XXI Congreso de Ecuaciones Diferenciales y Aplicaciones XI Congreso de Maemáica Aplicada Ciudad Real, 2-25 sepiembre 29 (pp. 8) Numerical schemes for scalar conservaion laws wih siff source erms R. Dona, I. Higueras 2, A. Marínez-Gavara 3 Dp. de Maemàica Aplicada, Univ. de València, Apdo. 46, Burjasso. E-mail: dona@uv.es. 2 Dpo. de Ingeniería Maemáica e Informáica, Univ. Pública de Navarra, Apdo. 36, Pamplona. E-mail: higueras@unavarra.es. 3 Dpo. de Ecuaciones Diferenciales y Análisis Numérico, Univ. de Sevilla, Apdo. 42, Sevilla. E-mail: gavara@us.es. Palabras clave: siff source erm, conservaion laws Resumen We consider a simple model problem of a hyperbolic PDE wih a siff source erm, inroduced by Le Veque and Yee in [5], as he basic es case o explore he abiliy of Implici-Explici Runge-Kua schemes o produce and mainain non-oscillaory reacion frons. Since Srong Sabiliy conceps canno be applied in a direc manner o he model problem, we develop a Weak Sabiliy heory ha provides a convenien framework o sudy he non-oscillaory properies of numerically compued reacion frons.. Inroducion Many physical problems are governed by hyperbolic conservaion laws wih non vanishing siff source erms. These problems could describe he effec of relaxaion as in he kineic heory of gases, chemical reacions, elasiciy wih memory, waer waves, raffic flows, ec. In some problems he source erms depend only on he soluion, i.e. s(x, u) = s(u) and ye he soluion naurally develops srucures in which he source erms are nonzero, and possibly large, only over very small regions in space. This ofen happens if he source erms model chemical reacions beween differen species (reacing flow) in cases where he reacions happen on ime scales much faser han he fluid dynamic ime scales. Then, soluions can develop hin reacion zones where he chemical-kineics aciviy is concenraed. Such problems are said o have siff source erms, in analogy wih he classical case of siff ordinary differenial equaions (ODEs).
R. Dona, I. Higueras, A. Marínez-Gavara Numerical difficulies ofen appear when he fas reacions are in near-equilibrium during mos of he compuaion. Some ime scales, ypically hose driving he reacion erms, are several orders of magniude faser han he scale on which he soluion is evolving and on which one would like o compue. Wih many numerical mehods, including all explici mehods, aking a ime sep appropriae for he slower scale of ineres can resul in violen numerical insabiliy, caused by he fas scales. Sabiliy, meaning he absence of violen oscillaory behavior, can be achieved by using implici mehods. A variey of excellen implici mehods have been developed for solving siff sysems of ODEs, and many of he same echniques can be applied when he source erms are siff in order o obain sable resuls. A differen ype of numerical difficuly is, however, also encounered in numerical simulaions concerning hyperbolic PDEs wih siff source erms: he occurrence of frons propagaing a he wrong speeds. I is observed in [5] ha he essenial numerical difficulies o be encounered in he numerical approximaion of convecion-reacion problems wih siff reacion erms can be idenified and sudied mos easily by looking a he equaion u + u x = µu(u )(u ) < x < >, () 2 where he parameer µ > conrols he siffness of he problem. The soluion wih piecewise consan iniial daa {, if x < xd, u(x, ) = (2), if x > x d, is simply u(x, ) = u(x, ). Here we revisi LeVeque and Yee s model problem: a linear advecion equaion wih a parameer dependen source erm. In order o make more precise he relaion beween he occurrence of he pahological reacion frons, he specific form of he source erm and he various parameers involved in he numerical simulaions, we sar by considering very basic firs order echniques which allow us o analyze he source of he pahology. We obain an explici expression for he discrepancy beween he rue speed of propagaion of he reacion fron and he speed of he numerical fron produced by he scheme, and his allows us o quanify he observaion, made by Le Veque and Yee, ha i is precisely he inclusion of non-equilibrium values, a a discree shock profile, wha induces he pahology. Our analysis highlighs he imporance of ensuring ha he compued numerical profiles are monoone. 2. Mehod of lines discreizaions The applicaion of he mehod of lines o he model problem of he previous secion reduces he PDE o an iniial value problem for a sysem of ordinary differenial equaions (ODEs), U = L(U()) + S(U()), U() = (u(x, ), u(x 2, ),..., u(x N, )) T, (3) for he vecor U() = (U (), U 2 (),..., U N ()) T wih componens U i () u(x i, ). Due o he naure of he problem, differen operaors are assigned o he convecive derivaive and he source erm. 2
