Runge-Kutta discontinuous Galerkin finite element method for one-dimensional multimedium compressible flow

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1 SSN , Eglad, K Joural of formato ad Computg Scece Vol. 3, No. 3, 008, pp. 5-4 Ruge-Kutta dscotuous Galerk fte elemet metod for oe-dmesoal multmedum compressble flow Rogsa Ce + Departmet of Matematcs, Saga versty, Saga 00444, P. R. Ca Receved Ja. 0, 008, accepted July, 008) Abstract. Ruge-Kutta dscotuous Galerk RKDG) fte elemet metod for yperbolc coservato laws s a g order metod, wc ca adle complcated geometres flexbly ad treat boudary codtos easly. ts paper, we propose a ew umercal metod for treatg terface usg te advatages of RKDG fte elemet metod. We use level set metod to track te movg terface. every tme step, a Rema problem at te terface s defed. Te two cells adjacet to te terface are computed usg te Rema problem solver. f te terface crosses a cell te ext tme step, te values of te flow varables of te cell crossed are modfed troug lear terpolato. Otewse, we do otg. Keywords: fte elemet, umercal aalyss, Euler equato. troducto Te orgal dscotuous Galerk DG) fte elemet metod was troduced by Reed ad Hll [5] for solvg te eutro trasport equato. Ruge-Kutta dscotuous Galerk RKDG) fte elemet metod for o-lear yperbolc coservato laws was proposed by Cockbur et al. [6, 5, 4, 7]. Ts metod was costructed by usg a explct g order Ruge-Kutta tme dscretzato ad pecewse lear DG metod for space dscretzato. t ca be desged for ay order of accuracy space ad tme, adle flexbly complcated geometres, ad treat easly boudary codtos. Above all, te computato of ay elemet depeds oly o te formato of tself ad ts mmedate egbors, so t ca be used easly for te effcet parrel mplemetato. RKDG fte elemet metod performed very well for computato of te sgle-medum compressble flow. Varous umercal scemes suc as te total varato dmsg TVD) scemes, te essetally o-oscllatory ENO) or te wegted essetally o-oscllatory WENO) scemes ave bee developed to solve sgle-medum flow, wc ca usually aceve g order accuracy wt sarp ad essetally o-oscllatory sock trasto. However, we tose scemes s drectly appled to mult-medum flow, opyscal oscllatos usually occur te vcty of te materal terface. order to prevet oscllatos ear te terface, may metods were preseted to treat te terface. Te orgal gost flud metod proposed by Fedkw et al [8]. offers a farly smple ad flexble way to treat mult-medum terface ad s easly exteded to mult-dmeso. te GFM, oe flow-feld wt two medum s separated to two flow-feld by level set metod, eac of wc as a sgle medum ad ca be computed solely. t was foud to be less effcet we appled to gas-water flow or a strog sock wave teractg wt te terface. Ts as resulted may modfed GFM [,, 9, 7]. A Rema problem based metod for te resoluto of compressble mult-materal flow was proposed by Cocc et al. [3]. Te umercal algortm cosst of a predctor step ad a correcto step. every tme step, correcto of te dffused odes s carred out o bot sdes of te terface. Several autors exteded RKDG fte elemet metod to mult-medum compressble flow. RKDG fte elemet for two-medum flow smulatos oe ad two dmeso wt te orgal GFM ad te modfed GFM were vestgated by Qu et al. [3]. Aoter mportat work of Qu [4] s to use te RKDG fte elemet metod for two-medum flow computato wt coservatve treatmet of te movg materal terface. R.S.Ce et al. computed mult-medum compressble flow by RKDG fte elemet wt a ew modfed GFM [, ]. ts paper, we use RKDG fte elemet metod for oe dmeso mult-medum compressble flow + Emal address: rsce@yaoo.c Publsed by World Academc Press, World Academc o

