Well-Balanced Discontinuous Galerkin Methods for the Euler Equations Under Gravitational Fields

Transcription

1 J Sci Comput 6 67:49 5 DOI.7/s Well-Balanced Discontinuous Galerkin Methods for the Euler Equations Under Gravitational Fields Gang Li Yulong Xing Received: April 5 / Revised: Jul 5 / Accepted: 4 August 5 / Published online: September 5 Springer Science+Business Media New York 5 Abstract Euler equations under gravitational field admit hdrostatic equilibrium state where the flu produced b the pressure is eactl balanced b the gravitational source term. In this paper, we present well-balanced Runge utta discontinuous Galerkin methods which can preserve the isothermal hdrostatic balance state eactl and maintain genuine high order accurac for general solutions. To obtain the well-balanced propert, we first reformulate the source term, and then approimate it in a wa which mimics the discontinuous Galerkin approimation of the flu term. Etensive one- and two-dimensional simulations are performed to verif the properties of these schemes such as the eact preservation of the hdrostatic balance state, the abilit to capture small perturbation of such state, and the genuine high order accurac in smooth regions. ewords Euler equations Runge utta discontinuous Galerkin methods Wellbalanced propert High order accurac Gravitational field Introduction Hdrodnamical evolution in a gravitational field arises in man applications from astrophsics and climate. The are usuall modeled b the Euler equations governing the conservation of mass, momentum and energ, coupled with a source term due to the gravitational field. In one space dimension, the take the form of B Yulong Xing ing@ucr.edu Gang Li gangli978@6.com School of Mathematical Sciences, Qingdao Universit, Qingdao 667, Shandong, People s Republic of China Department of Mathematics, Universit of California Riverside, Riverside, CA 95, USA

2 494 J Sci Comput 6 67:49 5 ρ t + ρu =, ρu t + ρu + p = ρφ, E t + E + pu = ρuφ,. where ρ denotes the fluid densit, u is the velocit, p represents the pressure, and E = ρu + ρe e is internal energ is the non-gravitational energ which includes the kinetic and internal energ of the fluid. γ is the ratio of specific heats and φ = φ is the time independent gravitational potential. The ideal gas law p = γ E ρu /,. is considered to close this sstem. The Euler equations under gravitational field. belong to the class of hperbolic equations with source terms also referred as hperbolic balance laws, which takes the general form of U t + FU = SU,,. where U is the solution vector with the corresponding flu FU,andSU, is the source term. This sstem usuall admits non-trivial stead state solutions, in which the source term is eactl balanced b the flu gradient. One main challenge in the numerical simulation of such balance laws is that a standard numerical method ma not satisf the discrete version of this balance eactl at or near the stead state, and ma introduce spurious oscillations, unless the mesh size is etremel refined. To save the computational cost, well-balanced methods, which preserve eactl these stead state solutions up to machine accurac, are speciall designed to ensure accurate simulations and ehibit essential stabilit properties on relativel coarse meshes. Another prototpical eample considered etensivel in the literature for hperbolic balance laws is the shallow water equations with a non-flat bottom topolog. Man researchers have developed well-balanced methods for the shallow water equations using different approaches, see, e.g. [,,8,,,8,9,,5] andthe references therein. For the Euler equations. under static gravitation potential φ, there eists the hdrostatic equilibrium state, also called mechanical equilibrium, where the eternal forces such as gravit are balanced b the pressure gradient force: ρ = ρ, u =, p = ρφ..4 It is difficult to design well-balanced numerical methods which can accuratel preserve all the solutions of.4. Two important special stead state are the constant entrop isentropic and constant temperature isothermal hdrostatic equilibrium states [9]. Techniques required to balance each equilibrium can be different. In this paper, we onl consider the isothermal hdrostatic balance with a constant temperature T. For an ideal gas satisfing p = ρ,.5 where R is the gas constant, the stead state solution.4 becomes ρ = ρ ep φ, u =, p = ρ = ρ ep φ,.6 with a constant ρ after some simple calculation. The simplest and most commonl encountered case is the linear gravitational potential field with φ = g, and the corresponding isothermal hdrostatic balance takes the form of

