The discontinuous Galerkin method with Lax Wendroff type time discretizations

Transcription

1 Comput. Methods Appl. Mech. Engrg. 94 (5) The discontinuous Galerkin method with La Wendroff tpe time discretizations Jianian Qiu a,, Michael Dumbser b, Chi-Wang Shu c, *, a Department of Mechanical Engineering, National Universit of Singapore, Singapore 96, Singapore b Institut fuer Aero- und Gasdnamik, Pfaffenwaldring, 755 Stuttgart, German c Division of Applied Mathematics, Brown Universit, Bo F, Providence, RI 9, USA Received 3 Januar 4; received in revised form 5 October 4; accepted 9 November 4 Abstract In this paper we develop a La Wendroff time discretization procedure for the discontinuous Galerkin method (LWDG) to solve hperbolic conservation laws. This is an alternative method for time discretization to the popular total variation diminishing (TVD) Runge Kutta time discretizations. The LWDG is a one step, eplicit, high order finite element method. The limiter is performed once ever time step. As a result, LWDG is more compact than Runge Kutta discontinuous Galerkin (RKDG) and the La Wendroff time discretization procedure is more cost effective than the Runge Kutta time discretizations for certain problems including two-dimensional Euler sstems of compressible gas dnamics when nonlinear limiters are applied. Ó 4 Elsevier B.V. All rights reserved. Kewords: Discontinuous Galerkin method; La Wendroff tpe time discretization; Runge Kutta method; Limiter; WENO scheme; High order accurac * Corresponding author. addresses: mpeqj@nus.edu.sg (J. Qiu), michael.dumbser@iag.uni-stuttgart.de (M. Dumbser), shu@dam.brown.edu (C.-W. Shu). The research of the author is supported b NUS Research Project R The research of this author is supported b NNSFC grant 83 while he is in residence at the Department of Mathematics, Universit of Science and Technolog of China, Hefei, Anhui 36, PR China. Additional support is provided b ARO grant DAAD and NSF grant DMS /$ - see front matter Ó 4 Elsevier B.V. All rights reserved. doi:.6/j.cma.4..7

2 J. Qiu et al. / Comput. Methods Appl. Mech. Engrg. 94 (5) Introduction In this paper, we stud an alternative method for time discretization, namel the La Wendroff tpe time discretization [8], to the popular TVD Runge Kutta time discretization in [3,8], for the discontinuous Galerkin methods in solving nonlinear hperbolic conservation law sstems u t þrfðuþ ¼; ð:þ uð; Þ ¼u ðþ: The first discontinuous Galerkin (DG) method was introduced in 973 b Reed and Hill [4], in the framework of neutron transport (stead state linear hperbolic equations). A major development of the DG method was carried out b Cockburn et al. in a series of papers [9,8,7,6,], in which the established a framework to easil solve nonlinear time dependent hperbolic conservation laws (.) using eplicit, nonlinearl stable high order Runge Kutta time discretizations [3] and DG discretization in space with eact or approimate Riemann solvers as interface flues and total variation bounded (TVB) limiter [7] to achieve nonoscillator properties for strong shocks. These schemes are termed RKDG methods. In RKDG method, DG is a spatial discretization procedure, namel, it is a procedure to approimate the spatial derivative terms in (.). The time derivative term there is discretized b eplicit, nonlinearl stable high order Runge Kutta time discretizations [3]. An alternative approach could be using a La Wendroff tpe time discretization procedure, which is also called the Talor tpe referring to a Talor epansion in time or the Cauch Kowalewski tpe referring to the similar Cauch Kowalewski procedure in partial differential equations (PDEs) [33]. This approach is based on the idea of the classical La Wendroff scheme [8], and it relies on converting all the time derivatives in a temporal Talor epansion into spatial derivatives b repeatedl using the PDE and its differentiated versions. The spatial derivatives are then discretized b, e.g. the DG approimations. RKDG methods have the advantage of simplicit, both in concept and in coding. The also enjo good stabilit properties when the TVD tpe Runge Kutta or multi-step methods are used [3,8]. Thus RKDG methods are ver popular in applications. To obtain provable nonlinear stabilit for scalar cases, a TVD time discretization [3,8] is preferred. There is however an order barrier for TVD Runge Kutta methods with positive coefficients: the cannot be higher than fourth order accurate [5]. Even if we ignore the TVD requirement for the time discretization, fifth and higher order Runge Kutta methods need more stages than the order of accurac, hence the efficienc would be reduced for ver high order time discretizations. The second approach, the La Wendroff tpe time discretization, which is also referred to as the Talor Galerkin method for finite element methods, usuall produces the same high order accurac with a smaller effective stencil than that of the first approach, and it uses more etensivel the original PDE. However, the formulation and coding of this procedure could be quite complicated, especiall for multi-dimensional sstems. We first review briefl the Talor Galerkin methods in the literature. A high order Talor Galerkin scheme to solve initial-boundar value problems for first order, linear hperbolic sstems was developed b Safjan and Oden [6]. A predictor corrector tpe and unconditionall stable higher order Talor Galerkin method scheme was presented b Youn and Park [38]. In Tabarrok and Su [3], a semi-implicit Talor Galerkin finite element method was developed for solutions of incompressible flows with heat transfer. In Colin and Rudgard [], the authors developed a third order low dissipation Talor Galerkin finite-element scheme within an unstructured/hbrid parallel solver for unstead large edd simulation. In Lukácová-MedvidÕová and Warnecke [9], a La Wendroff tpe second order evolution Galerkin method for multi-dimensional hperbolic sstems was discussed. The references quoted above were all based on continuous finite element methods. In [,3], Choe and Holsapple developed a Talor Galerkin method based on discontinuous finite elements. Their method is a one-step, eplicit finite element scheme, second order

