An efficient implementation of the divergence free constraint in a discontinuous Galerkin method for magnetohydrodynamics on unstructured meshes

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1 An efficient implementation of te divergence free constraint in a discontinuous Galerkin metod for magnetoydrodynamics on unstructured meses Cristian Klingenberg 1, Frank Pörner 1, and Yinua Xia 2, 1 Department of Matematics, University of Würzburg, Emil-Fiscer-Str. 40, Würzburg, Germany 2 Scool of Matematical Sciences, University of Science and Tecnology of Cina, Hefei, Anui , P.R. Cina Abstract. In tis paper we consider a discontinuous Galerkin discretization of te ideal magnetoydrodynamics (MHD) equations on unstructured meses, and te divergence free constraint ( B = 0) of its magnetic field B. We first present two approaces for maintaining te divergence free constraint, namely te approac of a locally divergence free projection inspired by locally divergence free elements [19], and anoter approac of te divergence cleaning tecnique given by Dedner et al. [15]. By combining tese two approaces we obtain a efficient metod at te almost same numerical cost. Finally, numerical experiments are performed to sow te capacity and efficiency of te sceme. AMS subject classifications: 65M12, 65M20, 65M60, 35L65. Key words: Ideal magnetoydrodynamics equations, discontinuous Galerkin metod, divergence free constraint, locally divergence free projection, divergence free cleaning tecnique. 1 Introduction Many pysical problems arising in a modeling process can be described by te magnetoydrodynamic (MHD) equations wic model te dynamics of electrically conducting fluids (e.g. plasma). At ig temperature due to ionization, all gases will cange to plasma. Terefore te MHD equations are important in many pysical applications. Corresponding autor. addresses: klingen@matematik.uni-wuerzburg.de (C. Klingenberg), frank.poerner@matematik.uni-wuerzburg.de (F. Pörner), yxia@ustc.edu.cnl (Y. Xia) ttp:// Global Science Preprint

2 2 Since te equations are igly nonlinear, analytic solutions are not available for te problem. We will focus on te numerical solutions of te ideal MHD equations, represented by an yperbolic conservation law. Also, an additional involution constraint B =0 for its magnetic field B is needed. Besides te numerical callenges wen solving suc a nonlinear system, tis constraint introduces additional difficulties. On te analytic level te involution constraint is always fulfilled, but numerical experiments indicate tat negligence in dealing wit te divergence constraint may lead to numerical instability and nonpysical solutions. Many numerical approaces ave been developed to solve conservation laws, e.g. finite volume metod (FVM) and finite element metod (FEM). Eac of tem as its advantages and disadvantages. We will focus on te discontinuous Galerkin (DG) metod, wic combines te flexibility of FEM wit te numerical fluxes from FVM. Te DG metod uses piecewise basis functions wic are discontinuous on te boundary of te elements. Normally tey are cosen to be piecewise polynomials. Due to te discontinuity of te basis function across cell boundaries, te sceme is very flexible compared to standard continuous finite element metod, suc as its ability to deal wit arbitrary unstructured grids wit anging nodes. Additionally, eac cell can ave its own polynomial degree independent of its neigbors. Furtermore, te DG sceme admits extremely local data structure (elements only communicate wit its immediate neigbors) wic leads to ig parallel efficiency. Te first discontinuous Galerkin metod was introduced by Reed and Hill [28] in Tey were interested in te neutron transport problem, i.e. a time independent linear yperbolic equation. In a series of papers [9 13] Cockburn and Su developed a framework to solve nonlinear time dependent problems, like Euler or MHD equations. For time integration tey are using explicit, nonlinearly stable ig order Runge-Kutta time discretizations [30] and for te spatial DG discretization tey apply exact or approximate Riemann solvers as interface numerical fluxes. To avoid numerical ascillations near socks tey suggested to apply total variation bounded nonlinear limiters [29]. Due to its good properties, suc as ig order accuracy and parallel efficiency, te discontinuous Galerkin metod as found rapid applications in diverse areas as aeroacoustics, electro-magnetism, gas dynamics and many more. Several numerical results establis te good convergence beavior and reveal an excellent level of details in numerical runs, see e.g. [1]. Wit regards to te numerical influence of te divergence constraint, several modifications and ideas ave been developed to satisfy te constraint or at least reduce te negative impact on te numerical solution. One of te first persons to notice te impact of nonzero divergence to te stability of te numerical scemes were Brackbill and Barnes. In [5], tey proposed a global projection metod to stabilize teir sceme. Te projection needs to solve a global elliptic partial differential equation at eac time step. Anoter approac is given by Powell [25, 27]. Te derivation of one-dimensional