3 J Sci Comput 6 67: ρ = ρ ep gρ /p, u =, p = p ep gρ /p..7 Man astrophsical problems involve nearl stead state flows in a gravitational field, therefore it is essential to correctl capture the effect of gravitational force in these simulations, especiall if a long-time integration is involved, for eample in modeling star and gala formation. Improper treatment of the gravitational force can lead to a solution which either oscillates around the equilibrium, or deviates from the equilibrium after a long time run. In recent ears, well-balanced numerical methods for the Euler equations with gravitational fields have attracted much attention. LeVeque and Bale [] etended the quasi-stead wave-propagation methods for the Euler equations under a static gravitational field. Finite volume well-balanced discretizations with respect to dominant hdrostatics has been proposed b Botta et al. [] for the nearl hdrostatic flows in the numerical weather prediction. Xu and his collaborators [,7,6] have etended the gas-kinetic scheme to the multidimensional gas dnamic equations to develop well-balanced numerical methods, where the gravitational potential was modeled as a piecewise step function with a potential jump at the cell interface. Finite volume well-balanced methods for the isentropic hdrostatic equilibrium are proposed b appeli and Mishra [9]. High order finite difference well-balanced methods for the isothermal equilibrium are introduced in [] b Xing and Shu. Other related work on well-balanced methods for the Euler equations with gravitational field can be found in [4,7,7]. All of the works mentioned above are finite difference or finite volume methods. During the past few decades, high order discontinuous Galerkin DG method has gained great attention in solving hperbolic conservation laws. DG method is a class of finite element methods using discontinuous piecewise polnomial space as the solution and test function spaces see [5] for a historic review. It combines advantages of both finite element and finite volume methods, and can achieve high order of accurac easil with the use of high order polnomials within each element. Several advantages of the DG method, including its accurac, high parallel efficienc, fleibilit for hp-adaptivit and arbitrar geometr and meshes, make it useful for a wide range of applications. The main objective of this paper is to develop high order accurate well-balanced Runge utta DG RDG methods for the isothermal hdrostatic balance of Euler equations with gravitation field. This will be the first paper to achieve this goal, to our best knowledge. To achieve well-balanced propert, we first rewrite the source terms in an equivalent special form using the hdrostatic balance solution.6. The are then discretized to be both high order accurate for general solutions and eactl well balanced with the pressure gradient at the equilibrium state. The proposed method is a generalization of well-balanced RDG methods [] designed for balancing the stead state solutions of the shallow water equations. This paper is organized as follows. In Sect., we first present the novel one-dimensional high order well-balanced DG method, which can preserve the isothermal hdrostatic balance solution.6 eactl, and at the same time is genuinel high order accurate for the general solutions. We then etend the proposed well-balanced method to multi-dimensional problems. Section contains etensive numerical simulation results to demonstrate the behavior of our well-balanced DG methods for one- and two-dimensional Euler equations under gravitational field, verifing high order accurac, the well-balanced propert, and good resolution for smooth and discontinuous solutions. Some conclusions are given in Sect. 4.

4 496 J Sci Comput 6 67:49 5 Well-Balanced RDG Methods In this section, we present high order well-balanced DG methods for the stead state solution satisfing.6. To better illustrate the ke well-balanced idea, we start b presenting numerical methods that balance a simplified version of the stead state.6, which takes the form of ρ = c ep g, u =, p = c ep g,. in conjunction with the linear gravitational potential filed, i.e., φ = g.. Etension to general stead state solution.6 will be discussed later in Sect... We will confine our discussion to one dimensional problem first, again for ease of presentation, and discuss the generalization to high dimensional case at the end of this section.. Notations We start b presenting the standard notations. We divide the interval I =[a, b] into N subintervals and denote the cells b =[ j, j+ ] for j =,...,N. The center of each cell is j = j + j+, and the mesh size is denoted b h j = j+ j, with h = ma j N h j being the maimal mesh size. The piecewise polnomial space Vh k is defined as the space of polnomials of degree up to k in each cell,thatis, } Vh {v k = : v P k, j =,,...,N.. Note that functions in Vh k are allowed to have discontinuities across element interfaces. For an unknown u, its numerical approimation in the DG methods is denoted b u h, which belongs to the finite element space Vh k.wedenotebu h + and u j+ h the limit j+ values of u h at j+ from the right cell + and from the left cell, respectivel. The usual notations [u h ]=u + h u h and {u h}= u+ h + u h are used to represent the jump and the average of the function u h at the element interfaces.. Well-Balanced Methods for the Simplified Stead State. Man well-balanced methods, including DG methods, have been designed for the shallow water equations. To achieve the well-balanced propert, the ke idea is to introduce a numerical discretization of the source term, which mimics the approimation of the flu term, so that the eact balance between the source term and the flu can be achieved at the stead state numericall. There eist two commonl used approaches in the literature to design well-balanced DG methods for the shallow water equations. The first approach [,]often rewrite the equations into an equivalent wa and introduce a non-standard discretization of the source term based on that. The second approach [,,4] emplos the idea of hdrostatic reconstruction [] to modif the approimation of numerical flu while keeping a simple source term approimation. Due to the nonlinear dependence of the isothermal stead state. on the eternal gravitational field φ, it is not eas to etend the hdrostatic reconstruction idea directl. Here, we would like to follow the first approach.

