( ) κ ( f j +1. ( ) 1 + κ ( f j +2. ( ) j +1/ 2 ) = Uf j +1/ 2. = r j +1/ 2. Ψ( r) = r + r 1 + r. Ψ( r) = minmod(r,1) j 1/ 2 ( F n +1/ 2 F n

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1 ttp://users.wpi.edu/~gretar/me6.tml! Numerical Metods! or Hyperbolic! Equatios VIII! Grétar Tryggvaso! Sprig! Upwid or oliear systems:!!flux vector splittig!!goduovʼs metod! Higer Order Metod!!Higer order upwid!!higer order Goduov metods! Higer Resolutio mootoic scemes!!basic Idea!!Flux Limiters! To prevet oscillatios ear socks we usig ig order scemes, we ave already talked about artiicial viscosity were we attempt to smoot out oscillatios aroud te socks.! Te more moder approac is to prevet te apparece o oscillatios by eiter:! Limit te slopes we te variables are extrapolated to te cell boudaries! Limit te luxes ear socks to prevet uder or over soot! Higer Order Upwid! Near socks te liear recostructio leads to over ad udersoots.! To prevet oscillatios, apply LIMITERS! + + -!! +! +! May possible limiters ca be desiged! Limitig te variables:! Predictor step! L Variables! +/ Fid:! Fial step! +/ = Δt F +/ F / +/ = + ΨL +/ +/ ( ) R +/ +/ = + ΨR +/ +/ + F +/ +/ = F ( L ) +/, R +/ ( +/ +/ ) + = Δt F +/ +/ ( +/ F / ) κ = Limitig te Fluxes! Predictor step! Variables! Fid:! Fial step! +/ = Δt F +/ F / L +/ F +/ +/ = F + = +/ = + +/ ( L ) +/, R +/ ( +/ +/ ) ([ F +/ + Ψ L +/ ( F +/ F +/ )] +/ [ F / + Ψ L / ( F +/ F / )]) / Δt +/ R +/ +/ = + +/ +/ + κ =

2 It is possible to derive a umber o properties tat te limiters sould satisy, bot to preserve secod order accuracy i smoot parts o te low ad to satisy te etropy coditios.! Va Leer Limiter! Ψ( r) = r + r + r Ψ( r) Ψ = r Ψ = r Witi tose limitatios, owever, several limiters are possible:! Ψ( r) Ψ = r Ψ = r 3 r Te limiters must lie i te saded area! Lax-Wedro! Beam-Warmig! r = + Mimod limiter! mi(r,) i r > Ψ( r) = i r or! Ψ( r) = mimod(r,) were! x i x < y ad xy > mimod(x,y) = y i x > y ad xy > i xy < 3 r Example: Liear Advectio Equatio! + U x = F = U L F +/ = U +/ r +/ = r +/ = + Ψ( r) = r + r + r Ψ = i + = -! +! +! Predictor step! +/ = Δt F +/ F / Variables! Fid! + = L +/ R +/ = + κ = + κ + F +/ Δt +/ = F + + κ ( + ) + κ ( + + ) ( L ) +/, R +/ ( +/ +/ ) ([ F +/ + Ψ L +/ ( F +/ F +/ )] +/ [ F / + Ψ L / ( F +/ F / )]) /.5 upwid! Secod order! It ca be sow tat or a airly large class o metods lux limiters ad slope limiters are essetially te same. Limiters ave also bee used to derive artiicial viscosity.!.5 Limited! Tus, altoug limiters are peraps ot as artiicial as addig artiicial viscosity, tey are basically te same remedy, amely we add selective dissipatio aroud socks.!