2 6 R. Ce, et al: Ruge-Kutta dscotuous Galerk fte elemet metod ad use level set metod to keep track of te terface locato. every tme step, a Rema problem at te terface s defed. Te two cells adjacet to te terface s computed usg te solver of te Rema problem smlar to te GFM, te oter cells are computed by sgle-medum metod. f te terface crosses a cell ceter at te ext tme step, te values of te cell crossed are modfed by terpolato. Oterwse, we do otg. Te paper s dvded as follows. secto, Euler equatos, equato of state EOS),ad level set equato are provded. secto 3, we frst revew te RKDG fte elemet metod, ad te we descrbe detal te metod of terface treatmet. Tests o gas-gas ad gas-lqud flow are gve secto 4. secto 5, a bref cocluso s preseted.. Equatos.. Euler Equatos Te Euler equatos for oe-dmesoal compressble flow are wrtte as F ) + = 0 t x wt = ρ, ρu, E), F ) = ρu, ρu + p, E + p) u). Here ρ s te desty, u s te velocty, p s te pressure, ad E s te total eergy for ut volume. Te total eergy s gve as E = ρ e + ρ u... Equato of State EOS) For closure of te system, te EOS s requred. We wll use te followg equato of state ts work.. Te stffeed gas equato of state s te form.) For γ -law gas, p = 0. p = γ )ρe γp..).3. Level Set Equato Te level set metod used o track te terface ca be wrtte as Φ t x, + u x, Φ x x, = 0.3) Here u s te velocty of flud. geeral, Φ x, starts out as a dstace fucto. But over several tme steps, t wll ot be equal to te dstace fucto. order to keep t as te sged dstace fucto, te retalzato step s ecessary. ts work, te Rema solver at te terface provdes te accurate velocty of te terface, so we replace te u x, by terface velocty ad te re-talzato s ot mplemeted. Te level set equato s solved by te tegral-averagg sceme proposed by Lu et al. [0]. 3. RKDG fte elemet metod 3. Descrpto RKDG fte elemet metod Te computatoal doma R s dvded to N cells, R =, were = x /, x+ / ), x + x =,,, N. Deote te cell ceters by + / x = / ad te cell szes by x = x+ / x / ). Let us defe = V = V = k k { p BV L : p P )} k Were s soluto space, V s test fucto space, ad P ) s te space of polyomals of degree k o. We use a local ortogoal bass over,{ x), l = 0,,, k}. We coose for example v l JC emal for cotrbuto: edtor@jc.org.uk

3 Joural of formato ad Computg Scece, 3 008) 3, pp Te approxmate soluto ca be wrtte as Were l { } k l = 0 0 ), ), ) 3, / / x x x x x = v x = v x = x x v 3.) are te degrees of freedom. k l x, = v x) for x. 3.) l= 0 We substtute 3.) to.), multply by x), tegrate over a cell ad tegrate by parts: + / t ± ±, + / = x+ / t d dt l v l l d + F x, ) vl x) dx x /l ) dx + F x+ /, ) F x /, )) = 0, 3.3) l = 0,,, k. + Te flux F x, )) s usually replaced by a mootoe umercal flux H, ), were ), resultg te sceme: + / + / d l d + F x, ) vl x) dx dt x /l ) dx H + /, + / ) H /, / )) = 0, 3.4) l = 0,,, k. We use te local Lax-Fredrcs flux H + /, + / ) = [ F + / ) + F + / ) α + / + / + / )], wt p) + p) λ λ ) α, + / = max + /, p 3 p) F were λ ± + /, p =,, 3, are te 3 real egevalues of te Jacoba ) ±. = / + At last, te sem-dscrete sceme ca be wrtte as d = L ). 3.5) dt We dscrete 3.5) by Ruge-Kutta metod. order to aceve varato stablty, a slope lmter s used after eac Ruge-Kutta stage. For a complete dscusso of te metod, te reader s referred to [6]. 3. Treatmet of terface Te cell szes are uform, we deote te cell ceters by, deote by te level set fucto value of x at t = t, deote x by te terface locato at t =. Assumg te terface x s betwee x ad + / t x+ t = t x x x + x x + x x+ x+, te left of terface s flud, ad te rgt of terface s flud. At stace, tree stace may occur: ) s betwee ad, as sow Fg. a); ) s betwee ad x, as sow Fg. b); 3) s betwee ad, as sow Fg. c). Supposed tat te flow states at bee kow, te followg steps are take to obta te respectve quattes at te ext step: x Φ t = t ave JC emal for subscrpto: publsg@wa.org.uk