5 J Sci Comput 6 67: Reformulation of the Equation We first reformulate the original governing equations as follows ρ t + ρu =, ρu t + ρu + p = ρ epgep g, E t + E + pu = ρug,.4 where we replace ρg b ρ epgep g in the second equation, following the idea in our recent finite difference work []. B writing in this special form, we hope to create the derivative term in the source term, which can be treated in the similar wa as the flu term at the stead state. to achieve the well-balanced propert. We do not change the source term in the last equation, since the well-balanced propert for this equation can be easil obtained when u = at the stead state. For brevit, we rewrite the Eq..4 in a concise vector form U t + FU = S, where U = ρ, ρu, E T with the superscript T denoting the transpose, FU represents the phsical flu and S is the source term. Denote the DG approimation to the solution U b U h Vh k. The standard semi-discrete DG methods for.4 are defined as follows: for an test function v Vh k, U h is given b U h t vd FU h v d + ˆF j+ v ˆF I j+ j v + = Svd,.5 j j where F j+ = f U h, t, U j+ h +, t,.6 j+ and f a, a is a numerical flu. One eample is the simple La-Friedrichs flu f a, a = Fa + Fa αa a,.7 where α = ma λu with λu being the eigenvalues of the Jacobian F U, andthemaimum is taken over the whole region... Novel Source Term Approimation The standard DG methods.5 alone do not have the well-balanced propert. To preserve the stead state solution., we need to introduce a non-standard approimation to the source term integral. The function ep g appears in the source term S, and we introduce the notation b = ep g, for ease of presentation. We first decompose the integral of the source term in the second equation as

6 498 J Sci Comput 6 67:49 5 S vd = ρ epg ep g vd = = ρ j b v b b j j+ j+ ρ + b ρ j b vd. b j ρ b b vd.8 v + bv j d + j In Remark below, we provide some eplanations on wh such decomposition is needed for the purpose of well-balancedness, and introduce some other alternative equivalent decompositions. We now project b into the piecewise polnomial space Vh k, to obtain the polnomial b h, using the standard L projection. Our numerical approimation to the source term.8 takes the form of ρh S vd ρ h j b h b h j + ρ h j {b h } b h j b h vd j+ v {b j+ h } j v + b j h v d,.9 where ρ, b are replaced b the DG approimations ρ h, b h, and the boundar values of b h are replaced b the average of b h at the cell interface, denoted b {b h }, to be consistent with the numerical flu ˆF j±/. For the integral of the source term in the third equation, we simpl approimate it b S vd ρu h gvd,. where the standard quadrature rule is used to evaluate this integral... Well-Balanced Numerical Flues The last piece in designing well-balanced DG methods is to modif the numerical flu ˆF j+/. The term αa a in the La-Friedrichs flu.7 contributes to the numerical viscosit term, which is essential for this nonlinear conservation laws. However the will destro the well-balanced propert at the stead state. We propose to modif it as F j+/ = F where the coefficient α is defined as U h j+/, t + F U h + j+/, t α U h + j+/, t U h j+/, t,. b h + j+/ b h j+/ α = α ma b h,. to maintain enough artificial numerical viscosit. This modification does not affect the accurac, but at the stead state., the term U h /b h becomes a constant. Therefore, the effect

7 J Sci Comput 6 67: of these viscosit terms becomes zero and the numerical flu now reduces to a simple form F j+ = [ ] F U, t + F U +, t.. j+ j+..4 Well-Balanced Methods All these together lead to a well-balanced DG method for the Euler equations, as outlined in the following proposition. Proposition For the Euler equations. with the linear gravitational potential field., the semi-discrete DG methods.5, combined with.9,. and., are wellbalanced for the stead state solution.. Proof At the stead state., we have ρ h = cb h, u =, p h = cb h. Eas to observe that the well-balanced propert holds for the first and third equations, as both the flu and source term approimations in these equations become zero. For the momentum equation, we have ρ h j /b h j = ρ h /b h c, and the source term approimation.9 becomes S vd c {b h } j+ v {b j+ h } j v + b j h v d..4 Since u =, the flu term F = ρu + p reduces to p. Utilizing. and the equilibrium p h = cb h, its numerical approimation takes the form of ˆF j+ v ˆF j+ j v + F j v d = c{b h } j+ v c{b j+ h } j v + cb j h v d..5 We can conclude that the flu and source term approimations balance each other, which leads to the well-balanced propert of our methods. Remark The choice of ρ h j /b h j in.9 is not unique, and can be replace b an other term that can recover constant c at the stead state.6, for eample, ρ h j /b h j.we would like to comment that although ρ epg = c at the stead state., this equalit does not hold for the product of two polnomials which approimate these functions, i.e., ρ h epg h = c everwhere pointwise where epg h stands for the L projection of epg into the space V k h. Remark A straightforward wa to approimate the source term.8 b: ρ epg ep g vd I i = ρ i+/ v i+/ ρ i / + v i / + ρ ep g I i ep g v d,.6