3 453 J. Qiu et al. / Comput. Methods Appl. Mech. Engrg. 94 (5) accurate in both time and space, and is developed for the computation of weak solutions of nonlinear hperbolic conservation laws in one dimension. Mied finite element methodolog is used to treat the second derivative terms in the La Wendroff procedure, which limits the method to second order accurac to avoid a global solver. More recentl, Dumbser [3] developed the ADER (Arbitrar high order schemes using DERivatives, see [34]) discontinuous Galerkin method for solving the linear aeroacoustics sstems. ADER methods also use the La Wendroff procedure to convert time derivatives to spatial derivatives, so the method in [3] is essentiall the same as our method in this paper for the linear case. Dumbser and Munz [4] are also etending the ADER discontinuous Galerkin method to the nonlinear case using generalized Riemann solvers [36]. The La Wendroff tpe time discretization was also used in high order finite volume schemes [5,34,35] and finite difference schemes []. In this paper we eplore the La Wendroff tpe time discretization procedure for DG spatial discretizations up to third order (P case). Unlike in [,3], we do not use a mied finite element formulation to treat the higher order derivatives, hence in principle our methodolog can be easil generalized to arbitrar order of accurac. The resulting schemes, however, are more comple to formulate and to code than RKDG methods using the TVD Runge Kutta time discretizations, hence unless there is an advantage in CPU timing for the same accurac, the will not be competitive. Fortunatel, we demonstrate numericall that the La Wendroff time discretization procedure adopted in this paper is more cost effective and produces sharper discontinuit transition than RKDG methods for certain problems including two-dimensional Euler sstems of compressible gas dnamics, when nonlinear limiters are used. Of course, this procedure becomes progressivel more complicated with more complicated PDEs and/or higher order time accurac, hence we do not epect it to be alwas cost effective relative to the standard RKDG methods. The eact break-even point depends on man factors, such as the specific PDE (how complicated the time derivatives can be represented b spatial derivatives via the PDE), the relative cost of the limiter to the core DG procedure, the specific implementation including efficient cache usage, etc. In this paper we do not address the important issue of time discretization for PDEs with diffusion terms and/or with stiff source terms, which calls for hbrid eplicit/implicit time discretization. There are good Runge Kutta methods to easil achieve this, see, e.g. [4]. The La Wendroff procedure in this paper can also be adapted for such purpose, through careful Talor epansions. The organization of this paper is as follows. In Section, we describe in detail the construction and implementation of the high order DG method with a La Wendroff tpe time discretization, for one and two-dimensional scalar and sstem equation (.). In Section 3 we provide etensive numerical eamples to demonstrate the behavior of the schemes and to perform a comparison with RKDG methods. Concluding remarks are given in Section 4.. Construction and implementation of the scheme In this section we describe in detail the construction and implementation of the discontinuous Galerkin method with a La Wendroff tpe time discretization, for one- and two-dimensional scalar and sstem conservation laws... One-dimensional case Consider the one-dimensional scalar conservation laws: u t þ f ðuþ ¼ ; uð; Þ ¼u ðþ: ð:þ