4 8 R. Ce, et al: Ruge-Kutta dscotuous Galerk fte elemet metod. Defe Rema problem F ) + = 0 t x, x < x x, t ) = +, x > x Solve te Rema problem, we ca get u te velocty of terface), p te pressure of terface), ρ L te desty of te left of terface), ρ te desty of te rgt of terface), te etropy of te left of terface), S R R te etropy of te rgt of terface). We deote L = ρ, ρ u, E ), = ρ, ρ u, E ). L L L R R R R. Te sobarc fx tecque ca be used to reduced te overeatg errors [7]. Ts s doe by S = S L, S + = S R 3. For te cells,,,, use EOS of Flud ; For te cells,,,, use EOS of Flud. We + + N, + N + ca compute te flow at,,,,, as for sgle-medum flow. But te cells ad must be treated specally. We we compute flow at cell, we ca t use te flow state at drectly because tey belog to dfferet fluds. So we modfy t as below: f we compute at +, H, ) H, ). f we compute at, smlar as at + / + / = + / L + + +, H, ) H, ) + / + / = R + / 4. Solve te level set equato usg u, we ca obta x, t ). [ 5. f + x, x+, we do otg. f terpolato x ] x [ x, x ] Φ + + S L + 3.6) +, we obta by terpolato: for costat JC emal for cotrbuto: edtor@jc.org.uk

5 Joural of formato ad Computg Scece, 3 008) 3, pp or for lear terpolato, + [ +, + = + + R R + = Φ + Φ + + f x x x ], we obta by terpolato:, 3.7) R. 3.8) for costat terpolato, +, 3.9) = L or for lear terpolato, + Te soluto ca be advaced from to. 4. Numercal results L + = Φ + Φ L. 3.0) Fg. Example, 00 cells Fg.3 Example, 400 cells JC emal for subscrpto: publsg@wa.org.uk

6 0 R. Ce, et al: Ruge-Kutta dscotuous Galerk fte elemet metod ts secto, fve umercal expermets are preseted. For te computatos, CFL s set to 0., computatoal doma s [0, ], te sold le s te exact soluto ad te pots are te computed soluto. A trd order accurate RKDG fte elemet metod s used to solve te Euler equato. Example 4. Ts s a gaseous sock tube problem take from [7] ad te tal codtos are,0,,.4,0), f 0 x < 0.5, ρ, u, p, γ ) = 0.5,0,0.,.,0), f 0.5 < x. Te computatoal tme s t = 0.. Fg. sows te result of 00 mes pots, ad Fg.3 sows te result of 400 mes pots. From te Fgs. -3, we see tat tere are o-pyscal oscllato ear te terface ad te computed solutos cocur wt te exact solutos. Example 4. We cosder a two-pase gas-lqud Rema problem wc s take from [6].Te tal states are defed as.4,0,.753,.4,0), 0.99,0, ,5.5,.505), ρ, u, p, γ ) = 4 f f 0 x < 0.5, 0.5 < x. Fg.4 Example, 00 cells Fg.5 Example, 400 cells JC emal for cotrbuto: edtor@jc.org.uk

7 Joural of formato ad Computg Scece, 3 008) 3, pp 5-4 Te results of 00 mes pots ad 400 mes pots at t = 0. are sow Fgs.4-5. Te computatoal solutos are coverget to pyscal solutos as te mes refed. Example 4.3 A gas-lqud sock tube problem s take from [9]. Te tal codtos are 0.5,00,0000,.5,0), f 0 x < 0.5, ρ, u, p, γ ) =,0,,7.5,3309), f 0.5 < x. Sow Fgs.6-7 are te respectve results obtaed by 00 mes pots ad 400 mes pots at tme t = 0.00 tat agree wt te exact solutos. Fg.6 Example 3, 00 cells Fg.7 Example 3, 400 cells Example 4.4 We cosder strog sock mpactg o a gas-water terface wc s take from [7]. Te tal posto of sock ad terface are te same at. Te tal codto are as follows: ,9.88,000,.4,0), f 0 x < 0.5, ρ, u, p, γ ) =,0,,7.5,3309), f 0.5 < x. Te results at t = are preseted Fgs.6-7. Te posto of sock ad terface are predcted accurately. JC emal for subscrpto: publsg@wa.org.uk

8 R. Ce, et al: Ruge-Kutta dscotuous Galerk fte elemet metod Example 4.5 Jet mpact o te gas-water terface s cosdered. Ts s take from [7]. Te tal codto are gve as,90,,.4,0), f 0 x < 0.5, ρ, u, p, γ ) = 000,0,,7.5,3309), f 0.5 < x. Ts s a very dffcult problem. Te results of 00 mes pots ad 400 mes pots at t = 0.05 are obtaed, sow Fgs.7-8. Te results sow tat te sock wave ad te terface are well located. Fg.8 Example 4, 00 cells Fg.9 Example 4, 400 cells JC emal for cotrbuto: edtor@jc.org.uk