Numerical schemes for scalar conservaion laws wih a siff source erm The erm L(U) in (3) is he spaial discreizaion operaor of he convecive derivaive erm, u x. The erm S(U) represens he discree approximaion of he source erm, which will always be defined in his work as S(U) i = s(u i ). I is widely acceped ha siff source erms should be handled in an implici fashion, in order o avoid sabiliy problems relaed wih he fas scales. On he oher hand, here are a number of robus, and raher specialized, numerical flux funcions ha can be used if disconinuous, or nearly disconinuous, soluions need o be compued. These observaions lead, in a raher naural way, o consider IMplici-EXplici (IMEX) Runge-Kua (RK) schemes for he ime inegraion of MOL discreizaions (see [6]). The abiliy o rea he convecive par in an explici fashion, while sill mainaining an implici handling of he source erms gives a disinc advanage when designing a general purpose high order, high resoluion numerical scheme. As expeced, he occurrence of frons moving a wrong speeds canno be avoided by using any paricular numerical flux funcions in he discree convecive derivaive. The cause of he delay mechanism can be undersood, and quanified, by considering firs he simples numerical echniques for he model problem. In [3], we examine he limiaions ha are encounered when using he simples Eulerype firs order schemes for he ime discreizaion of he MOL sysem (3). For he sake of numerical illusraion, we shall consider he linear upwind discreizaion of he convecive derivaive in () L(U) j = U j U j. (4) x This echnique is paricularly simple because he only boundary condiions needed for he ime sepping process are he physical inflow boundary condiions, which, for he model problem, we ake as U n =, n. The Explici Euler mehod All explici mehods aking a ime sep appropriae for he slower scale of ineres can resul in violen numerical insabiliy caused by he faser scales. Typically, he compuaions end o become very inefficien because he imesep sizes dicaed by he sabiliy requiremens are much smaller han hose required by accuracy consideraions for he slowly varying soluion. The numerical insabiliy become larger when µ increases, o he poin of rendering he useless. In order o obain a free of unwaned oscillaions, he ime sep has o be reduced (on he same mesh). However, he reducion of only guaranees an effecive conrol over he oscillaions developed in he, bu he approximae soluion obained migh sill be qualiaively wrong. In figure, we display resuls for he numerical simulaion wih µ = on a mesh wih x =,. In figure -(a) we show a ypically oscillaory behavior, which is obained for CF L = / x =,4 (i.e. µ = 4, for larger values of µ more violen oscillaions are observed). In figure -(b) we have considered CF L =,2 (i.e. µ = 2). Here we see ha he soluion looks reasonable, however he disconinuous profile is delayed wih respec o ha of he rue soluion. This delay persiss when we reduce he ime sep even furher, as can be observed in figure -(c), where we show he numerical profile for CF L =, ( µ =,). 3
R. Dona, I. Higueras, A. Marínez-Gavara.2.2.8.8.8.6.6.6.4.4.4.2.2.2.2..2.3.4.5.6.7.8.9.2..2.3.4.5.6.7.8.9.2..2.3.4.5.6.7.8.9 (a) CF L =,4 (b) CF L =,2 (c) CF L =, Figura : Numerical soluion for model problem a =,3 for µ = using he firs order explici scheme. x =,. The Implici Euler mehod In he numerical reamen of siff ODEs, i is cusomary o apply implici echniques in order o bypass he sric requiremens on he ime sep imposed by he siffness of he problem. This echnique was proposed by Ahmad and Berzins in [], as a firs sep in order o sudy he effec of neglecing various erms in he nonlinear solvers involved in higher order implici discreizaions of (). The vecor of unknowns U n+ = (U n+, U2 n+,..., U n+ N ) can be compued from he known soluion a ime n, U n = (U n,..., U N n ) by applying a Newon procedure on he sysem ( Uj n+ = Uj n + U j n+ Uj n+ ) + s(uj n+ ), j =,, N (5) x The Jacobian marix of he sysem is of he form JG(V ) = I L(V ) V S(V ) V. (6) The expression of JL := L depends on he advecive erms. Clearly, i migh be very V complicaed or even impossible o compue if a nonlinear high order scheme is used (see [] for a discussion of alernaives in his case). As expeced, he s are non-oscillaory. When he siffness of he model is increased even furher, he occurrence of frons moving a he wrong speeds is observed again. In figure 2, a fron moving a he wrong speed is displayed, due o lack of proper spaial resoluion. In boh cases, x =, is used and CF L =,9 for he lef plo while CF L =,4 for he righ plo. The numerical ess carried ou in his secion clearly display he wo main problems ha may arise when he balance law () is solved numerically: he presence of oscillaions and a numerical delay in he shock profile. Our numerical experimens wih firs order discreizaions indicae ha implici or semi-implici reamens of he siff source erms can conrol he numerical oscillaions in a sraighforward manner, bu will no solve he problem of he numerical delay in he shock profiles, which sill demands an adequae spaial resoluion. The source of he observed delay can be fully analyzed for hese firs order schemes, as we show in he following secion. 4