8 5 J Sci Comput 6 67:49 5 and then change ρ, ep g to ρ h, b h, and replace the cell boundar value of ρ h b {ρ h }, seems to work b repeating the proof of Proposition. One main concern in taking this source term approimation is that the derivative of the unknown ρ appears in the source term approimation, and the flu {ρ h } is introduced at the cell interface to communicate with neighboring cells, which ma violate the La-Wendroff theorem, hence affect the convergence towards the weak solution when discontinuous solutions appear. For eample, when no source term presents, i.e., g =, the source term approimation.9 vanishes, while the approimation.6 does not if one replaces the cell boundar value of ρ h b {ρ h } in Temporal Discretization and Slope Limiter For the temporal discretization, high order total variation diminishing TVD Runge utta methods [5] can be used. In the numerical section of this paper, we appl the third order Runge utta methods: U h = Uh n + tf Uh n U h = 4 U n h + 4 U n+ h = U n h + U h + tf U h + tf U h U h,.7 with FU h being the spatial operator. When the solution contains discontinuities, slope limiter procedure is usuall needed for the DG methods. The are applied after each inner stage of the Runge utta methods. Man different choices of slope limiters have been presented in the literature. In this paper, we consider the total variation bounded TVB limiter presented in [6,4]. This limiter procedure itself might destro the well-balanced propert, and violate the eact preservation of the stead state.. Here we follow the idea presented in [4] and propose the following well-balanced wa to perform the TVB limiter. Usuall, when we perform the TVB limiter on the unknowns U h, it involves two steps. The first step is to check whether an limiting is needed in the cell based on the cell averages Ū h j, Ū h j± and U h j+/, U h + j /. If the answer is positive, the second step is to appl the TVB limiter on the variables U h in this cell. To present well-balanced slope limiter procedure, we propose to first check if the limiting is needed based on the cell averages U h /b h j, U h /b h j± and U h /b h j+/, U h/b h + j /. If one cell is determined as needing limiting, we appl the actual TVB limiter on U h as usual. Note that U h /b h becomes constant at the stead state.. When the limiting procedure is implemented this wa, if the stead state is reached, no cell will be flagged as requiring limiting, hence we do not appl an TVB limiter and therefore the well-balanced propert is maintained.. Well-Balanced Methods for the General Stead State.6 Well-balanced DG methods have been designed for the special stead state. in the previous subsection. In the subsection, we etend these methods to the more general stead state.6 with the gravitational field φ. We first rewrite the governing Eq..as

9 J Sci Comput 6 67: ρ t + ρu =, ρu t + ρu + p φ = ρ ep ep φ, E t + E + pu = ρug,.8 which is the analog of the Eq..4 for the special stead state.. The semi-discrete well-balanced DG methods still take the form of.5, but with a modified flu and source term approimations outlined below. For ease of presentation, we introduce the notation d = ep φ, and denote its projection in the space Vh k as d h. Following the same technique as stated above, we decompose the integral of the source term in the second equation as φ S vd = ρ ep ep φ ρ vd = d d vd = ρ j d v d + v + dv d j j+ j+ j j d + and approimate it b S vd + ρ h j {d h } d h j ρ d ρ j d j ρh ρ h j d h d h j j+ v d vd,.9 j+ d h vd {d h } j v + d j h v d,. where ρ, d are replaced b the DG approimations ρ h, d h, and the boundar values of d h are replaced b the cell average {d h }. The integral of the source term in the third equation is still approimated b., and the numerical flu takes the same form as in., with b h replaced b d h. Following the same proof, we can show that the semi-discrete DG methods.5, combined with.,. and. with b h replaced b d h, are well-balanced for the general stead state solution.6 with the gravitational field φ..4 Etension to Multi-dimensional Case In this subsection, we etend well-balanced DG methods to multi-dimensional Euler equations with the gravitational field φ. The governing equations have the following formulation ρ t + ρu =, ρu t + ρu u + pi d = ρ φ, E t + E + pu = ρu φ,.