4 J. Qiu et al. / Comput. Methods Appl. Mech. Engrg. 94 (5) We denote the cells b I i ¼½ i ; iþ Š, the cell centers b i ¼ i þ iþ and the cell sizes b D i ¼ iþ i. Let Dt be the time step, t n = t n Dt. B a temporal Talor epansion we obtain uð; t þ DtÞ ¼uð; tþþdtu t þ Dt u tt þ Dt3 6 u ttt þ: ð:þ If we would like to obtain (k )th order accurac in time, we would need to approimate the first k time derivatives: u t ;...; oðkþþ u. We will proceed up to third order in time in this paper, although the procedure can be naturall etended to an higher orders. ot kþ The temporal derivative terms in (.) can be replaced with the spatial ones using the governing equation (.): u t ¼ fðuþ ¼ f ðuþu ; u tt ¼ ðf ðuþu t Þ ¼ f ðuþu u t f ðuþu t ; u t ¼ f ðuþðu Þ f ðuþu ; u ttt ¼ ðf ðuþðu t Þ þ f ðuþu tt Þ : Then we can rewrite the approimation to (.) up to third order as: uð; t þ DtÞ ¼uð; tþ DtF : ð:3þ with F ¼ f þ Dt f ðuþu t þ Dt ðf ðuþðu 6 t Þ þ f ðuþu tt Þ. The standard discontinuous Galerkin method is then used to discretize F in (.3), described in detail below. The DG solution as well as the test function space is given b V k h ¼fp : pj I i P k ði i Þg, where P k (I i ) is the space of polnomials of degree 6 k on the cell I i. We adopt a local orthogonal basis over I i, fv ðiþ l ðþ; l ¼ ; ;...; kg, namel the scaled Legendre polnomials v ðiþ ðþ ¼; vðiþ ðþ ¼ i ; v ðiþ D ðþ ¼ i i D i ;...: Other basis functions can be used as well, without changing the numerical method, since the finite element DG method depends onl on the choice of space V k h, not on the choice of its basis functions. The numerical solution u h (,t) in the space V k h can be written as: u h ð; tþ ¼ Xk l¼ u ðlþ i ðtþv ðiþ l ðþ; for I i ; ð:4þ and the degrees of freedom u ðlþ i ðtþ are the moments defined b u ðlþ i ðtþ ¼ Z u h ð; tþv ðiþ l ðþd; l ¼ ; ;...; k; a l I i where a l ¼ R I i ðv ðiþ l ðþþ d are the normalization constants since the basis is not orthonormal. In order to determine the approimate solution, we evolve the degrees of freedom u ðlþ i : Z u ðlþ i ðt nþ Þ¼u ðlþ i ðt n Þþ a l F d I i d vðiþ l ðþd þ ^F iþ= v ðiþ l ð iþ= Þ ^F i = v ðiþ l ð i = Þ ¼ ; l ¼ ; ;...; k; where ^F iþ= is a numerical flu which depends on the values of the numerical solution u h and its spatial derivatives at the cell interface i/, both from the left and from the right. This numerical flu is related ð:5þ