9 Joural of formato ad Computg Scece, 3 008) 3, pp Fg.0 Example 5, 00 cells Fg. Example 5, 400 cells 5. Coclusos ts paper, RKDG fte elemet metods wt a ew treatmet of te movg terface ave bee developed to smulate oe-dmesoal mult-medum compressble flow. At every tme step, a Rema problem at te terface s defed, te two cells ear terface are computed usg te solver of Rema problem. f te terface crosses a cell, we modfy te values of te cell crossed by terpolato. Ts metod s very smple. Compared wt GFM metod, t cost less. Numercal results for gas-gas ad gaslqud flow oe dmeso sow te preset metod s robust. Ogog work s to exted tese metods for two dmeso. 6. Refereces [] R. S. Ce, X. J. Yu, A g order accurate RKDG fte elemet metod for oe dmesoal compressble multcompoet Euler equato. Cese J. Comput. Pys. 006, 3: [] R. S. Ce, X. J. Yu, A RKDG fte elemet metod for two dmesoal compressble multmeda fluds. Cese J. Comput. Pys. 006, 3 : [3] J.-P. Cocc, R. Saurel, A Rema problem based metod for te resoluto of compressble multmateral flows. J. Comput. Pys. 997, 37: JC emal for subscrpto: publsg@wa.org.uk

10 4 R. Ce, et al: Ruge-Kutta dscotuous Galerk fte elemet metod [4] B. Cockbur, S. Hou, C.-W. Su, TVB Ruge-Kutta local projectg dscotuous Galerk fte elemet metods for coservato laws V: te multdmesoal case. J. Comput. Mat. Comp. 990, 54: [5] B. Cockbur, S.-Y. L, C.-W. Su, TVB Ruge-Kutta local projectg dscotuous Galerk fte elemet metods for coservato laws : oe dmesoal systems. J. Comput. Pys. 989, 84: [6] B. Cockbur, C.-W. Su, TVB Ruge-Kutta local projectg dscotuous Galerk fte elemet metods for coservato laws : geeral framework. Mat. Comp. 989, 5: [7] B. Cockbur, C.-W. Su, TVB Ruge-Kutta local projectg dscotuous Galerk fte elemet metods for coservato laws V: multdmesoal systems. J. Comput. Pys. 998, 4: [8] R. P. Fedkw, B. T. Aslam, S.oser, A o-oscllatory Eulera approac to terfaces multmateral flows te gost flud metod), J. Comput. Pys. 999, 5: [9] X. Y. Hu, B. C. Koo, A terface teracto metod for compressble multfluds. J. Comput. Pys. 004, 98: [0] R. X. Lu, X. P. Lu, L. Zag, Z. F. Wag, Trackg ad rescotructo metods for movg-terface. Appl. Mat. Mec. 004, 5: [] T. G. Lu, B. C. Koo, C. Wag, Te gost flud metod for compressble gas-water smulato. J. Comput. Pys. 005, 04: 93. [] T. G. Lu, B. C. Koo, K. S. Yeo, Gost flud metod for strog sock mpactg o materal terface. J. Comput. Pys. 003, 90: [3] J. X. Qu, T. Lu, B.C.koo, Smulatos of compressble two-medum flow by Ruge-Kutta dscotuous Galerk metods wt te gost flud metod. Commu. Comput. Pys. 008, 3: [4] J. X. Qu, T. G. Lu, B. C. Koo, Ruge-Kutta Dscotuous Galerk metods for compressble two-medum flow smulatos: oe-dmesoal case. J. Comput. Pys. 007, : [5] W. H. Reed, T. R. Hll, Tragular mes metods for te eutro trasport equato. Tec. rep. Los Alamos Scefc Laboratory Report LA-R-973). [6] K. M. Syue, A effcet sock-capturg algortm for compressble multcompoet problems. J. Comput. Pys. 998, 4: [7] C. W. Wag, T. G. Lu, B. C. Koo, A real-gost flud metod for te smulato of mult-medum compressble flow. SAM J. Sc. Comput. 006, 8: JC emal for cotrbuto: edtor@jc.org.uk