3 3 fluxes (for a finite volume sceme) is based on te symmetrizable form of te MHD equations. In order to symmetrize MHD, we ave to add source terms proportional to B, see [26]. It was discovered later tat te robustness of a MHD code can be improved by adding tese so called Powell-source term, see [34]. In 2002 Dedner et al. [15] introduced teir yperbolic divergence cleaning tecnique wic as several advantages over te Powell-source term. Tey introduced a generalized Lagrangian multiplier to te MHD equations along wit some control parameters. Wile te Powell-source terms only propagates te divergence wit te fluid velocity, Dedner s metod allows it to control te speed of its movement and additionally smoots te divergence. Bot Powell s and Dedner s metods are in general not able to reduce te divergence to zero, like te projection metod of Brackbill and Barnes does, but tese metods are easily implemented into te DG framework. In [20, 21] Li et al. introduces a DG sceme wic is based on a local reconstruction and preserves te zero divergence of te magnetic field globally. Tis is done by first discretizing te normal components of te magnetic field along te edges of te elements, followed by a locally divergence free reconstruction inside te elements. Li et al. also introduced in [19] a DG sceme wic uses a locally divergence free basis for te magnetic field, wic is natural wit regard to te te DG-formalism. Anoter big class of numerical scemes preserving te divergence of te magnetic fields are scemes using constrained transport. Te main idea of te constrained transport approac is to use a special discretization of te evolution equation for te magnetic field. It uses a finite volume formalism to evolve te density, momentum, and energy, and exploits Stokes teorem and uses a face-averaged representation of te magnetic fields to enforce B =0. (see [33]). In tis article, we will discuss an implementation of te divergence free constraint on unstructured meses based on te DG metod. Using locally divergence free elements (see [19]) is simple, but it is not trivial to find a basis wic results in a well-beaved problem for ig polynomial order. Furtermore, we still need to control te global divergence. To acieve tis, we first evolve te magnetic field wit te standard piecewise polynomial basis, wic are not locally divergence free. Ten, we project te magnetic field into te subspace of locally divergence free vector fields as a pre-process (LDFP). Te result does not differ muc from te use of locally divergence free elements. And it also decouples te control of te divergence from te Runge-Kutta (RK) step. To control te global part of te divergence, we adopt te approac of Dedner et al. (DEDNER). Here we ave two parameters to control bot te advection speed and te smooting factor of te divergence. We implement tis approac as a post-process procedure. To control bot te local and te global part of te divergence we combine bot metods (ProjDED), were we apply te local projection as a pre-process step, and for te global part we apply te mixed approac of Dedner as a post-process step. Tis paper is organized as follows. In Section 2, we introduce te MHD equations

4 4 along wit its involution constraint and te DG-sceme for a two dimensional system of conservation laws. Section 3 is dedicated to te divergence constraint and its numerical treatment. We present te divergence cleaning tecnique and te locally divergence free basis projection. We combine tose two metods to obtain an efficient metod ProjDED illustrated in Section 4. Subsequently, te numerical verification and experimental validation are discussed wit several test problems: accuracy, Orszag-Tang, blast problem and te magnetic rotor. Finally, conclusions are drawn in Section 5. 2 Discontinuous Galerkin metod for te MHD equation 2.1 Te equations of ideal MHD Te dynamics of electrically conductive fluids is not fully described by te Euler equations. By coupling te Euler equations wit Maxwell s equations, we obtain te ideal MHD equations t ϱ + (ϱv) = 0, t (ϱv) + (ϱvv T +(p+ 12 ) BT B)I BB T = 0, t B + (vb T Bv T ) = 0, ( t E + (E+ p+ 1 ) 2 BT B)v (B T v)b = 0, (2.1) togeter wit te involution constraint B =0. (2.2) Here ϱ,v, B,E denotes te density, velocity, magnetic field and total energy, respectively. Te MHD equations are closed by connecting te pressure p wit te conserved variables p = (γ 1)(E 1 2 ϱvt v 1 2 BT B). Te ratio of specific eats is denoted wit γ >1. For a derivation of system (2.1) we refer to [32]. If te initial magnetic field satisfies (2.2), te exact solution of system (2.1) will fulfill (2.2) for all time. 2.2 Discontinuous Galerkin Finite Element Metod Te discontinuous Galerkin (DG) metod is a finite element metod (FEM) wit discontinuous basis functions wic uses te idea of a numerical flux taken from finite volume