3 Te moder discussio o ig resolutio scemes is ote based o te cocept o Total Variatio Dimiisig (TVD) Scemes sice it ca be sow tat te Total Variatio sould dimiis or yperbolic systems! TV = dx x Flux Vector Splittig! For a more extesive discussio see! Hirsc, Volume, Capter! Wesselig, Capter 9! For upwid scemes, it is ecessary to determie te upstream directio. For systems wit may caracteristics ruig bot let ad rigt, tere is ot oe upstream directio! Upwid Sceme - Revisited! Geeralized Upwid Sceme (or bot U > ad U < )! Deie! UΔt ( ), U > UΔt ( ), U < + = + = + U + = ( U + U ), U = ( U U ) Te two cases ca be combied ito a sigle expressio:! [ ] + = Δt U + ( ) + U ( + ) Flux Splittig! Flux Splittig! A system o yperbolic equatios! For a system o equatios tere are geerally waves ruig i bot directios. To apply upwidig, te luxes must be decomposed ito let ad rigt ruig waves! + F x = ca be writte i te orm! + [ A] x = ; Te system is yperbolic i! Steger-Warmig (979)! [ A] = F [ T ] [ A] [ T] = [ λ]; [ T ] = T q T q N

4 Flux Splittig! F = [ A] = T Te matrix o eigevalues Hece! [ A] = A + [ ][ λ] [ T] [ λ] [ λ] = [ λ + ] + [ λ ] [ ] + [ A ] = [ T ][ λ + ] T is divided ito two matrices! [ ] T [ ] + [ T] λ [ ] Deie! F = F + + F F + = [ A + ], F = [ A ] Coservatio law becomes! + F + x + F x = Flux Splittig! Example: -D Hyperbolic Equatio! Leadig to! v t w t + c c x = v x w x = = v w ; F = c w v ; v = ; w = x [ A] = F = c Flux Splittig! [ T ] = q T = c T λ + q c ; [ ] = c [ A + ] = [ T ] [ λ + ][ T] ; [ A ] = [ T] [ λ ][ T ] F + = [ A + ]; F = [ A ] Leadig to:! F + = cv c w ; F = cv c w v + cw v cw c [ λ ] = For a oliear system o equatios suc as te Euler equatios, tere is some arbitrariess i ow te system is split! ttp://users.wpi.edu/~gretar/me6.tml! Numerical Metods! or te! Euler Equatios! Grétar Tryggvaso! Sprig! Te Euler Equatios! Te Euler Equatios! Caracteristics! Oe dimesioal sock tube-exact solutio! For te Euler equatios:! ρ ρu ρu + ρu + p = x ( / ) ρe ρu E + p ρ Numerical solutios! Upwid/lux splittig! Lax-Wedro/artiicial viscosity! were! Deie! E = e + u / H = + u /; = e + p/ρ Two dimesioal problems! Ideal Gas:! p = ρrt; e = e( T ); c v = de /dt = ( T); c p = d /dt R = c p c v ; γ = c p /c v ; p = ( γ )ρe

5 Te Euler Equatios! Te Euler Equatios! ρ ρu ρu + ρu + p = x ( / ) ρe ρu E + p ρ p = ( γ )ρe c = γp/ρ Expadig te derivative ad rearragig te equatios! ρ u ρ ρ u + u /ρ u p ρc x = + A or! x = u p Te Caracteristics or te Euler Equatios are oud by idig te eigevalues or A T! det(a T λi) = Fid! det(a T λi) = or:! Te Euler Equatios! or! ( c ) = ( u λ) u λ ( u λ) = λ = u (( u λ) c ) = u λ = ±c λ = u ± c u λ det ρ u λ ρc = /ρ u λ Tereore! λ = u c; λ = u + c; λ 3 = u Te Rakie-Hugoiot coditios! For a yperbolic system! + F x = Te speed o a discotiuity (sock) is oud by movig to a rame were a sock movig wit speed s is statioary!! s x + F x = Itegratig across te sock yields! [ ] = [ F] s For te Euler equatios:! ρ ρu ρu + ρu + p = x ( / ) ρe ρu E + p ρ Te Rakie-Hugoiot coditios are:! s( ρ L ρ R ) = (( ρu) L ( ρu) R ) s( ( ρu) L ( ρu) R ) = (( ρu + p) L ( ρu + p) R ) s( ( ρe ) L ρe ) = (( ρu(e + p /ρ)) L ρu(e + p/ρ) ) R R Te sock tube problem! u L,, ρ L! u R, p R, ρ R! Desity! L! 5! 3!! R! Pressure! Expasio Fa! Cotact! Sock! Expasio Fa! Cotact! Sock!