Numerical schemes for scalar conservaion laws wih a siff source erm.9.9.8.8.7.7.6.6.5.5.4.4.3.3.2.2....2.3.4.5.6.7.8.9..2.3.4.5.6.7.8.9 (a) CFL=,9 (b) CFL=,4 Figura 2: Numerical soluion for model problem wih µ = a =,3, using he firs order fully implici scheme. x =,. CFL= x. 3. Wave speed analysis The analysis below, which is based on simple consideraions on he wave speed of he fron, shows ha he incorrec speed of propagaion of he numerical fron is, in fac, a by-produc of he spaial discreizaion, ha will always be presen as long as here are values of he unknown ha lie sricly beween and. The area under he disconinuous soluion for he model problem () wih iniial condiion (2) φ() := u(x, )dx (7) Hence φ () =, which is he speed of propagaion of he rue soluion. On he oher hand, if he soluion o he sysem of ODEs (3) is a monoone profile, i is naural o define he area under he discree profile, a ime, as φ x () := x N U i (). (8) For a MOL discreizaion, as esablished in (3), he ime variaion of his quaniy can also be easily compued: d d φ x() = x N i= i= d d U i() = µ x N i= U i (U i )(U i ). (9) 2 Clearly, d d φ x() represens he velociy of propagaion of he discree fron, hence he relaion above implies ha he discree profile can move a a speed ha can be quie differen from ha of he rue profile. In fac, he discrepancy is equal o he delay facor α(µ x, U) = µ x N i= U i (U i )(U i ). () 2 A he fully discree level, where he is given by a sequence (U n i ) i, 5
R. Dona, I. Higueras, A. Marínez-Gavara.8.6.4.2.2.4.6.8.5..5.2.25.3.8.6.4.2.2.4.6.8.5..5.2.25.3 (a) µ x =, CFL=, (b) µ x =, CFL=,2 Figura 3: α(µ x, U n ) as a funcion of. Fully explici scheme..8.6.4.2.2.4.6.8.8.6.4.2.2.4.6.8.8.6.4.2.2.4.6.8.5..5.2.25.3.5..5.2.25.3.5..5.2.25.3 (a) µ x =, CFL=,7 (b) µ x =, CFL=,2 (c) µ x =, CFL=,7 Figura 4: α(µ x, U n ) as a funcion of. Fully implici scheme. a each ime sep, he corresponding discree equivalen o (8) is N φ n x := x Ui n, () so ha, he discree wave speed defined by LeVeque and Yee in [5] becomes ws(n) = x Uj n U n j = φn+ x φn x, n >. (2) x j j The discree wave speed, ws(n) in (2) can easily be compued for he fully discree Explici Euler (EE) and Implici Euler (IE). A sraighforward compuaion easily leads o ws(n) EE = + α(µ x, U n ), ws(n) IE = + α(µ x, U n ). The funcion s(u) = u(u )(u,5) saisfies s(u) 5 2 for u [, ] and in a ypical simulaion here are only a small number of poins ha conribue o he sum, precisely he poins a he discree disconinuous profile, hence i is o be expeced ha α(µ x, U) = O(µ x 2 ), bu non-zero. Thus, he simples way o ensure a correc speed of propagaion for he fron is o ensure ha µ x is below a securiy hreshold. In figures 3 and 4, we show plos of α(µ x, U n ) wih respec o n for he firs order schemes (EE, IE). We noice ha he behavior of his funcion depends on he CFL number, and ha i migh be oscillaory, bu in all cases α(µ x, U n ) = O(5 2 µ x), i= 6
Numerical schemes for scalar conservaion laws wih a siff source erm as expeced and provided ha he soluion is a monoone profile wih only a finie (and small) number of non-equilibrium poins. Hence, for hese schemes we can only aemp o obain a qualiaively correc speed of propagaion for he numerical fron if we ensure ha µ x is below a securiy hreshold. If he is a monoone profile wih values in [, ], wih a finie, and small, number of ransiion values, we have ha for µ x, he delay becomes O( 2 ), which would render i undeecable upon visual examinaion. I should be noiced ha in order o be compuaionally meaningful, ws(n) in (2) needs o be compued on a monoone (i.e. non-oscillaory) discree profile. For homogeneous conservaion laws, he use of TVD consrains is a general procedure ha can be used o ensure a sable, and oscillaion free,. However, for balance laws, TVD consrains migh be oo srong, as i is indeed he case for he model problem. Hence, we seek o enforce weaker forms of sabiliy, and his issue will be sudied in he following secion. 