Adaptation and Assessment of a High Resolution

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1 Adaptatio ad Assessmet of a High Resolutio Semi-Discrete Numerical Scheme for Hyperbolic Systems with Source Terms ad Stiffess R. Naidoo, ad S. Baboolal Departmet of Mathematics ad Physics, M.L. Sulta Techiko, P.O. Box, Durba, South Africa aidoor@yoda.cs.udw.ac.za Departmet of Computer Sciece, Uiversity of Durba-Westville, Private Bag X5, Durba, South Africa sbab@pixie.udw.ac.za Abstract. I this work we outlie the details required i adaptig the third-order semi-discrete umerical scheme of Kurgaov ad Levy SIAM J. Sci. Comput. () to hadle hyperbolic systems which iclude source terms. The performace of the scheme is the assessed agaist a fully discrete scheme, as well as referece solutios, o such problems as shock propagatio i a Broadwell gas ad shocks i gas dyamics with heat trasfer. Itroductio This paper is cocered with the umerical itegratio of u(x, t) t + f (u) x = g (u), () ε a oe-dimesioal hyperbolic system of partial differetial equatios. Here u(x, t) is the ukow m-dimesioal vector fuctio, f(u) is the flux vector, g(u) is a cotiuous source vector fuctio o the right had side (RHS), with x the sigle spatial coordiate ad t the temporal coordiate ad the parameter ε > distiguishes betwee stiff systems (ε << ) ad stadard, o-stiff oes (ε = ). Such equatioss ca be used to model may physical systems, icludig fluids ad gases. I the last decade, particularly followig the work of Nessyahu ad Tadmor, a family of fully-discrete, high-resolutio, Riema-solver-free schemes have bee produced i order to umerically solve hyperbolic systems such as the aforemetioed. More recetly, also based o the same Riemasolver-free approach, secod ad third order semi-discrete schemes were devised by Kurgaov ad Tadmor ad Kurgaov ad Levy. Oe advatage of the latter is that they ca be applied o o-staggered grids ad thus ease the implemetatio of boudary coditios. Here we are particularly iterested i the details of adaptig the latter so that it be ca applied to systems with source P.M.A. Sloot et al. (Eds.): ICCS, LNCS, pp. 5 6,. Spriger-Verlag Berli Heidelberg

2 Adaptatio ad Assessmet of a High Resolutio Semi-discrete Numerical Scheme 5 terms icludig those that are stiff such as () above. I order to assess the performace of this scheme we examie its merits agaist a adaptatio followig 5 for o-staggered grids, of the fully-discrete scheme of for systems with source terms, as well as agaist exact or referece solutios for two prototype problems. Oe such is the problem of shock propagatio i a Broadwell gas 6, 7 ad the other is that of shocks i a model of gas dyamics with heat trasfer. The modified umerical scheme. Derivatio with source term Here the umerical itegratio of problem () is cosidered o some uiform spatial ad temporal grids with the spacigs, x=x j+ x j ; t=t + t (with j ad beig suitable iteger idices). For the oliear homogeeous case of (), Kurgaov ad Levy obtai the third-order semi-discrete scheme where ad dū j dt = f(u + (t)) + f(u (t)) f(u + (t)) f(u (t)) x j+ j+ j j a j+ (t) x a j± u + (t) u (t) a j (t) u + (t) u (t), j+ j+ x j j ( ( ) ( )) f f = max ρ u (u (t)), ρ j± u (u+ (t)), () j± u + := P j± j+ (x j±,t ); u := P j± j (x ±,t ). () I the above, the forms () are respectively the left ad right itermediate values at x j+ of a piecewise polyomial iterpolat P j(x, t ) that fit a already computed or kow cell average values {ū j } at time level. Also ρ(.) deotes the spectral radius of the respective Jacobia, defiig the maximum local propagatio speeds a. j± They also obtai a extesio of the above whe the RHS of () is of the form Q x where Q(u(x, t),u x(x, t)) is a dissipatio flux satisfyig a parabolicity coditio. However, to allow for a arbitrary source term, say, g(u(x, t)) i () (omittig for coveiece the stiffess parameter) we must proceed as outlied i ad follow through the costructio of the scheme with this added detail. Thus, employig the above metioed uiform spatial ad temporal grids ad itegratig () over the cell I(x) :={ξ ξ x x }gives ū t + x f(u(x + x x,t)) f(u(x,t)) () =ḡ (5)