10 5 J Sci Comput 6 67:49 5 where R l l =, is the spatial variable, ρ, u, p denote the densit, velocit, and pressure. E = ρ u + p/γ is the non-gravitational energ. The operators, and are the gradient, divergence and tensor product in R l, respectivel. Here we are interested in maintaining the stead state solution with the constant temperature T and the zero velocit, given b ρ = ρ ep φ, u =, p = ρ = ρ ep φ.. In the special case of linear gravitational potential field φ = g, the corresponding stead state solutions takes the form of ρ = ρ ep ρ g /p, u = v =, p = p ep ρ g /p. Let T τ be a famil of partitions of the computational domain parameterized b τ>. We do not specif the mesh element here. In two dimension, well-balanced DG methods proposed below work for both rectangular and triangular meshes. For an element T τ, we define τ := diam and τ := ma τ. For each edge e i T i =,...,m of,we τ denote the outward unit normal vector b ν i and the area of the element b. Let us denote the multi-dimensional Euler equations.b U t + FU = S, where U = ρ, ρu, E T, FU is the flu and S is the source term. The DG approimation U h belongs to the finite dimensional space Vτ k {w L ; w P k T τ },. where P k denotes the space of polnomials on the element with at most k-th degree. The semi-discrete DG method is given b m t Uw d FU wd + F e i ν i w ds = Sw d,.4 i= where w is a test function from the test space Vτ k. The numerical flu F is defined b F e i ν i U = F int i, U et i,ν i..5 where U int i and U et i are the approimations to the values on the edge e i obtained from the interior and the eterior of. The simple global La-Friedrichs flu takes the form of Fa, a,ν= [Fa ν + Fa ν αa a ]..6 We can etend the well-balanced DG methods designed in Sect.. to multiple space dimensions. Following the steps in one-dimensional case, we first introduce the notation d = ep φ,.7 and rewrite the source term in the momentum equation of.as φ ρ φ = ρ ep ep φ = ρ d..8 d e i

11 J Sci Comput 6 67: The well-balanced approimation to the integral of this source term follows an analogue of the decomposition.9, which leads to ρ S w d = d w d.9 d = ρ d m d int i= e i i ν i w ds d w d ρ + d ρ d d w d, where stands for the middle point or arbitrar point within the element that is eas to evaluate. We approimate the source term.9b ρ h S w d ρ h d d h d h h w d + ρ h m {d d h h }ν i w ds d h w d,. i= e i where ρ, d are replaced b the DG approimations ρ h, d h, and the boundar values of d h are replaced b the cell average {d h }. The last piece in designing the well-balanced DG methods is to replace the La-Friedrichs numerical flu.5.6 b: F e i ν i = F α U et i d h et i where the coefficient α is defined as U int i ν i + F U int d h i int i U et i ν i.. α = α ma d h,. to maintain enough artificial numerical viscosit. All these together lead to a well-balanced DG method for the Euler equations, as outlined in the following proposition. Proposition For the multi-dimensional Euler equations. with the gravitational potential field φ, the semi-discrete DG methods.4, combined with. and., are well-balanced for the stead state solution.. Numerical Results In this section, we carr out etensive one- and two-dimensional numerical eperiments to demonstrate the performance of the proposed well-balanced RDG methods. In all the computations, we use the third order TVD Runge utta methods.7, coupled with third order finite element DG methods i.e., k =. The CFL number is taken as.8.

12 54 J Sci Comput 6 67:49 5. n= n=.8 n= n= Densit Velocit n= n=. n= n= Energ.5 Pressure Fig. The numerical solutions of the shock tube problem under gravitational field in Sect.. at time t =.. Top left densit distribution, top right velocit distribution, bottom left energ distribution, bottom right pressure distribution. One-Dimensional Shock Tube Problem Under Gravitational Field In this standard Sod test, the discontinuous initial conditions are given b {,, if.5, ρ, v, p =.5,,. otherwise, on a unit computational domain [, ], with a constant gravitational field g = φ = acting in the negative direction. We compute this problem up to t =.. We present the numerical results compared with the reference solutions obtained with a much refined uniform cells in Fig.. Due to the presence of the gravitational force, the densit distribution is pulling towards the left direction, and negative velocit appears in some regions. B comparing the results in these figures, we can clearl observe that the numerical results capture sharp discontinuit transition even on a relativel coarse mesh with cells, and agree well with the reference solutions.