6 Te Sock-Tube Problem! Exact Solutio!! 3!! give:!, ρ L, u L, p R, ρ R, u R c L = γ ρ L c R = γ p R ρ R α = γ + γ Cosider te case > p R :!!Sock separates R ad!!cotact discotiuity separates ad 3!!Expasio a separates 3 ad R!! 3!! give:!, ρ L, u L, p R, ρ R, u R Te Rakie-Hugoiot coditios give a oliear P = p relatio or te pressure ump across te sock! P = p L ( γ ) ( c R / c L )(P ) p R γ ( γ + ( γ + ) ( P ) ) wic ca be solved by iteratio! γ /( γ ) p R Te speed o te sock is! γ + ( γ +)P s sock = u R + c R γ /! 3!! x sock = x + s sock t Te speed o te cotact is! s cotact = u 3 = u = u L + c γ L γ P p γ R x cotact = x + s cotact t Te let ad side o te a moves wit speed! s L = c L x L = x + s L t Te rigt ad side o te a moves wit speed! s R = u c L x R = x + s R t! 3!!! 3!! α = γ + γ Let uiorm state! give:!, ρ L, u L, c L = γ ρ L I te expasio a ()! γ + (s x) γ ρ = ρ c L α t u = u L + s x γ + t + (s x) p = p c L α t γ γ Beid te cotact (3)! p 3 = p u 3 = u ρ 3 = ρ L P p R /γ Beid te cotact (3)! p 3 = p u 3 = u ρ 3 = ρ L P p R /γ Beid te sock ()! + αp ρ = α + P ρ R p = P p R u = u L + c L γ P p R γ γ Rigt uiorm state! give:! p R, ρ R, u R c R = γ p R ρ R

7 Te speed o soud is give by! c = γ p ρ Te Mac umber is deied as te ratio o te local velocity over te speed o soud! Ma = u c Ma < subsoic! Ma > supersoic! Test case:! Socktube problem o G.A. Sod, JCP 7:, 978! = 5 ; ρ L =.; u L = p R = ; ρ R =.5; u R = t ial =.5 x =.5 Subsoic case! x 3 5 velocity t=.5! Velocity! Pressure! Desity! Mac umber 3 5 Mac Number! 3 5 desity x pressure Test case:! Pressure! Socktube problem o G.A. Sod, JCP 7:, 978! Desity! velocity Mac umber 3 5 = 5 ; ρ L =.; u L = p R = ; ρ R =.; u R = t ial =.5 x =.5 Supersoic case! Velocity! Mac Number! Solve usig irst order upwidig wit lux splittig! Here we solve te oe-dimesioal Euler equatio usig te va Leer vector lux splittig! Va Leer! F + = ρ (γ )u + c (u + c) c γ [(γ )u + c] (γ ) F = ρ (γ )u c (u c) c γ [ c (γ )u ] (γ )