3 5 R. Naidoo ad S. Baboolal where ad ū(x, t) := u(ξ,t)dξ (6) x I(x) ḡ := g(u(ξ, t)) dξ (7) x I(x) Now assumig the {ū j } are already computed or kow cell-averages of the approximate solutio at time t = t we itegrate as i over the cotrol volumes x j,r,x j,l t,t +,x j,r,x j+,l t,t + ad x j+,l,x j+,r t,t + where x j±,l := x j± a j± t; x j±,r := x j± + a j± t (8) with the piecewise polyomial form i the cell I j take as P j (x, t )=A j +B j (x x j )+ C j(x x j ). (9) where the costats A j,b j,c j are evaluated as i. These the result respectively, i the weighted averages w +, w j j +, w + which differ from those i j+ oly i the respective additive source terms a j t xj,r t + x j,l ( ) x t a + a j j+ t g dx dt, () xj+,l t + x j,r t gdxdt () ad a j+ t xj+,r t + x j+,l t g dx dt. () are recostructed third order piece- The from the cell averages w + j+ wise polyomials take as ad w + j w + = j± Ãj± + B j± (x x j± )+ C j± (x x j± ), w + j (x) w + j. () where the costats Ãj, B j ad C j are evaluated as i. The ew cell averages o the ustaggered grids are obtaied from these polyomials by ū + j = x xj,r w + j x j xj+,l dx + x j,r The semi-discrete form is the defied by the limit xj+ w j + dx + x j+,l w + j+ dx ()

4 Adaptatio ad Assessmet of a High Resolutio Semi-discrete Numerical Scheme 55 dū j (t) ū + j ū j = lim. (5) dt t t Proceedig with () ad () as i, the coefficiets i the polyomial form simplify resultig i dū j dt = f(u + (t)) + f(u (t)) f(u + (t)) f(u (t)) x j+ j+ j j a j+ (t) u + (t) u (t) a j (t) u + (t) u (t) x j+ j+ x j j + lim t x t t + j,r t + lim t x j,l gdxdt+ lim t x t t( x t(a j+ + a j )) t + j+,r t x j+,l t + j+,l t x j,r gdxdt g dx dt. (6) We ote that the o-smooth parts of the solutio are cotaied over spatial widths of size a j± t. Full details with clear sketches are give i. Now, whe the limits are take o the source itegrals, the first two vaish as the Riema fas shrik to zero. At the time, sice ū =ū(t)(ad hece g) isa costat over this cell, it ca be show for the other that lim t t( x t(a j+ + a j )) t + j+,l t x j,r gdxdt=g(u j). Hece the modified semi-discrete scheme with source term g(u(x, t)) is a j+ (t) x dū j dt = x f(u + (t)) + f(u (t)) f(u + (t)) f(u (t)) j+ j+ j j u + (t) u (t) j+ j+ a j (t) x u + (t) u (t) + g(u j j j (t)). (7) where the rest of the terms are as i ()-(). To compute with (7), it is coveiet to use ODE system solvers, such as Ruge-Kutta formulae. For istace, writig (7) i the form, du j dt = F j, (8) where F j is the vector of the RHS, we ca employ the secod-order (i time) Ruge-Kutta (RK) scheme for it as: U () = U + tf (U ) RK : U () = U + U () + tf (U () ) (9) U + = U ()

5 56 R. Naidoo ad S. Baboolal where the U deotes the vector of compoets u j, the superscript ad + deote successive time levels, whilst the other (,) deote itermediate values. We shall refer to the scheme (7) with RK (9) as the SD scheme. We ote that such a scheme is geerally third order except i regios of steep gradiets whe it degrades to order two. Sice also, RK is secod order i time, it will make sese whe we compare its performace to that of the fully discrete scheme (NNT) for systems with source terms for itegratio o ustaggered grids : ū + j = (ū j+ +ū j +ū j ) 6 + t λ 8 ( u xj+ u xj ) 8 ( ) ( ) ( ) ( g u j+ +g u j +g u j +g u + j+ ( f j+ fj ) ( + f + j+ f + ) j u + u + xj+ xj ) +g ( u + j ) + g ( u + j where λ = t/ x ad which will be used used with a UNO derivative approximatio,. ) (). Implemetatio details The implemetatio of the NNT scheme () above follows previous reports,, where i particular we metio that the source term ca make the scheme implicit. The latter the requires a iteratio at each grid poit at every time level. The implemetatio of the modified semi-discrete scheme SD (7) follows closely the prescriptio give i where i particular we employ the costats give i their equatio (.9) for the o-oscillatory piece-wise polyomial (9). I additio, it is required to compute at every time step the spectral radii () of the Jacobias of the flux terms, which we obtaied exactly for the small test systems to follow. Fially, with chose iitial ad boudary coditios, the solutio is advaced with the explicit Ruge-Kutta scheme (8)-(9). The codes were writte i 6-bit real precisio Fortra 77, employig real-time graphics to depict evolvig profiles. They were compiled with the Salford FTN95/wi versio.8 compiler ad ru o a PC uder MS Widows ad NT. Applicatios ad tests. Shocks i a Broadwell gas Here we solve the goverig equatios for a Broadwell gas 6, 7, ρ t + m x =, m t + z x =, z t + m x = ( ρ +m ρz ), ε

