Higher-Order Compact Schemes for Numerical Simulation of Incompressible Flows

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1 NASA/CR ICASE Report No Higher-Order Compact Schemes for Numerical Simulation of Incompressible Flows Robert V. Wilson and Ayodeji O. Demuren Old Dominion University Mark Carpenter NASA Langley Research Center Institute for Computer Applications in Science and Engineering NASA Langley Research Center Hampton, VA Operated by Universities Space Research Association National Aeronautics and Space Administration Langley Research Center Hampton, Virginia Prepared for Langley Research Center under Contract NAS February 1998

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11 $t Zt{s$Bt B 8 PB Y {B8}s{ PB 8Zjs$Bt $R IZ{ I Ak Ps{B R BP 1 PB Y PBZ YTB I R{Y 8 sti ; PB Y R$-YTB I R{Y 8 $t {B8}s $RBt B -}j${$ { t sj I$PP t{ R{Y 8 R BP Y Rs8 B I qsaj E ]B8}s $RBt BP -}j${$ { t sj I$PP t{ sti $8}j${$ {B8}s{ s}} B-$8s$BtR BP Y R {BtI I $%s$% 7{Y 8 q Zt{s$Bt B 7 t{$j 7$9 PBZ YTB I { t sj PBZ YTB I {B8}s{ R$-YTB I { t sj R$-YTB I {B8}s{ ( ) ( 8 6! )(? ) 4 φ 6 ( ) ( 3.6 6! )(? ) 4 φ 6 ( ) ( 72 8! )(? ) 6 φ 8 ( ) ( ! )(? ) 6 φ 8 x n x 3B tbtt} $BI${ ABZtIs $ R Bt TR$I I P$t$ I$PP t{ R s JZ$ I B {jbr Y RkR 8 BP JZs$BtR V $) dsti $) WsY$ ITB I {B8}s{ ABZtIs k R{Y 8 $R ZR IU w1dd φ 1 + α KX φ (? ) 2 y KX7 φ by ) dd sti ) dn ) T1 ) dx sti ) Td s Y {B PP${$ tr BP Y Y$ IT α KX y KX1 y KX2 y KX3 y KX4 B I ABZtIs k R{Y 8 3B Y R$-YTB I R{Y 8 s t s ABZtIs k R{Y 8 $R JZ$ I s $ ) 1 sti $ ) WTd qy PBZ YTB I $t $B R{Y 8 $R ZR I s Y R }B$tR R$t{ Btjk s Y T}B$t R t{$j $R t I I V R$8$js stsjkr$r BP Y dt* I$PPZR$Bt JZs$Bt $R ZR I B $t% R$s Y RBjZ$Bt JZsj$$ R BP Y } BT }BR I {B8}s{ s}} B-$8s$Bt B Y R {BtI I $%s$% qy $I$sBtsj {B8}s{ R{Y 8 } R t I Ak ej w1(d k$ jir s tz8 ${sj bs% tz8a U w11d * ( > ) (? ) 2 2y[ 1 cos( >? )] + -- [ 1 cos( 2>? )] α cos( >? ) K qy I$RR$}s$Bt B R PB Y -}j${$ R {BtITB I { t sj I$PP t{ sti PBZ YT sti R$-YTB I {B8T }s{ R{Y 8 R s RYBbt $t 3$ n G R Ys Y tz8 ${sj bs% tz8a PB Y {B8}s{ R{Y 8 8B {jbr jk s}} B-$8s R Y -s{ bs% tz8a B% s b$i st BP bs% tz8a R +Zst$s$% 8 srz R BP RBjZ$Bt }Bb PB Y %s $BZR R{Y 8 R s $% t $t qsaj er$8s R BP Y G RYBb Ys BZYjk b${ sr 8stk }B$tR s JZ$ I PB Y -}j${$ R {BtITB I { t sj I$PP t{ B } BIZ{ Y Rs8 s{{zt s{k sr Y {B8}s{ R{Y 8 d(