8 va Leer vector lux splittig! ρu ρu + p = ρ (γ )u + c (u + c) ρu(e + p ρ ) c γ ρ (γ )u c (u c) c γ [(γ )u + c] [ c (γ )u ] (γ ) (γ ) For example, te mass lux:! ρ c (u + c) ρ (u c) c = ρ c = ρ c u + uc + c u + uc c uc = ρu Several oter splittig scemes are possible, suc as:! Steger-Warmig! F + = ρ γ (γ )u + c (γ )u + (u + c) (γ )u 3 + (u + c)3 + 3 γ (u + c)c (γ ) F = ρ (u c) u c γ (u c) + 3 γ c γ Rewrite te lux terms i terms o Mac umber:! ρ (γ )u + c (u + c) c γ = ρc u c + c γ + γ u c [(γ )u + c ] c (γ ) γ + γ u c = ρc c (M +) γ + γ M c γ + γ M + F + x + F - x = ρ ρu + ρc c (M +) x γ + γ M + x ρc c (M ) + γ γ M ρe c γ + γ M c γ γ M + F+ x + F- x = Solve by! + = Δt F + + F - F + Were! Δt - - F + F = ρc c (M +) γ + γ M ; F = ρc c (M ) + γ γ M c γ + γ M c γ γ M or istep=:maxstep or i=:x,c(i)=sqrt( gg*(gg-)*(re(i)-.5*(ru(i)^/r(i)))/r(i) );ed or i=:x,u(i)=ru(i)/r(i);ed; or i=:x,m(i)=u(i)/c(i);ed or i=:x- %upwid r(i)=r(i)-(dt/)*(... (.5*r(i)*c(i)*(m(i)+)^) - (.5*r(i-)*c(i-)*(m(i-)+)^)+... (-.5*r(i+)*c(i+)*(m(i+)-)^) - (-.5*r(i)*c(i)*(m(i)-)^) ); ru(i)=ru(i)-(dt/)*(... (.5*r(i)*c(i) *(m(i)+)^) *(( +.5*(gg-)*m(i)) **c(i) /gg) -... (.5*r(i-)*c(i-)*(m(i-)+)^)*(( +.5*(gg-)*m(i-))**c(i-)/gg) +... (-.5*r(i+)*c(i+)*(m(i+)-)^)*((-+.5*(gg-)*m(i+))**c(i+)/gg) -... (-.5*r(i)*c(i) *(m(i)-)^) *((-+.5*(gg-)*m(i)) **c(i) /gg) ); re(i)=re(i)-(dt/)*(... (.5*r(i)*c(i) *(m(i)+)^) *((+.5*(gg-)*m(i))^ **c(i)^ /(gg^-)) -... (.5*r(i-)*c(i-)*(m(i-)+)^)*((+.5*(gg-)*m(i-))^**c(i-)^/(gg^-)) +... (-.5*r(i+)*c(i+)*(m(i+)-)^)*((-.5*(gg-)*m(i+))^**c(i+)^/(gg^-)) -... (-.5*r(i)*c(i) *(m(i)-)^) *((-.5*(gg-)*m(i))^ **c(i)^ /(gg^-)) ); ed

9 Eect o resolutio! Socktube problem o G.A. = 5 ; ρ L =.; u L = Sod, JCP 7:, 978! p R = ; ρ R =.5; u R = x Pressure 8 6 Fial time:.5! Desity.8.6 Solve usig secod order Lax-Wedro! x=6; maxstep=3! x=56; maxstep=8! 6 8. x=6; maxstep=3! x=8; maxstep=6!. x=56; maxstep=8! x=5; maxstep=56! 6 8 For te luid-dyamic system o equatios (Euler equatios):! ρ ρu ρu + ρu + p = x ( / ) ρe ρu E + p ρ were! E = e + u / ; Add te artiicial viscosity to RHS:! p = ( γ ) ρe u u = α ρ Proect #3! x x x u Solutios o te D Euler equatio usig Lax-Wedro! * +.5 Δt ( F ) = = Δt F * * + F F' = F α ρ u F + Wit a artiicial viscosity term added to te corrector step! u u x x Test case:! Eect o α! Socktube problem o G.A. Sod, JCP 7:, 978! = 5 ; ρ L =.; u L = p R = ; ρ R =.5; u R = Fial time:.5! x=8; α =.5 desity x=8; desity α =.5 α =. α =.5 α = x pressure α =.5 α =. α =.5 α =