6 Adaptatio ad Assessmet of a High Resolutio Semi-discrete Numerical Scheme 57 where ε is the mea free path ad ρ(x, t),m(x, t),z(x, t) are the desity, mometum ad flux respectively. The rage ε =.. 8 cover the regime from the o-stiff to the highly stiff. I particular, the limit ε = 8 requires a reormalizatio of the variables such as i the form x = ε x, t = ε t followed by computatios o a equivalet fier grid (see for example 8). We observe that i the limit ε we arrive at z = z E (ρ, m) = ρ (ρ +m ) which leads to the equilibrium solutio of the goverig equatios above which the reduce to the Euler equatios. The SD (7) ad NNT scheme () were applied to the above with the two sets (Rim ad Rim) of iitial coditios correspodig to several Riema problems, each distiguished by aspecific ε-value: { ρ=,m=,z =; x<xj. Rim : ρ=,m=.96,z=; x>x J. Rim : { ρ=,m=,z =; x<xj. ρ=.,m=,z =; x>x J. I all calculatios absorbig boudary coditios were employed, where i particular, the boudary values were obtaied by quadratic extesios of iteral poit values o a fixed spatial grid, over a itegratio domai o the X-axis. Results obtaied are depicted i Figure. Other parameters used here were x =., t =.5, x J = 5 i (a) ad (b) ad x =., t =., x J = i (c) ad (d) for both methods. We observe that i virtually all cases, the semi-discrete scheme give better results tha the modified NNT scheme.. Shocks i a Euleria gas with heat trasfer Here we solve the Euler equatios for the oe-dimesioal flow of a gas i cotact with a costat temperature bath 9: (ρe) t (ρu) t + ρ t + (ρu) x =, + (ρu +p) x (ρue + up) x =, = Kρ(T T ).

7 58 R. Naidoo ad S. Baboolal SD (a) NNT (b) (c) (d) Fig.. Broadwell gas shock solutios with (a) ε =(Rim), (b) ε =. (Rim), (c) ε = 8 (Rim) ad (d) ε = 8 (Rim). Here the curve labelled ρ, z, m.the sap-shot time is t =.5iall cases. The dotted lies are computed solutios ad the solid oes are exact (refied grid) solutios.

8 Adaptatio ad Assessmet of a High Resolutio Semi-discrete Numerical Scheme 59 The ρ, u, T, e, E = e + u,p=(γ )ρe are the desity, flow velocity, temperature i uits of e, iteral eergy, total eergy ad pressure respectively with K>the heat trasfer coefficiet ad T the costat bath temperature. The iitial coditios used were: { ρ=.5,u=.,p=.; x<5. Rim : ρ =.,u=.,p=.; x>5. where differet K =, 5,, are employed. Computed results with SD ad NNT are show i Figure. SD K=, NNT K= K= K= Fig.. Shocks i a gas with heat trasfer. The curve labelled ρe, ρ, ρu. The grid legths of x =.ad t =. were used for both methods. The output time is t =.iall cases.

9 6 R. Naidoo ad S. Baboolal I these we observe SD captures the shocks sigificatly better tha does NNT. Although the exact (grid refied oes) are ot show here we metio that they are close to the SD oes, except for the oscillatios see i them. We believe the oscillatios are due to the derivative calculatios from the fitted piecewise polyomials, ad ca be improved by takig differet smoothig coefficiets, as suggested i. Further, the NNT curves show poor resolutio of the shocks i compariso, ad also show far more dissipatio, as expected. Coclusio We have idicated i this work, how the third-order semi-discrete umerical scheme of Kurgaov ad Levy ca be suitably modified to iclude source terms i oe-dimesioal hyperbolic systems. Results obtaied with it o shock propagatio i a Broadwell gas ad i a gas dyamics model with heat trasfer show its clear superiority over a fully discrete modificatio for systems with source terms. Ackowledgemets The authors wish to thak Professors D. Levy, G. Russo ad Shi Ji for their helpful correspodeces. Refereces. Nessyahu, H., Tadmor, E.: No-oscillatory cetral differecig for hyperbolic coservatio laws. J. Comput.Phys. 87 (99) Kurgaov, A., Tadmor, E.: New high-resolutio cetral schemes for oliear coservatio laws ad covectio-diffusio equatios. J. Comput. Phys. 6 () -8.. Kurgaov, A., Levy, D.: A third-order semi-discrete cetral scheme for coservatio laws ad covectio-diffusio equatios. SIAM J. Sci. Comput. () Naidoo, R., Baboolal, S.: Modificatio of the secod-order Nessyahu-Tadmor cetral scheme to a o-staggered scheme for hyperbolic systems with source terms ad its assessmet. Submitted to SIAM J. Sci. Comput. 5. Jiag, G.-S., Levy, D., Li, C.-T., Osher, S., Tadmor, E.: High-resolutio ooscillatory cetral schemes with o-staggered grids for hyperbolic coservatio laws. SIAM J. Numer. Aal. 5 (998) Caflisch, R.E., Ji, S., Russo, G.: Uiformly accurate schemes for hyperbolic systems with relaxatio. SIAM J. Numer. Aal. (997) Ji, S.: Ruge-Kutta methods for hyperbolic coservatio laws with stiff relaxatio terms. J. Comput. Phys. (995) Naldi, G., Pareschi, L.: Numerical schemes for hyperbolic systems of coservatio laws with stiff diffusive relaxatio. SIAM J. Numer. Aal. 7 () Pember, R. B.: Numerical methods for hyperbolic coservatio laws with stiff relaxatio II. Higher-order Goduov methods. SIAM J. Sci. Comput. (99) 8-89.