13 J Sci Comput 6 67: Table L errors for different precisions for the stead state solution. in Sect.. N Precision ρ ρu E Single.8E-7.E E-7 Double.76E-5.77E-5.4E-5 Single.E-7.4E-7 4.E-7 Double.99E-5.6E-5.84E-5. One-Dimensional Isothermal Equilibrium Solution In this test case, used in [,7,], we test the well-balanced propert of the proposed DG methods for an ideal gas with γ =.4 under the linear gravitational field φ = g =. The isothermal stead state solution is given b ρ = p = ep and u =,. which is in the form of the special stead state.. The computational domain is set as [, ]. We first show an eample to demonstrate the well-balanced propert of the proposed DG methods. The initial condition is taken as the stead state solution. which should be eactl preserved b an well-balanced method. In order to demonstrate that the stead state is indeed maintained up to the round-off error, we use single precision and double precision respectivel to perform the computation. We compute the solution until t = using both and uniform mesh cells, and present the L errors of numerical solutions in Table. It can be clearl observed that the numerical errors are all at the level of round-off error for different precisions, which verifies the desired well-balanced propert accordingl. Net, we demonstrate the advantage of well-balanced methods b simulating a small perturbation of the isothermal stead state solution.. We keep the densit and velocit, but modif the initial pressure state to p, t = = p + η ep.5, where η is a non-zero perturbation parameter. Two cases, η =. and η =., have been considered. In Fig., we present the pressure perturbations at t =.5 on a mesh with cells, and a reference solution obtained with a much refined cells. The initial pressure perturbation is also included as a dashed line. In addition, we run the same numerical test using the non-well-balanced DG methods, with a straightforward integration of the source term, and show their results in Fig. for comparison. It is obvious that the results of wellbalanced DG methods are in good agreement with the reference solutions for both cases, while non-well-balanced DG methods onl provide good results for the big perturbation, but fail to capture the small perturbation with cells. This demonstrates the importance of well-balanced methods in capturing small perturbations to equilibrium states.. One-Dimensional Gas Falling into a Fied Eternal Potential Net, we consider a more general gravitational field, which takes the sine wave form: φ = φ L π sin π L,

14 56 J Sci Comput 6 67:49 5 Pressure perturbation.5..5 Initial state n= n= non-wb Pressure perturbation.5. 5E-5 Initial state n= n= non-wb E Fig. The pressure perturbation of a hdrostatic solution in Sect... The result of the well-balanced method with and cells, and that of the non-well-balanced denoted b non-wb method with cells. Left η =., right η =. Table L errors for different precisions for the stead state solution. in Sect.. N Precision ρ ρu E Single.98E-7.86E-7.E-7 Double.9E-5 5.7E-6 8.8E-6 Single.4E-7.65E-7.5E-7 Double.49E E-6.E-5 where L is the computational domain length and φ is the amplitude. This test case was first considered in [6] and later used in [7,]. The general stead state takes the following form ρ = ρ ep φ, u = and p = ρ ep φ,. with a constant temperature T. We first verif the well-balanced propert of the the proposed DG methods. For an ideal gas with γ = 5/, the initial conditions are defined in. with parameters ρ =, R =, T =.6866, L = 64 and φ =.. We compute the eample up to t = 5 using both and uniform cells. We appl both single and double precisions, to carr out the computation. L errors of ρ, ρu and E are presented in Table, where we can clearl observe that the errors are all at the level of round-off error for different precisions. In the following test case, we impose a small perturbation to the stead state.6, and let the solution run for a long time. Eventuall, it will converge to an isothermal hdrostatic state. We would like to compare the performance of well-balanced and non-well-balanced methods for this test. We define the initial data as φ ρ = ρ ep, u =, p = ρ ep φ +. ep, with the same parameters used in the well-balanced test. We run the simulation for,, time steps with 64 uniform cells, and show the numerical results at the final time in Fig.. For comparison, we also plot the numerical results b the non-well-balanced DG methods. Eas to observe that the constant velocit and constant temperature distributions of the

15 J Sci Comput 6 67: Densit well-balanced non-wb Velocit 6E-5 4E-5 E-5 -E-5 well-balanced non-wb.8-4e E Pressure.9 well-balanced non-wb Temperature well-balanced non-wb Fig. The numerical solutions of well-balanced method solid line and non-well-balanced method square bo, denoted b non-wb for the convergence test in Sect.. after,, time steps. Top left densit distribution, top right velocit distribution, bottom left pressure distribution, bottom right temperature distribution equilibrium state are well-captured b the proposed well-balanced DG methods, while the non-well-balanced methods fail to achieve this..4 Two-Dimensional Accurac Test In this two-dimensional eample, we test the convergence rate of well-balanced DG methods for the Euler equations. with a linear gravitational field φ = φ =. For such linear field, a time-dependent eact solution has been proposed in [], which takes the form of ρ,, t = +.sinπ + tu + v, u,, t = u, v,, t = v, p,, t = p + tu + v +.cosπ + tu + v /π, on a square domain [, ] [, ]. The constants are set as u = v = andp = 4.5 in this test case. The eact solutions are taken as the boundar condition when needed. We run the simulation up to t =.. The L errors and orders of accurac are shown in Table. We can clearl observe that the epected high order accurac is achieved for the proposed well-balanced DG methods.