5 453 J. Qiu et al. / Comput. Methods Appl. Mech. Engrg. 94 (5) to the so-called generalized Riemann solvers [36]. In this paper, we use the following simple La Friedrichs flu ^F iþ= ¼ F iþ= þ F þ iþ= aðuþ iþ= u iþ= Þ ; where u iþ= and F iþ= are the left and right limits of the discontinuous solution uh and the flu F at the cell interface i/, and a = ma u jf (u)j. For the sstem case, the maimum is taken for the eigenvalues of the Jacobian f (u). The integral term in (.5) can be computed either eactl or b a suitable numerical quadrature accurate to at least O(D kl ). In this paper we use two and three point Gaussian quadratures for k = and k = respectivel. An important component of DG methods for solving conservation laws (.) with strong shocks in the solutions is a nonlinear limiter, which is applied to control spurious oscillations. Although man limiters eist in the literature, e.g. [9,8,7,6,,,4], the tend to degenerate accurac when mistakenl used in smooth regions of the solution. In [ 3], we initialized a stud of using weighted essentiall nonoscillator (WENO) and Hermite WENO (HWENO) methodolog as limiters for RKDG methods. The idea is to first identif troubled cells, namel those cells where limiting might be needed, then to abandon all moments in those cells ecept the cell averages and reconstruct those moments from the information of neighboring cells using a WENO or HWENO methodolog. This technique works quite well in our one and two-dimensional test problems [ 3]. In this paper we adopt the WENO limiter developed in [] for k = and HWENO limiter in [,3] for k = to maintain the compactness of the DG method. The limiting procedure is performed in ever inner Runge Kutta stage for the RKDG method. Thus, when we use the third order TVD Runge Kutta time discretization [3], the limiting procedure is performed three times for one time step; but we onl need to perform the limiting procedure once per time step for the one step LWDG methods. This actuall accounts for a major saving of computational cost of the LWDG methods over the RKDG methods. For sstems of conservation laws (.), u(,t) =(u (,t),...,u m (,t)) T is a vector and f(u) =(f (u,...,u m ),...,f m (u,...,u m )) T is a vector function of u. As before, the time derivatives in (.) are replaced b the spatial derivatives using the PDE. The DG discretization is then performed on each component. In order to achieve better qualities at the price of more complicated and costl computations, we use a local characteristic field decomposition in the limiting procedure. For the details of such local characteristic field decompositions, we refer to [9]. The limiter and the WENO and HWENO reconstructions within the limiters are all performed under local characteristic projections. We note that the second and higher order time derivatives, when converted to spatial derivatives as before, involve epressions like f (u) which is a matri (the Jacobian), f (u) which is a 3D matri (tensor), etc., which could become ver complicated. The smbolic manipulator Maple Ò [6] is used to avoid mistakes. The code is also quite long and mess compared with codes using Runge Kutta time discretizations. However, we will see in the net section that one can save CPU time b this approach for certain problems... Two-dimensional cases Consider the two-dimensional conservation laws: u t þ f ðuþ þ gðuþ ¼ ; uð; ; Þ ¼u ð; Þ: ð:6þ B a temporal Talor epansion we obtain uð; ; t þ DtÞ ¼uð; ; tþþdtu t þ Dt u tt þ Dt3 6 u ttt þ:

6 J. Qiu et al. / Comput. Methods Appl. Mech. Engrg. 94 (5) For eample, for third order accurac in time we would need to reconstruct three time derivatives: u t, u tt, u ttt. We again use the PDE (.6) to replace time derivatives b spatial derivatives. u t ¼ fðuþ gðuþ ¼ f ðuþu g ðuþu ; u tt ¼ ðf ðuþu t Þ ðg ðuþu t Þ ¼ ðf ðuþu u t þ f ðuþu t þ g ðuþu u t þ g ðuþu t Þ; u t ¼ ðf ðuþðu Þ þ f ðuþu þ g ðuþu u þ g ðuþu Þ; u t ¼ ðf ðuþu u þ f ðuþu þ g ðuþðu Þ þ g ðuþu Þ; u ttt ¼ ðf ðuþðu t Þ þ f ðuþu tt Þ ðg ðuþðu t Þ þ g ðuþu tt Þ : Then we rewrite the approimation to (.6) up to third order as: uð; t þ DtÞ ¼uð; tþ DtðF þ G Þ; with ð:7þ F ¼ f þ Dt f ðuþu t þ Dt 6 ðf ðuþðu t Þ þ f ðuþu tt Þ; G ¼ g þ Dt g ðuþu t þ Dt 6 ðg ðuþðu t Þ þ g ðuþu tt Þ: The standard discontinuous Galerkin method is then used to discretize F and G in (.7). For sstems of conservation laws (.6), the time derivatives are replaced b the spatial derivatives using the PDE. The DG discretization is then performed on each component. Again the limiter and the WENO, HWENO reconstructions are all performed under local characteristic projections. To avoid mistakes, we again use the smbolic manipulator Maple Ò [6] to obtain the complicated time derivative terms for the sstem case. 3. Numerical results In this section we present the results of our numerical eperiments for the LWDG schemes, LWDG for P and LWDG3 for P, developed in the previous section, and compare them with RKDG schemes, RKDG for P with second order Runge Kutta method and RKDG3 for P with third order Runge Kutta method in [5 ]. We have used both uniform and nonuniform meshes in the numerical eperiments, obtaining similar results. We will onl show results with uniform meshes to save space. We first remark on the important issue of CPU time of LWDG and RKDG methods. In general the LWDG methods have smaller CPU costs for the same mesh and same order of accurac in our implementation for one and two-dimensional test cases, when the nonlinear limiters are applied, even though the CFL number of LWDG methods is smaller than that of RKDG methods for linear stabilit. The linear stabilit limit for RKDG is.333 and for LWDG it is.3. For the third order case, this linear stabilit limit is.9 for RKDG3 and it is.7 for LWDG3. Similar to that for the RKDG method [], the CFL limits for the LWDG method are obtained b standard von Neumann analsis with a numerical eigenvalue solver. In our numerical eperiments, the CFL numbers are taken as. and. for LWDG and LWDG3, and as.3 and.8 for RKDG and RKDG3, respectivel. In Table, we provide a CPU time