10 8 6 x=6; x=8; x=56; time=.5 x pressure Eect o resolutio! α = desity Outlie o L-W program! or istep=:! or i=:x,p(i)=..; ed! or i=:x-!!!%predictio step! r(i)=..! ru(i)=..! re(i)=..! ed! or i=:x,p(i)=..; ed! or i=:x-!!!%correctio step! r(i)=..! ru(i)=..! re(i)=..! ed! or i=:x,u(i)=ru(i)/r(i);ed! or i=:x-!!!%artiicial viscosity! ru(i)=..! re(i)=..! ed! time=time+dt,istep! ed! Summary! ttp://users.wpi.edu/~gretar/me6.tml! Numerical solutios o te oedimesioal Euler equatios! Upwid/lux splittig! Lax-Wedro/artiicial viscosity! Multi-dimesioal low! Grétar Tryggvaso! Sprig! Multidimesioal yperbolic problems are geerally doe usig SPLITTING, were eac coordiate directio is treated separately! Split versus usplit advectio! + u x + v y = Usplit! + i, = i, + uδt ( i, i, ) + vδt i, ( i, )

11 Split versus usplit advectio! (-v)δt! vδt! uδt! (-u) Δt! + i, = i, + uvδt = i, + ( vδt) uδt + uδt + uvδt + u x + v y = Fully usplit! ( i, i, ) ( i, i, ) + ( uδt) vδt ( i, i, ) ( i, i, ) + vδt ( i, i, ) i, i, ( i, + i, ) + u x + v y = Time splittig! First! te! * i, = i, + uδt + i, = * i, + vδt ( i, i, ) * * ( i, i, ) 3 5 % two-dimesioal timesplit advectio usig irst order upwid % =3;m=3;step=35;legt=.;=legt/(-); dt=.3*;=zeros(,m);o=zeros(,m);=zeros(,m);time=.; u=.; v=.; cx=.35;cy=.35; 5 5 or i=:-, or =:m- r=sqrt((cx-*(i-))^+(cy-*(-))^); i(r <.) (i,)=.;ed ed,ed Time-split! Usplit! or l=:step,l,time old o;mes(); axis([ m.5]);pause; o=; or i=:-, or =:m- (i,)=o(i,)-(dt*u/)*(o(i,)-o(i-,)); ed,ed or i=:-, or =:m- (i,)=(i,)-(dt*v/)*((i,)-(i,-)); ed,ed time=time+dt; ed; =igure; cotour(,);axis square; Te Euler Equatios i twodimesios! For te Euler equatios:! ρ ρu ρv ρu + ρu + p + ρuv = ρv x ρuv y ρv + p ρe ρu(e + p /ρ) ρv(e + p/ρ) were! E = e + (u + v ) /; p = (γ )ρe I vector orm! + F x + G y =

12 Two-dimesioal problems are early exclusively solved by splittig were we apply oe-dimesioal metods to eac directio i sequece! Flux vector splittig! F ± = ± ρ c u ± c ( γ )u ± c γ v v [ + ( γ )u ± c ] ( γ ) + F + x + F x + G + y + G y = G ± = ± ρ c v ± c u ( γ )v ± c γ u [ + ( γ )v ± c ] ( γ ) Results rom a review paper:! P. Woodward ad P. Colella:! Computatioal domai! Sock! =/3 Te Numerical Simulatio o two- Dimesioal Fluid Flow wit Strog Socks.! J. Comput. Pys. 5 (98), 5-73.! Give ilow! 3! wall!! Outlow! =/6 =/ Goduovʼs metod! MacCormacʼs metod! =/3 =/6 =/ Eormous progress as bee made i solutio teciques or yperbolic systems wit socks i te last twety years. Advaced metods are ow able to resolve complex socks witi a grid space or two, eve i multidimesioal situatios or a large rage o goverig parameters ad pysical complexity.! Here, we ave oly examied relatively elemetary aspects o metods or yperbolic systems, but tis sort itroductio sould ave taugt you metods to solve suc systems ad itroduced you to literature.! MUSCL! PPM!