16 58 J Sci Comput 6 67:49 5 Table L errors and numerical orders of accurac for the eample in Sect..4 Cells ρ ρu ρv E L error Order L error Order L error Order L error Order 8 8.E-4.9E-4.9E-4.84E E-5..E-5.7.E-5.7.8E-5.6.4E-6.9.7E-6.7.7E-6.7.6E E E E E E-8.9.5E E E E E E E-9. Table 4 L errors for different precisions for the stead state solution. in Sect..5 Precision ρ ρu ρv E Single.5E-8.E-8.7E-8.5E-8 Double 8.7E-4 6.4E-4 8.5E-4.7E-4.5 Two-Dimensional Isothermal Equilibrium Solution We use this test case, taken from [], to demonstrate the well-balanced propert and the capacit of the proposed methods for capturing the small perturbation of an isothermal equilibrium solution in the two-dimensional case. Consider an ideal gas with γ =.4 and the linear gravitational field φ = φ = g. On a unit square domain, we are interested in the following isothermal equilibrium state ρ = ρ ep ρ g +, p u, = v, =, p, = p ep ρ g +,. p with ρ =., p = andg =. We first test the well-balanced propert b using this equilibrium state as the initial data. We compute the solution up to t = on a mesh with 5 5 uniform cells. In order to demonstrate that the stead state is indeed maintained up to round-off error, we use single precision and double precision to carr out the computation. The L errors of ρ, ρu, ρv and E are shown in Table 4, where the well-balanced propert can be easil observed. Net, we demonstrate the advantage of well-balanced methods b imposing a small perturbation to the pressure state of the isothermal equilibrium solution: p,, t = = p ep ρ g + + η ep ρ g. +., p p where η =. is a non-zero perturbation constant. We compute the eample with both wellbalanced DG methods and non-well-balanced methods with a straightforward calculation of the source term, up to t =.5 with 5 5 cells and simple transmissive boundar conditions. The contour plots of their pressure perturbation are shown in Fig. 4 and the D figures of the pressure perturbation are shown in Fig. 5. We also include the densit perturbations in Fig. 6. From these figures, we can observe that non-well-balanced DG

17 J Sci Comput 6 67: Fig. 4 The contours of the pressure perturbation of a two-dimensional hdrostatic solution in Sect..5 at time t =.5 with 5 5 cells. uniforml spaced contour lines from. to.. Left results based on well-balanced method, right results based on non-well-balanced method Z Z Y Y X X Pressureperturbation Pressureperturbation Fig. 5 The D figure of the pressure perturbation of a two-dimensional hdrostatic solution in Sect..5 at time t =.5 with 5 5 cells. Left results based on well-balanced method, right results based on non-well-balanced method Fig. 6 The contours of the densit perturbation of a two-dimensional hdrostatic solution in Sect..5 at time t =.5 with 5 5 cells. uniforml spaced contour lines from. to.. Left results based on well-balanced method, right results based on non-well-balanced method

18 5 J Sci Comput 6 67: Fig. 7 The contours of the pressure perturbation of a two-dimensional hdrostatic solution in Sect..5 at time t =.5 with cells. uniforml spaced contour lines from. to.. Left results based on well-balanced method, right results based on non-well-balanced method methods are not capable of capturing such small perturbation on the coarse mesh, while the well-balanced ones can resolve it ver well. At the end, we also run the both methods on a refined mesh with uniform cells, and show their pressure perturbation results in Fig. 7. The results of non-well-balanced DG methods are improving on the refined mesh, and the difference between well-balanced and non-well-balanced methods becomes smaller, which is what we epected..6 Two-Dimensional Eplosion Problem In this last test case, we test a two-dimensional circular eplosion problem following the setupin[4]. We consider an ideal gas γ =.4 under the linear gravitational field with φ =, φ = g =.8, and the initial conditions ρ,, t = =, u,, t = =, v,, t = =, p,, t = = g + {.5, if <.,, otherwise, on the computational domain [, ] [, ]. Simple transmissive boundar conditions are used in all directions. This test can also be viewed as a small perturbation of the stead state solution. We compute the solutions using both well-balanced and non-well-balanced DG methods, and compare their performance. Due to the circular pressure perturbation near the center of the domain, a shock wave will be developed, and propagate to the boundar. A uniform mesh of computational cells is used, and we perform the simulation until t =.4. In Figs. 8 and 9, we plot the densit ρ and velocit u + v of both well-balanced and non-well-balanced methods at times t =.,.8 and.4, where we can easil observe the big numerical error of non-well-balanced methods in the velocit plots. We would like to comment that although the underline stead state in this test does not have the form of., therefore our well-balanced methods are not designed to capture this stead state solution, but we can still observe the good behavior in capturing small perturbation of this equilibrium state.