7 4534 J. Qiu et al. / Comput. Methods Appl. Mech. Engrg. 94 (5) Table CPU time (s) for the LWDG and RKDG methods to compute the double Mach reflection problem in Eample 3.8 for the two meshes of 3 and 4 6 cells Schemes LWDG RKDG k = k = k = k = M comparison between LWDG and RKDG methods for the two-dimensional Euler equations, double Mach reflection test case in Eample 3.8. The constant M in the table is the TVB constant in the limiter, see [7, 3] for details. The computation is performed on a Compaq Digital personal workstation, 6au alpha-599 MHz with 56 MB RAM. We can see that in general the RKDG methods cost about 3% more in CPU time than the LWDG methods for this problem, even though the CFL number for the RKDG method used in the computation is.5 times of those for LWDG methods because of the limitation on linear stabilit. We remark that for linear problems, both LWDG and RKDG can be implemented b efficient local matri vector multiplications with pre-computed local matrices and vectors containing the coefficients of the solution when represented b local basis functions in the stencil of the scheme (see, e.g. [39]), hence the per time stage cost of LWDG and RKDG is the same for such linear cases, thus LWDG is more efficient than RKDG for such linear cases because LWDG is a one step method, see, e.g. [3]. 3.. Accurac tests We first test the accurac of the schemes on linear scalar problems, nonlinear scalar problems and nonlinear sstems. We onl show the results of two-dimensional nonlinear scalar and one and two-dimensional nonlinear sstem problems to save space. We define the standard L and L error norms b sampling the errors at 4 equall spaced points inside each cell in each direction, emulating the L and L norms of the error function which is defined everwhere. Eample 3.. We solve the following nonlinear sstem of Euler equations u t þ f ðuþ ¼ ; ð3:þ with u ¼ðq; qv; EÞ T ; f ðuþ ¼ðqv; qv þ p; vðe þ pþþ T : Here q is the densit, v is the velocit, E is the total energ, p is the pressure, which is related to the total energ b E ¼ p þ c qv with c =.4. For the epression of the Jacobian f (u) for this eample we refer to [5]. A similar (but longer) epression eists for the second derivative f (u). The initial condition is set to be q(, ) =.sin(p), v(, ) =, p(, ) =, with a -periodic boundar condition. The eact solution is q(,t) =.sin(p( t)), v(,t) =, p(,t) =. We compute the solution up to t =. The errors and numerical orders of accurac of the densit q for the LWDG scheme in comparison with the results of the RKDG scheme are shown in Table. We can see that both schemes achieve their designed orders of accurac with comparable errors for the same mesh. Eample 3.. We solve the following nonlinear scalar Burgers equation in two dimensions: u t þ þ ¼ ; u u ð3:þ

