Advection! Discontinuous! solutions shocks! Shock Speed! ! f. !t + U!f. ! t! x. dx dt = U; t = 0

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1 p:// Advecio Discoiuous soluios socks Gréar Tryggvaso Sprig Discoiuous Soluios Cosider e liear Advecio Equaio + U = Te aalyic soluio is obaied by caracerisics d d = U; d d = ; <, = > > Discoiuiy o soluio is allowed d d = U Discoiuous Soluios Iviscid Burgers Equaio + = Caracerisics d d = ; = ; d d d =?? d <, = > > Sock Wave Discoiuous Soluios A slig variaio o e iiial codiio ormaio o sock < a +, = * > a a & + a < < a Sock Wave Discoiuous Soluios Sock Speed a = a a =

2 Te speed o e sock Discoiuous Soluios Wrie: = C = + Subsiue io: + F = C + F = were = C C + F d = & * d + C * d + & & F * d = & Discoiuous Soluios C & + d + * F & + d = * C + F F = C = F F akie- Hugoio relaios Eample Discoiuous Soluios + = C = F F = F = = + Discoiuous Soluios Coservaive Scemes are guaraied o give e correc Sock Speed sice ey correspod o a direc applicaio o e coservaio priciples. No-coservaive scemes may or may o do so. C = + Coservaive Meod Coservaio ad sock speed Eample: iviscid Burgers equaio: & + = or + = I capurig e correc soluio beavior or discoiuous Iiial daa, coservaive meods are esseial a. Coservaive Meod Eample: or iviscid Burgers equaio wi discoiuous iiial daa Cosider upwid ad orward Euler sceme: No-coservaive orm + = + = = Coservaive orm = + + = = Never moves +

3 Te Eropy Codiios Iviscid Burgers Equaio + = Te rasormaio ; eaves e equaio ucaged bu resuls i a upysical soluio. Te eropy codiio is used o selec e correc soluio Te Eropy Codiios Te Eropy Codiios Te Eropy Codiios everse Sock? + = Caracerisics d d = ; = ; d d <, = < > Usable, eropy-violaig soluio areacio Wave pysically correc soluio + = Caracerisics d d = ; = ; d d <, = < > Te Eropy Codiios Eropy Codiio Weak soluios o yperbolic equaios may o be uique. How ca we id a pysical soluio ou o may weak soluios? I luid mecaics, e acual pysics always icludes dissipaio, i.e. i e orm o viscous Burgers equaio: + = Tereore, wa we are ruly seekig is e soluio o e viscous Burgers equaio i e limi o For a coservaio equaio Eropy Codiio: A discoiuiy propagaig wi speed C saisies e eropy codiio i Versio I: Versio II: F > C > F F F + = F F F Ad some oers C Sock Wave or

4 Eropy Codiio Give a coservaio equaio F + = ewrie i caracerisic orm + F = were: or: d d = F = F d ds =; Te Eropy Codiio saes a e caracerisics mus eer e discoiuiy. Tus, is speed C saisies mus saisy F > C > F d ds = F Sock Wave Eropy Codiio Similarly, e sock speed is give by C = F F Tus F F F F C Sock Wave or Meas a e ypoeical sock speed or values o bewee e le ad e rig sae mus give sock speeds a are larger o e le ad smaller o e rig. Eample Problem: iear Wave Equaio Advecig a sock wi several scemes + U =, < <, <, = > Eac Soluio: U Apply various umerical meods + = U + = + = U U U + & + = U + U Upwid + + = eap-rog a-wedro MacCormack Compariso: =.; d=.5;=4, ime=.75 Upwid eap-rog a-wedro MacCormack

5 Compariso: =.5; d=.5;=8, ime=.75 Compariso: =4 ad =8, ime=.75.5 Upwid.5 eap-rog.5 Upwid.5 MacCormack a-wedro.5 MacCormack.5 Upwid.5 MacCormack Observaio : Secod-order meods eds o capure saper soluio beer accuracy, bu ey produce wiggly soluios. Modiied Equaio Observaio : Firs-order meods are dissipaive ad less accurae, bu e soluio does o oscillae. preserves moooiciy. ookig a e srucure o e error erms by derivig e Modiied Equaio ca oe lead o isig io ow e approimae soluio beaves Derive modiied equaio or upwid dierece meod: + Usig Taylor epasio: = U + = =

6 Subsiuig &- +., / / / / /. + / 6 * + U &, * = Tereore, U U + U = I elps e ierpreaio i all erms are wrie i, + Takig urer derivaives irs i ime, e i space: U U + U = U U U U U = U = U & + U + & U + u + O U u + O + Similarly, we ge = U = U + O, + O, = U + O, Fial orm o e modiied equaio: U + U = U 6 U = + +O,,, By applyig upwid dierecig, we are eecively solvig: U + U = U Numerical dissipaio diusio U Also oe a e CF codiio = < esures a posiive diusio coeicie Dissipaio Dispersio Cosider: = ook or soluios o e orm:, = ae ik Subsiue o ge da = ik a d solve a = a o e ik For eve we ge diusio, or odd we ge dispersio Firs-Order Meods ad Diusio Upwid: U + U = a-friedrics: U 6 + U U + U = & + Dissipaive = Smearig U =

7 Secod-Order Meods ad Dispersio a-wedro: + U = U Beam-Warmig: + U = U 6 U U 8 U = 8 d=.5*.5.5 a-wedro Upwid Dispersive = Wiggles Firs-Order Meods ad Diusio Upwid: U + U = U 6 + Dissipaive = Smearig = U Ariicial Viscosiy a-wedro: U U + U = 8 Dispersive = Wiggles Ariicial Viscosiy Use ceered dierece meod e.g. a-wedro - Secod order accuracy - Oscillaio ear discoiuiy I order o damp ou oscillaio, we ca eier - Use implici umerical viscosiy upwid or - Add a eplici umerical viscosiy Ariicial Viscosiy Vo Neuma ad icmyer 95 Cosider: eplace e lu by: F + = F = F ; = D O coeicie + F = & = D Givig: & as - Simulaes e eec o e pysical viscosiy o e grid scale, coceraed aroud discoiuiy ad egligible elsewere. - is ecessary o keep e viscous erm o iger order.

8 Ariicial Viscosiy Eample: iear Advecio Equaio: + U = D Discreizaio by a-wedro + = U + *, +, D + + & +/ U & & & D + + / - /./ Approimae: D gives & +/ + = U + D * Ariicial Viscosiy = D U & , + + Ariicial Viscosiy Ariicial Viscosiy.5 Ariicial Viscosiy by awedro =6; sep=5; leg=4.;=leg/-;d=.5*; y=zeros,;=zeros,;=;ime=;.5 upwid or m=:sep,m old o;plo,liewid,; ais[ -.5,.5];old o; plo[,d*m-/+.5,d*m-/+.5,],[,,,],r,liewid,;pause; y=; ime=ime+d; or i=:-, i=yi-.5*d/*yi+-yi *d*d//*yi+-.*yi+yi *d/*absyi+-yi*yi+-yi-... absyi-yi-*yi-yi- ; ed; ed; a-wedro D= a-wedro D= Noliear advecio equaio.5 Upwid a-wedro Ariicial viscosiy ca be used wi mos oer ceered dierece scemes ad was, or a wile, THE way aeroauical compuaios were doe. Oer ypes, icludig iger order, ave bee used