19 J Sci Comput 6 67: Fig. 8 The contours of the densit and the velocit u + v of the eplosion problem in Sect..6 b well-balanced DG methods with cells at times t =.left, t =.8middleandt =.4right. Ten uniforml spaced contour lines from.9955 to for densit and ten uniforml spaced contour lines from.89 to.88 for velocit. Top densit, bottom velocit Fig. 9 The contours of the densit and the velocit u + v of the eplosion problem in Sect..6 b non-well-balanced DG methods with cells at times t =. left, t =.8 middle and t =.4 right. Ten uniforml spaced contour lines from.9955 to for densit and ten uniforml spaced contour lines from.89 to.88 for velocit. Top densit, bottom velocit

20 5 J Sci Comput 6 67: Concluding Remarks In this paper, we have constructed well-balanced DG methods for the isothermal equilibrium state solution of the Euler equations under static gravitational field. Special attention has been paid to the approimation of the source term, to achieve the well-balanced propert. We have demonstrated that the proposed DG methods can balance the general isothermal equilibrium state eactl, and at the same time maintain the high order accurac for the general solutions. Etensive numerical eamples are provided to demonstrate the well-balanced propert, accurac, and good resolution of the proposed numerical methods for both continuous and discontinuous solutions. Another interesting equilibrium state of the Euler equations is the isentropic hdrostatic state. How to design high order well-balanced methods for such equilibrium and more general stead state constitutes our future work. Acknowledgments The research of the first author is supported b the National Natural Science Foundation of P.R. China No. 54, 4 and the Project for Scientific Plan of Higher Education in Shandong Providence of P.R. China No. JLI8. This work was partiall performed at the State e Laborator of Science/Engineering Computing of P.R. China b virtue of the computational resources of Professor Li Yuan s group. The first author is also thankful to Professor Li Yuan for his kind invitation. The research of the second author is sponsored b NSF grant DMS-6454 and ORNL. The work was partiall performed at ORNL, which is managed b UT-Battelle, LLC, under Contract No. DE-AC5-OR75. References. Audusse, E., Bouchut, F., Bristeau, M.-O., lein, R., Perthame, B.: A fast and stable well-balanced scheme with hdrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 5, Bermudez, A., Vazquez, M.E.: Upwind methods for hperbolic conservation laws with source terms. Comput. Fluids, Botta, N., lein, R., Langenberg, S., Lützenkirchen, S.: Well-balanced finite volume methods for nearl hdrostatic flows. J. Comput. Phs. 96, Chertock, A., Cui, S., urganovz, A., Özcan, S.N., Tadmor, E.: Well-balanced central-upwind schemes for the Euler equations with gravitation. SIAM J. Sci. Comput., submitted 5. Cockburn, B., arniadakis, G., Shu, C.-W.: The development of discontinuous Galerkin methods. In: Cockburn, B., arniadakis, G., Shu C.-W. eds. Discontinuous Galerkin Methods: Theor, Computation and Applications. Lecture Notes in Computational Science and Engineering, Part I: Overview, vol., pp. 5. Springer 6. Cockburn, B., Shu, C.-W.: The Runge utta discontinuous Galerkin method for conservation laws V: multidimensional sstems. J. Comput. Phs. 4, Desveau, V., Zenk, M., Berthon, C., lingenberg, C.: A well-balanced scheme for the Euler equation with a gravitational potential. Finite Vol. Comple Appl. VII-Methods Theor. Asp. Springer Proc. Math. Stat. 77, Greenberg, J.M., LeRou, A.Y.: A well-balanced scheme for the numerical processing of source terms in hperbolic equations. SIAM J. Numer. Anal., appeli, R., Mishra, S.: Well-balanced schemes for the Euler equations with gravitation. J. Comput. Phs. 59, LeVeque, R.J.: Balancing source terms and flu gradients on high-resolution Godunov methods: the quasi-stead wave-propagation algorithm. J. Comput. Phs. 46, LeVeque, R.J., Bale, D.S.: Wave propagation methods for conservation laws with source terms. In: Proceedings of the 7th International Conference on Hperbolic Problems, pp Luo, J., Xu,., Liu, N.: A well-balanced smplecticit-preserving gas-kinetic scheme for hdrodnamic equations under gravitational field. SIAM J. Sci. Comput., Perthame, B., Simeoni, C.: A kinetic scheme for the Saint-Venant sstem with a source term. Calcolo 8, 4. Shu, C.-W.: TVB uniforml high-order schemes for conservation laws. Math. Comput. 49, 5 987

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