4. Weak sabiliy In general, when a PDE is solved numerically, i is naural o require ha he numerical soluion saisfies as many qualiaive properies of he as possible. Sabiliy requiremens sem from he desire o have numerical schemes ha preserve, a he discree level, cerain properies of he analyic soluion of he problem o be solved. For homogeneous hyperbolic conservaion laws, he fac ha he Toal Variaion (TV) of he soluion does no increase in ime has lead o he developmen of he so-called TVD (Toal-Variaion Diminishing) schemes, a highly successful sabiliy framework ha emerged from imposing his condiion on he numerical scheme. In he case of he balance law, he heorem 5. in [3] (see also [2]) esablishes a monooniciy propery ha can be used o enforce a weaker ype of sabiliy in he numerical schemes. In paricular in he LeVeque and Yee s model problem () wih he iniial profile (2), he applicaion of he heorem gives ha u(x, ), (3) and hence, i makes sense o require similar inequaliies for he, U n i, i =,, N n, (4) perhaps under cerain sepsize resricions. The numerical experimens done [3] show ha numerical oscillaions are always associaed o violaions of propery (4) and hence we may consider i as a key propery o avoid he producion of numerical oscillaions. We, hus, inroduce a weaker form of sabiliy ha seeks o comply wih propery (3). We refer o his propery as weak sabiliy (WS), as opposed o he srong sabiliy (SS) conceps, such as in [4], which prevens growh in a given norm (or semi-norm), ha canno be applied o he model problem due o he paricular form of he source erm. A numerical scheme is hen ermed weakly sable if i saisfies (4) provided U i. Wih he aim o provide useful ools ha lead us o expec a monoone behavior in he s, we analyze in [3] he weak sabiliy properies of firs order schemes 7
R. Dona, I. Higueras, A. Marínez-Gavara of he ype considered in he previous secion. As in he case of Srong Sabiliy, we expec o obain Weak Sabiliy only under cerain resricions on x and. The weak sabiliy propery of he EE scheme for he model problem, can be wrien in erms of CFL condiion as CF L = x 2 2 + µ x. (5) This is a sufficien condiion, however i seems o be raher sharp, according o our numerical experimens. For example, for µ x =, we obain weak sabiliy for CF L 2/3,66, while for µ x = weak sabiliy is guaraneed only for CF L,66. In he fully implici firs order scheme, we are led o expec monoone profiles for s provided hese soluions can be compued. Noice ha if s(u), he implici reamen of he convecive derivaive leads o a linear sysem wih a non-singular bi-diagonal marix. However, here is no guaranee in general ha U n+ can be compued independenly of for s(u) in he model problem. In pracice, we have observed ha an increase in he CFL number seems o lead o convergence problems in he Newon-like ieraive process. Conclusion. We have analyzed he phenomenon of waves propagaing a nonphysical speeds for MOL schemes obained from firs order upwind spaial discreizaion of he convecive derivaive in a model problem proposed by LeVeque and Yee in [5]. The use of he simple upwind discreizaion allows us o carry ou a deailed sudy of he delay facor. I is seen ha he delay or advance of he discree wave profile in a MOL discreizaion is conrolled mainly by µ x. Even hough we only showed numerical resuls for he firs order upwind discreizaion of he convecive erm, he behavior obained in his case is ypical of wha would be obained wih more sophisicaed, sae of he ar conservaive discreizaions for he soluion of he model IVP. The advanage of he firs order upwind discreizaion is ha i allows for a raher sysemaic analysis of he numerical soluion, which allowed us o esablish cerain bounds ha, for all pracical purposes, ensure monooniciy of he numerical profiles. Acknowledgemen The auhors acknowledge suppor from MTM28-974 and MTM28-785. Referencias [] I. Ahmad and M. Berzins. MOL solvers for hyperbolic PDEs wih source erms. Mah. Compu. Simulaion, 56(2):5 25, 2. Mehod of lines (Ahens, GA, 999). [2] A. Chalabi. On convergence of numerical schemes for hyperbolic conservaion laws wih siff source erms. Mah. Compu., 66(28):527 545, 997. [3] R. Dona, I. Higueras, and A. Marínez-Gavara. On sabiliy issues for IMEX schemes applied o hyperbolic equaions wih siff reacion erms. (in preparaion). [4] S. Golieb, W. C.Shu, and E. Tadmor. Srong sabiliy preserving high-order ime discreizaion mehods. SIAM Rev., 43()(TR-2-5):89 2, 2. [5] R. J. LeVeque and H. C. Yee. A sudy of numerical mehods for hyperbolic conservaion laws wih siff source erms. J. Compu. Phys., 86():87 2, 99. [6] L. Pareschi and G. Russo. Implici-Explici Runge-Kua schemes and applicaions o hyperbolic sysems wih relaxaion. J. Sci. Compu., 25(-2):29 55, 25. 8