8 J. Qiu et al. / Comput. Methods Appl. Mech. Engrg. 94 (5) Table Euler equations N LWDG RKDG L error Order L error Order L error Order L error Order P.7E E 3.3E E 3 6.5E 4..4E E 4.4.7E E E E E E 5..45E E 5.3.4E E E E E E E E E 6.99 P 6.E 4.5E 3.78E 3 4.5E E E E E E E E E E E E E E E E E E E E E 8.8 q(, ) =.sin(p), v(, ) =, p(, ) =. LWDG comparing with RKDG. Local La Friedrichs flu, using N equall spaced cells. t =.L and L errors of the densit q. with the initial condition uð; ; Þ ¼:5 þ sinðpð þ Þ=Þ and a 4-periodic boundar condition in both directions. When t =.5/p the solution is still smooth. The errors and numerical orders of accurac for the LWDG scheme in comparison with the results of the RKDG scheme are shown in Table 3. We can see again that both schemes achieve their designed orders of accurac with comparable errors for the same mesh. Eample 3.3. We solve the following nonlinear sstem of Euler equations n t þ f ðnþ þ gðnþ ¼ ; with n ¼ðq; qu; qv; EÞ T ; f ðnþ ¼ðqu; qu þ p; quv; uðe þ pþþ T ; gðnþ ¼ðqv; quv; qv þ p; vðe þ pþþ T : Here q is the densit, (u,v) is the velocit, E is the total energ, p is the pressure, which is related to the total energ b E ¼ p þ c qðu þ v Þ with c =.4. For the epression of the Jacobians f (u) and g (u), we refer to [5]. Similar (but longer) epressions eist for the second derivatives f (u) and g (u). The initial condition is Table 3 Burgers equation u t (u /) (u /) = N N LWDG RKDG L error Order L error Order L error Order L error Order P 7.9E 5.37E 6.4E 6.66E.5E.5.99E.44.54E.6.47E E E E E E E E E E 4..33E E 4.5.4E 3.93 P.99E 4.96E.98E 5.3E.66E E E E E E E E E E 4.6.7E E E E E E 4.87 Initial condition u(,, ) =.5 sin(p( )/) and periodic boundar conditions. LWDG comparing with RKDG. Local La Friedrichs flu, t =.5/p. L and L errors. Uniform meshes with N N cells. ð3:3þ

9 4536 J. Qiu et al. / Comput. Methods Appl. Mech. Engrg. 94 (5) set to be q(,,) =.sin(p( )), u(,,) =.7, v(,,) =.3, p(,,) =, with a -periodic boundar condition. The eact solution is q(,,t) =.sin(p( (u v)t)), u =.7, v =.3, p =. We compute the solution up to t =. The errors and numerical orders of accurac of the densit q for the LWDG scheme in comparison with the results of the RKDG scheme are shown in Table 4. We can see that both schemes achieve their designed orders of accurac with comparable errors for the same mesh. 3.. Test cases with shocks We now test the performance of the LWDG method for problems containing shocks. We have also computed man more problems such as the two-dimensional forward facing step problem, but will not present all the results to save space. We onl plot cell averages of the solution in the graphs. Table 4 Euler equations N N LWDG RKDG L error Order L error Order L error Order L error Order P.59E 7.55E 3.48E 7.34E 8.76E E E E E 3.6.6E.76.E E E E E 4.37.E E E E E 4.65 P.E 3 7.4E E 3.39E.45E E E 4 4..E E E E E E E E E E E E E 6.97 Initial condition q(,,) =.sin(p( )), u(,,) =.7, v(,,) =.3, p(,,) = and periodic boundar conditions. LWDG comparing with RKDG. Local La Friedrichs flu, t =.. L and L errors for the densit q. Uniform meshes with N N cells. Eact LWDG RKDG Eact LWDG3 RKDG3 u.5 u Fig.. The Buckle Leverett problem: t =.4, N = 8 cells. Here and below onl the cell averages are plotted. Left: k = ; Right: k =.

10 J. Qiu et al. / Comput. Methods Appl. Mech. Engrg. 94 (5) Eample 3.4. We solve the nonlinear nonconve scalar Buckle Leverett problem! 4u u t þ ¼ ; ð3:4þ 4u þð uþ with the initial data u = when 6 6 and u = elsewhere. The solution is computed up to t =.4. The eact solution is a shock rarefaction contact discontinuit miture. We remark that some high order schemes ma fail to converge to the correct entrop solution for this problem. In Fig., the solutions of LWDG comparing with RKDG using N = 8 cells are shown. We can see schemes of all orders give good nonoscillator resolutions to the correct entrop solution for this problem..4.4 Eact LWDG RKDG Eact LWDG3 RKDG Fig.. La problem: t =.3; LWDG together with RKDG; cells; TVB constant M = ; densit. Left: k = ; Right: k = Reference LWDG RKDG 4 3 Reference LWDG3 RKDG Fig. 3. The shock densit wave interaction problem: t =.8; LWDG together with RKDG; cells; TVB constant M = 3; densit. Left: k = ; Right: k =.

