,& ,"* -&4 -*1,7- *A7(,#&1$5 G. ,"* ,+(0*+ ,+(1$)&+, *A7(,#&1 #$ $&38*/ *?)3#0#,396 ($ #$ 7$7(339 /&1*6 ,"*1 ,"* $0"*-* #$ 1&, `.+** $,+*(- )+*$*+8#12Q %*0(7$* &. ,"* $*-#4#-)3#0#, 0&++*0,#&1 ,*+-$ =(3,"&72" ,"*9 (+* 7$7(339 $-(33>5 S& .&+ 0&1$#$,*1096 &1* $"&73/ (3$& $&38* ,"* ,+(0*+ ,+(1$)&+, *A7(,#&1 $*-#4 #-)3#0#,39 4 5 6? (4l ):- < !)+M − !r )+M , =[5[O> (l 4))+M = (l 4))+M -B@ + K =*525 X(7+#,@*1 *, (36 KLL[>6 "&;*8*+6 ,"(, $**-$ )+&%3*-(,#05 I&+ *?(-)3*6 #. 4 #$ @*+& #1 $&-* (+*( (1/ ,"* $*-#4#-)3#0#, 0&++*0,#&1 ,*+-$ (+* 1&14@*+& #1 ,"(, (+*(6 ,"*1 ,+(0*+ -($$ ;#33 %* )+&/70*/ #1 (1 (+*( ;"*+* 4 $"&73/ %* @*+&5 !"7%7+1 *, (3 =KLML> )+*$*1, ( -*,"&/ ;"*+* ,"*9 /#$0+*,#@* (1 (3,*+1(,#8* .&+&. ,"* $*-#4#-)3#0#, 0&1,#17#,9 *A7(,#&15 !"+&72" ( $*+#*$ &. #,*+(,#&1$ ,"* $*-#4 #-)3#0#, 0&++*0,#&1 ,*+-$ 0(10*3 (1/ 0&1$#$,*109 %*,;**1 (#+ -($$ (1/ ,+(0*+ ,+(1$4 )&+, #$ &%,(#1*/5 I&+ -&+* /*,(#3$ $** !"7%7+1 *, (3 =KLML>5 KP[ X(7+#,@*1 *, (35 8.8 Final Remarks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ao C#3/ (1/ <+(,"*+6 KLLa>5 Acknowledgments !"(1'$ ,& B+5 S5 W(3-(+#1# =G1$,#,7,* .&+ F18#+&1-*1, (1/ S7$,(#1(%#3#,96 F74 +&)*(1 D&--#$$#&16 J
, Y*$*(+0" D*1,*+> (1/ B+5 :5 \('3(1&8 =B(1#$" U*,*&+&3&2#0(3 G1$,#,7,*> .&+ /*,(#3$ &1 ,"* F!Fl *?)*+#-*1,5 U(19 .+7#,.73 /#$07$$#&1$ ;#," B+5 B5X5 C#33#(-$&1 =ND:Y>6 B+5 <5 Y($0" =6 B+5 :5W*,,*3-(1 =ND:Y>6 B+5 C5 S'(-(+&0' =ND:Y>6 B+5 U5 !(93&+ =S(14 /#( N(,#&1(3 X(%&+(,&+#*$> (1/ B+5 Y5 U#,,(3 =ND:Y> (+* (0'1&;3*/2*/ ($ ;*33 ($ ,"* #1,*+1(3 +*8#*; )*+.&+-*/ %9 B+5 B5X5 C#33#(-$&1 (1/ B+5 D5 F+(,"6 (1/ ,"* (1&19-&7$ (1/ F/#,&+ =B+5 D5 J(%3&1&;$'# (1/ B+5 U5:5 !(93&+> +*8#*;$5 !"* (7,"&+$ 2+(,*.7339 (0'1&;3*/2* <+&.5 J5!"7%7+1Q$ $722*$,#&1$ &1 17-*+#0(3 -#?#12 ,*$,$5 !"* !+$, (1/ ,"#+/ (7,"&+$ ;*+* )(+,#(339 $7))&+,*/ %9 ,"* BHF \FY <+&2+(- 71/*+ (;(+/ BF4SDLLLMaR[5 References 8(1 :3%(/( WB6 8(1 X**+ \6 Y&%*+,$ CC =MO[K> : 0&-)(+(,#8* $,7/9 &. 0&-)7,(4 ,#&1(3 -*,"&/$ #1 0&$-#0 2($ /91(-#0$5 :$,+&1&-9 (1/ :$,+&)"9$#0$ ML[^hai[P :+,*%+(1, Y6 !&++#3"&1 U =KLL[> G10+*($#12 ,"* (007+(09 #1 3&0(339 /#8*+2*10*4 )+*$*+8#12 !1#,* 8&37-* $0"*-*$ .