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13 n 7BjZ$Bt BP Y _B$RRBt JZs$Bt PB } RRZ V R$t$P${st s8bzt BP Y Bsj {B8}ZsT $Btsj $8 JZ$ I PB Y RBjZ$Bt BP Y $t{b8} RR$Aj Ws%$ T7B, R JZs$BtR wsr 8Z{Y sr YsjPD $R I %B I B Y tpb { 8 t BP Y {Bt$tZ$k JZs$Bt\RBjZ$Bt BP } RRZ qy$r R 8R P B8 Y Ps{ Ys %BjZ$Bt JZs$BtR -$R PB Y % jb{$k {B8}Bt tr w$ Y 8B8 tz8 JZs$BtRD by$j tbt -$R PB Y } RRZ ytr si st jj$}${ JZs$Bt 8ZR A RBj% I PB Y } RRZ by${y $t%bj% R Y RBjZ$Bt BP s RkR 8 BP JZs$BtR sti $R -} tr$% 7BjZ$BtR 8 YBIR PB jj$}${ JZs$BtR t sjjk Psjj $tb bb {s BT $ R T I$ { B $ s$% *$ { 8 YBIR ZRZsjjk $t%bj% RB8 PB 8 BP pszrr$st j$8$ts$bt by Y {B PT P${$ t 8s $- $R P$ R Ps{B I $tb st Z}} sti jbb 8s $- sti Y t Y RBjZ$Bt $R {B8}Z I ZR$t As{, RZAR$Z$Bt qy B} s$bt {BZt sti 8 8B k JZ$ 8 tr PB Y$R } B{ IZ {st A } BY$A$$% jk js PB Y RBjZ$Bt BP RkR 8R $t%bj%$t s js tz8a BP Zt,tBbtR w 10 6 $t k}${sj nt* } BAj 8RD qy sj ts$% B s I$ { RBjZ$Bt $R st $ s$% } B{ IZ by st $t$$sj s}} B-$8s$Bt B Y RBjZ$Bt $R ZR I B k$ ji st $8} B% I RBjZ$Bt qy$r } B{ RR $R } s I Zt$j Y RBjZ$Bt $R {Bt% I B b$y$t s I R$ I Bj st{ qy B} s$bt {BZt sti 8 8B k JZ$ 8 tr BP 8BR $ s$% 8 YBIR s j RR Yst Ys BP pszrr$st j$8$ts$bt qy PB Y $ s$% RBjZ$Bt } B{ IZ $R ZR I $t Y$R RZIk B RBj% Y _B$RRBt JZs$Bt PB } RRZ qy I s$jr BP Y$R } B{ IZ s BZj$t I $t Y$R {Ys} nd *$R{ $9 I Ns}js{$st!} sb qy I$R{ _B$RRBt JZs$Bt PB } RRZ by${y bsr I $% I $t 7 { 1; $R $% t AkU w1xd - `7 δ?7 δ?7 h - - δ?7 I K qy N<7 BP ej w1xd } R tr s I$R{ $9 I Ns}js{$st B} sb {B8}BR I BP bb s}}j${s$btr BP Y P$ R I $%s$% B} sb δ?7 y $R b jj,tbbt Ys ZR$t bb P$ R I $%s$% B} sb R B } R t Y Ns}js{$st B} sb Bt tbttrs I $IR {st j si B st BIIT % t I {BZ}j$t BP Y RBjZ$Bt yti I b$y RstIs I R {BtITB I { t sj I$PP t{$t PB Y P$ R I $%s$% B} sb Y RBjZ$Bt s % t $I }B$tR {B8}j jk I {BZ}j R P B8 Y BII $I }B$tR j si$t B tbtt}ykr${sj RZjR!t 8 Ik $R B $t BIZ{ 8R BP Y Rs8 B I sr Y Zt{s$Bt B by${y $t PP { }js{ R Y bb P$ R I $%s$% B} sb R b$y s R$tj R {BtI I $%s$% B} sb qy$r {BZ}j R Y RBjZ$Bt s BII sti % t $I }B$tR by$j 8s$ts$t$t Y Rs8 PB 8sj B I BP s{{z s{k qy Ns}js{$st B} sb $R I$R{ $9 I ZR$t s R$tj R {BtI I $%s$% B} sb B } % t }BRR$Aj I {BZ}j$t sti ej w1xd A {B8 RU w1ed - `7 δ??7 h - - δ?7 I K by δ??7 h - } R tr Y I$R{ Ns}js{$st BP } RRZ sti $R I$R{ $9 I ZR$t Y {B8}s{ R {BtI I $%s$% B} sb $% t Ak ej w1(d qy PP { BP }js{$t Y bb P$ R I $%s$% B} sb R b$y Y R$tj R {BtI I $%s$% Ns}js{$st B} sb $R RYBbt Ak b $$t ej w1ed sr d1