11 4538 J. Qiu et al. / Comput. Methods Appl. Mech. Engrg. 94 (5) Eample 3.5. We solve the one-dimensional nonlinear sstem of Euler equation (3.). We use the following Riemann initial condition for the La problem: ðq; v; pþ ¼ð:445; :698; 3:58Þ for 6 ; ðq; v; pþ ¼ð:5; ; :57Þ for > : The computed densit q is plotted at t =.3 against the eact solution. In Fig., the solutions of LWDG together with RKDG using N = cells with TVB constant M = in the minmod limiter are shown. We can see that the resolution of the contact discontinuit b LWDG is slightl better than (for the k = case) or comparable with (for the k = case) that b RKDG. Eample 3.6. A higher order scheme would show its advantage when the solution contains both shocks and comple smooth region structures. A tpical eample for this is the problem of shock interaction with entrop waves [3]. We solve the Euler equation (3.) with a moving Mach = 3 shock interacting with sine waves in the densit, i.e. initiall Reference LWDG RKDG 4 3 Reference LWDG3 RKDG Fig. 4. Details of the oscillator portion of the solution between =.5 and =.5 for Fig. 3. Left: k = ; Right: k = Reference LWDG RKDG 6 Reference LWDG3 RKDG Fig. 5. The blast wave problem: t =.38; LWDG together with RKDG; 4 cells; TVB constant M = 3; densit. Left: k = ; Right: k =.

12 J. Qiu et al. / Comput. Methods Appl. Mech. Engrg. 94 (5) Reference LWDG RKDG 4 3 Reference LWDG3 RKDG Fig. 6. Detail of solution between =.5 and =.5 for Fig. 5. Left: k = ; Right: k = Fig. 7. Double Mach reflection problem: second order (k = ) LWDG (top) and RKDG (bottom) with WENO limiters; 9 48 cells; TVB constant M = ; 3 equall spaced densit contours from.5 to.7. ðq; v; pþ ¼ð3:85743; :69369; :333333Þ for < 4; ðq; v; pþ ¼ð þ e sinð5þ; ; Þ for P 4:

13 454 J. Qiu et al. / Comput. Methods Appl. Mech. Engrg. 94 (5) Here we take e =.. The computed densit q is plotted at t =.8 against the reference eact solution, which is a converged solution computed b the fifth order finite difference WENO scheme [7] with grid points. In Fig. 3, we plot the densities b LWDG and RKDG with WENO limiters using N = cells for the TVB constant M = 3, and in Fig. 4, we show an enlarged view of the oscillator portion of the solution between =.5 and =.5. We can see that LWDG has slightl better resolution than RKDG for the same mesh and same limiter. It seems to be generall true that LWDG often has slightl better resolution than RKDG for the same mesh and same limiter, probabl due to the fact that the former is a one-step method using the limiter onl once per time step. Eample 3.7. We consider the interaction of blast waves of Euler equation (3.) with the initial condition: ðq; v; pþ ¼ð; ; Þ for 6 < :; ðq; v; pþ ¼ð; ; :Þ for : 6 < :9; ðq; v; pþ ¼ð; ; Þ for :9 6 : A reflecting boundar condition is applied to both ends, which in the discontinuous Galerkin framework means that a mirror smmetric polnomial is prescribed in the ghost cell outside the boundar point. See [37,5]. The computed densit q is plotted at t =.38 against the reference eact solution, which is a converged solution computed b the fifth order finite difference WENO scheme [7] with grid points. In Fig. 5, we plot the densities b LWDG and RKDG with WENO limiters using N = 4 cells for the TVB Fig. 8. Double Mach reflection problem: Third order (k = ) LWDG (top) and RKDG (bottom) with WENO limiters; 9 48 cells; TVB constant M = ; 3 equall spaced densit contours from.5 to.7.

14 J. Qiu et al. / Comput. Methods Appl. Mech. Engrg. 94 (5) constant M = 3, we also show an enlarged view of the solution between =.63 and =.83 in Fig. 6. We can again see that LWDG has a slightl better resolution than (for the k = case) or is comparable with (for the k = case) RKDG for the same mesh and the same limiter. Eample 3.8 (Double Mach reflection). This problem is originall from [37]. The computational domain for this problem is [, 4] [, ]. The reflecting wall lies at the bottom, starting from ¼ 6. Initiall a rightmoving Mach shock is positioned at ¼ 6, = and makes a 6 angle with the -ais. For the bottom boundar, the eact post-shock condition is imposed for the part from = to ¼ 6 and a reflective boundar condition is used for the rest. At the top boundar, the flow values are set to describe the eact motion of a Mach shock. We compute the solution up to t =.. Two different uniform meshes, with 96 4 and 9 48 cells, and two different orders of accurac for the LWDG and RKDG using WENO limiters with the TVB constant M =, from k = to k = (second to third order), are used in the numerical eperiments. To save space, we show onl the simulation results on the most refined mesh with 9 48 cells in Figs. 7 and 8, and the zoomed-in figures around the double Mach stem to show more details in Fig. 9. All the figures are showing 3 equall spaced densit contours from.5 to.7. For this problem the resolutions of LWDG and RKDG are comparable for the same order of accurac and the same mesh Fig. 9. Double Mach reflection problem: 9 48 cells; TVB constant M = ; 3 equall spaced densit contours from.5 to.7. Zoomed-in region to show more details. Top: second order (k = ); bottom: third order (k = ). Left: LWDG; Right: RKDG.