&+ UTB5 J D&-)7, <"9$ KKh=a>^ZPLRiZPKh \(+," !6 I+*/*+#0'$&1 < =MOOL> T#2"*+4&+/*+ $&37,#&1 &. ,"* F73*+ *A7(,#&1$ &1 71$,+70,7+*/ 2+#/$ 7$#12 A7(/+(,#0 +*0&1$,+70,#&15 G1^ :G:: <()*+ OL4LLMZ \(+," !6 J*$)*+$*1 B =MO[O> !"* /*$#21 (1/ ())3#0(,#&1 &. 7);#1/ $0"*-*$ &1 71$,+70,7+*/ -*$"*$5 <+&0 :G:: Kh," :*+&$)(0* S0#*10*$ U**,#126 Y*1& \(,*$ JY6 U0B&1(3/ : =MO[K> U73,#)3947)$,+*(-6 $*-#4X(2+(12#(1 (/8*0,#8* $0"*-*$^ :1(39$#$ (1/ ())3#0(,#&1 ,& ( -73,#43*8*3 )+#-#,#8* *A7(,#&1 -&/*35 U&1 C*( Y*8 MML=MK>^M[ZMiM[PK \3&$$*9 S*3*0,#8* -&1&,&1#0#,9 )+*$*+8(,#&1 #1 $0(3(+ (/4 8*0,#&15 J D&-)7, <"9$ KKh=ML>^RMaLiRM[Z [ I#1#,*4E&37-* !+(1$)&+, S0"*-*$ KPO \&0'-(1 SI =MO[O> W*1*+(3#@#12 ,"* .&+-73( .&+ (+*($ &. )&392&1$ ,& -&-*1,$5 !"* :-*+#0(1 U(,"*-(,#0(3 U&1,"39 Oa^MZMiMZK \+($$*7+ W<6 T(7237$,(#1* B:6 C(3,*+$ S6 Y($0" UHf:Y!6 ( 23&%(3 0"*-#0(3 ,+(1$)&+, -&/*3 .&+ &@&1* (1/ +*3(,*/ 0"*-4 #0(3 ,+(0*+$^ M5 -&/*3 /*$0+#),#&15 J W*&)"9$ Y*$ MLZ^K[6KaRiK[6K[O D&3*33( <6 S*'&+( UB =KLL[> : 3#-#,*+ .&+ ))- ,"(, )+*$*+8*$ (007+(09 (, $-&&," *?,+*-(5 J D&-)7, <"9$ KKh^hLaOihLha D&3*33( <6 C&&/;(+/ !"* )#*0*;#$* )(+(%&3#0 -*,"&/ =< .&+ 2($4 /91(-#0(3 $#-73(,#&1$5 J D&-)7, <"9$ RP^MhPiKLM D&33#1$ CB6 Y($0" B*$0+#),#&1 &. ,"* ND:Y D&--71#,9 :,-&$)"*+* U&/*3 =D:U Z5L>5 ND:Y !*0"5 N&,*6 ND:YV!N4PaPzS!Y 8(1 B&) T6 ://#$ Y6 I+($*+ W6 W#+(+/# I6 W+(@#(1# W6 G1&7* c6 d*339 N6 d372 C6 d73-(3( :6 N&/&) d6 <+*,*3 J =MOO[> F,*?^ : F7+&)*(1 ,+(0*+ *?)*+#-*1,o &%4 $*+8(,#&1$6 /#$)*+$#&1 -&/*33#12 (1/ *-*+2*109 +*$)&1$*5 :,-&$)"*+#0 F18#+&14 -*1, ZK^PL[O i PLOP B&$;*33 D:G =MO[P> : '#1*-(,#0 (1(39$#$ &. .+&1,&2*1*$#$ ($$&0#(,*/ ;#," ( 1&1/#4 8*+2*1, 8&+,*?5 J :,-&$ S0# )) MKPKiMKP[ B7'&;#0@ Jd =MO[P> D&1$*+8(,#8* +*@&1#12 =+*-())#12> .&+ 2*1*+(3 A7(/+#3(,*+(3 -*$"*$5 J D&-)7, <"9$ RP^PMMiPKP B7'&;#0@ Jd6 \(7-2(+/1*+ JY =KLLL> G10+*-*1,(3 +*-())#12 ($ ( ,+(1$4 )&+,V(/8*0,#&1 (32&+#,"-5 J D&-)7, <"9$ MaL^ZM[iZZR B7'&;#0@ Jd6 d&/#$ JC =MO[h> :007+(,* 0&1$*+8(,#8* +*-())#12 =+*@&1#12> .&+ (+4 %#,+(+9 X(2+(12#(14F73*+#(1 0&-)7,(,#&1$5 SG:U J&7+1(3 &1 S0#*1,#!0 (1/ S,(4 ,#$,#0(3 D&-)7,#12 [=Z>^ZLRiZKM B7++(1 BB =MOOO> N7-*+#0(3 U*,"&/$ .&+ C(8* FA7(,#&1$ #1 W*&)"9$#0(3 I37#/ B91(-#0$5 S)+#12*+4E*+3(2 F37$@'#*;#0@ J6 T*-3*+ YS6 U("3-(1 JB6 \+7";#3*+ X6 !('(0$ XX =KLLL> S*1$#4 ,#8#,9 &. 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