14 w1d δ??7 h - - by st $ s$% RBjZ$Bt $R srrz8 I sti h $R Y {Z t $ s BP } RRZ sti p $R Y R$IZsj $8Asjst{ IZ B $t{b8}j {Bt% t{ qs,$t Y I$% t{ BP Y 8B8 tz8 JZs$Bt ej wnd sp Y $ s$% RBjZ$Bt BP Y } RRZ sti $8 si%st{ 8 t $% RU - `7 - δ?7 I K p 0 w1md 2 - δ?7h - `7 - δ?7 I K δ? ` K ZA s{$t ejr w1d P B8 w1md sti s st$t $% R Y -} RR$Bt PB Y I$% t{ U w1d δ?7`7 K δ? δ??7 h + p 7 ejzs$bt w1d RYBbR Ys Y $8Asjst{ BP Y {Bt$tZ$k JZs$Bt $R IZ B bb {Bt $AZ$BtR w$d Y I$PP t{ A b t Y bb I$PP t Ns}js{$st B} sb R by${y $R 5(? E ) by 8 $R s Yst Y B I BP s{{z s{k BP Y Ns}js{$st B} sb R w$ $R srk B RYBb Ys $ $R PBZ YTB I PB -}j${$ R {BtIT B I B} sb RD sti w$$d Y R$IZsj $8Asjst{ IZ B $t{b8}j $ s$% {Bt% t{ ejzs$bt w1d sjrb RYBbR Ys Y R$IZsj $8Asjst{ p IB R tb t I B A I $% t B 8s{Y$t 9 B Btjk A jbb Y j % j BP Y I$PP t{ $t Y bb Ns}js{$st B} sb R <Bb % Y {Bt% t{ Bj st{ PB 8$ts$t Y $ s$% RBjZ$Bt BP Y _B$RRBt JZs$Bt 8ZR A RZPP${$ tjk R8sjj PB Y RBjZ$Bt B A $tr tr$$% B Y RB}}$t { $ $Bt VR Y $I $R P$t I Y Bj st{ 8ZR A IZ{ I $t s{{b Ist{ b$y Y B I BP s{{z s{k BP Y R{Y 8 3B -s8}j by t ZR$t Y R$-YTB I R{Y 8 Bt js$% jk P$t $IR wr qsaj dmd $ $R t { RRs k B {Bt% Y RBjZ$Bt B 8s{Y$t 9 B wd( Td; D $t B I PB Y RBjZ$Bt B R B IZ{ s s R$-YTB I s ejzs$bt w1ed s sjj $t $B $I }B$tR RZjR $t s RkR 8 BP JZs$BtR Ys $R RBj% I s s{y RZATRs BP Y $8 si%st{ 8 t R{Y 8 3B R$8}j${$k Y RkR 8 BP JZs$BtR $R I R{ $A I PB Y 1T* _B$RRBt JZs$Bt b$y } $BI${ ABZtIs $ R Bt s Zt$PB 8 $I PB Y PBZ YTB I $I$sBtsj R{Y 8 I P$t I $t 7 { 1n qy RBjZ$Bt } B{ IZ $R sr$jk - ti I B Y nt* _B$RRBt JZs$Bt sr } R t I $t V}} ti$- V e- tr$bt B t sj {Z %$j$t s $IR $R } R t I $t V}} ti$- qy R BP ej w1ed {st A b $ t $t Y PB 8 BP s RkR 8 BP JZs$BtR sru wn(d 1 1 \h [\ 4???? + \ 4rr rr ]h B dn

15 U U (nj x nj block matrix), where \?? U U U U U 1 α α α 1 α α 1 α α α 1 (ni x ni scalar matrix) { α{ α{ \ rr α{ { α{ { α{ { α{, where is the ni x ni identity matrix α{ α{ { (nj x nj block matrix) p p p y 4?? -----, where 2 p _? p p p (nj x nj block matrix) (ni x ni scalar matrix) y 4 rr _ r 2{ { 2{ { 2{ { 2{ { 2{ (nj x nj block matrix) { 2{ { x h h 1 h 2 h zi, where h i h 1, i h 2, i h z7 i (nj x 1 block vector) (ni x 1 scalar vector) x x B B 1 B 2 B zi, where B i B 1, i B 2, i B z7, i (nj x 1 block vector) (ni x 1 scalar vector) by h sti B s Y } RRZ sti RBZ { 8 s Y 7i $I }B$t $` 7, δ i?7 [ I 7 + ( K ` ) ] 7, i R} {$% jk qy Rk8ABjR z7 ) ]? Ld zi ) ] r Ld I tb Y tz8a BP $I }B$tR $t Y? r I$ {$BtR R} {$% jk zsjz R s 7 ) ]? Ld sti i ) ] r Ld jb{s$btr s }js{ I Ak %sjz R s 7 ) d sti i ) d R$t{, x d;

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