15 454 J. Qiu et al. / Comput. Methods Appl. Mech. Engrg. 94 (5) Concluding remarks We have developed a La Wendroff time discretization procedure for the discontinuous Galerkin method to solve hperbolic conservation laws. This is an alternative method for time discretization to the popular TVD Runge Kutta time discretization in RKDG methods. The LWDG is a one step eplicit high order finite element method. The nonlinear limiter for controlling spurious oscillations is performed once per time step. The method relies on converting all the time derivatives in a temporal Talor epansion into spatial derivatives b repeatedl using the PDE and its differentiated versions. The spatial derivatives are then discretized b the DG approimations. As a result, LWDG is more compact than RKDG and the La Wendroff time discretization procedure is more cost effective than the Runge Kutta time discretizations, when the nonlinear limiters are used, for certain problems including two-dimensional Euler sstems of compressible gas dnamics. Acknowledgments The first author would like to thank Professor B.C. Khoo and Professor G.W. Wei for their support and help. References [] R. Biswas, K.D. Devine, J. Flahert, Parallel, adaptive finite element methods for conservation laws, Appl. Numer. Math. 4 (994) [] K.Y. Choe, K.A. Holsapple, The Talor Galerkin discontinuous finite element method an eplicit scheme for nonlinear hperbolic conservation laws, Finite Element Anal. Des. (99) [3] K.Y. Choe, K.A. Holsapple, The discontinuous finite element method with the Talor Galerkin approach for nonlinear hperbolic conservation laws, Comput. Methods Appl. Mech. Engrg. 95 (99) [4] A. Burbeau, P. Sagaut, C.H. Bruneau, A problem-independent limiter for high-order Runge Kutta discontinuous Galerkin methods, J. Comput. Phs. 69 () 5. [5] B. Cockburn, Discontinuous Galerkin methods for convection-dominated problems, in: T.J. Barth, H. Deconinck (Eds.), High-Order Methods for Computational Phsics, Lecture Notes in Computational Science and Engineering, vol. 9, Springer, 999, pp [6] B. Cockburn, S. Hou, C.-W. Shu, The Runge Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Math. Comput. 54 (99) [7] B. Cockburn, S.-Y. Lin, C.-W. Shu, TVB Runge Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional sstems, J. Comput. Phs. 84 (989) 9 3. [8] B. Cockburn, C.-W. Shu, TVB Runge Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Math. Comput. 5 (989) [9] B. Cockburn, C.-W. Shu, The Runge Kutta local projection P-discontinuous Galerkin finite element method for scalar conservation laws, Math. Model. Numer. Anal. (M AN) 5 (99) [] B. Cockburn, C.-W. Shu, The Runge Kutta discontinuous Galerkin method for conservation laws V: multidimensional sstems, J. Comput. Phs. 4 (998) [] B. Cockburn, C.-W. Shu, Runge Kutta Discontinuous Galerkin method for convection-dominated problems, J. Scient. Comput. 6 () [] O. Colin, M. Rudgard, Development of high-order Talor Galerkin schemes for LES, J. Comput. Phs. 6 () [3] M. Dumbser, ADER discontinuous Galerkin schemes for aeroacoustics, in: Proceedings of the Euromech Colloquium no. 449, Chamoni, France, 3. [4] M. Dumbser, C.-D. Munz, Arbitrar high order discontinuous Galerkin schemes, in: S. Cordier, T. Goudon, M. Gutnic, E. Sonnendrucker (Eds.), Numerical Methods for Hperbolic and Kinetic Problems, IRMA Series in Mathematics and Theoretical Phsics, de Gruter, in press. [5] A. Harten, B. Engquist, S. Osher, S. Chakravath, Uniforml high order accurate essentiall non-oscillator schemes III, J. Comput. Phs. 7 (987) 3 33.

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