5 5 teory to connect te elements. Te DG metod as become quite popular for yperbolic partial differential equations since it combines te flexibility of FEM wit te underlying pysical dynamics of te equations. We want to refer to te work of Su and Cockburn ( [9 14]) for a detailed analysis and for te numerical implementation to te book of Hestaven ( [18]). Here, we give a brief introduce of te DG metod for te following two dimensional conservation laws u t + f (u) x + g(u) =0, (2.3) y wit te flux functions f (u) = were u = (u 1,u 2 ) T is in te domain Ω R 2. ( ) ( ) f1 (u) g1 (u), g(u) =, f 2 (u) g 2 (u) Let T be a regular triangulation of our computational domain Ω wit sape-regular elements K, ten our test function space is given as ( ) v1 V = {v = : v i K P N (K); i =1,2; K T }, v 2 were P N (K) is te space of polynomial functions of degree at most N over K. Notice tat functions in V are allowed to be completely discontinuous across element interfaces. We multiply (2.3) wit a test function v V and integrate over K T to obtain [( ) ( ] f1 (u) f2 (u) 0= u t v dx v g 1 (u) 1 + ) v g 2 (u) 2 dx K K [( ) ] + f (u)n x +g(u)n y v ds, K were n = (n x,n y ) T is te outward pointing normal vector of K. We now replace u wit te approximate solution u V and te term f (u)n x +g(u)n y by a suitable numerical flux function (u ;n). In our simulations, we mainly use te local governmentrics flux (LLF). Te local Lax-Friedrics flux along some cell interface wit te outer normal n = (n x,n y ) T is given by LF (u ;n) := 1 ( ( f (u 2 )+ f (u+ )) n x + ( g(u )+g(u+ )) n y )+ α 2 (u +u+ ), were a superscript indicates te interior information of te cell and a superscript + refers to te exterior information. Te constant α is defined as ( ) α =max ( f u λ nx +gn y ). u

6 6 Here λ( ) indicates te eigenvalue of te Jacobi matrix, and te maximum is taken over te relevant range of u ±. It is enoug to test wit basis functions ψ = (ψ 1,ψ 2 ) T V and we end up wit te semi-discrete DG formulation for system (2.3) in two space dimensions: Find u V, suc tat K (u ) t ψ dx = [( ) f1 (u ) ψ g 1 (u ) 1 + K [ ] (u ;n) ψ ds, K ( ] f2 (u ) ) ψ g 2 (u ) 2 dx for all basis functions ψ V and for all K T. Te extension to more space dimensions and to systems wit more tan two components is straigt forward and is omitted ere. In tis paper we will focus on two space dimensions, yielding 6 components. Te metods can easily extend to tree space dimensions. For more details see [18]. By te metod of line, we formulate (2.4) as an ODE (for details see e.g. [18]) (2.4) ū t = L(ū,t), were L denotes te rigt anded side of (2.4) and ū is te coefficient vector of u V. To perform a time integration we consider an explicit Runge-Kutta sceme wit s stages v (0) = ū n i =1,...,s : v (i) = i 1 α ij v (j) +β ij tl(v (j),t n +γ j t) j=0 = v (s) ū n+1, (2.5) To be more precise, we use a strongly stability preserving RK-sceme of order 4 (see [16, 18] for details). Te time step is given as (compare to [14]) c t κ (N+1) 2, were κ (0,1] is a constant. Here is te minimal diameter of te triangles. Te wave speed c is obtained by computing te fast wave arising in te one-dimensional Riemann- Problem located at te interface of eac triangle. In our simulations we use κ = 0.2 independent of te polynomial degree N.

7 7 3 Handling te divergence of te magnetic field A suitable measurement for B is given by B div := B dx+ B ds, (3.1) K T K } {{ } local e E e } {{ } global were E contains all edges of our triangulation T and denotes te jump in te normal component across te edge e. Tis can eiter be derived by taking properties of te space H(div) into account ( [22]) or by a functional approac ( [8]). We see tat B div consists of two different parts wic we want to refer as local and global part. Our first aim is to analyze different tecniques and identify weter tey operate on te local or on te global part. Second we want to present a sceme wic works on bot parts. It is well known tat te standard DG-sceme does not preserve a divergence free magnetic field, wic is initially divergence free. Tis is not very surprising since te additional constraint is not used in te derivation of te sceme. Unfortunately te error of te divergence is accumulating in time due to numerical errors and may lead to numerical instabilities. Next we describe some metods tat deal wit tis negative influence. 3.1 Dedner s approac A well known metod is te divergence cleaning given by Dedner et al. ( [15]), were te divergence is coupled wit te evolution equation of te magnetic field wit a nonpysical variable ψ: B t + (vb T Bv T )+ ψ =0, D(ψ)+ B =0, and D is a linear differential operator. One possible coice is te yperbolic approac D(ψ) = 1 c 2 t ψ were ψ satisfies te wave equation 2 ttψ c 2 ψ =0. If we coose te parabolic approac D(ψ)= 1 c 2 p ψ, we obtain tat ψ satisfies te eat equation ψ t c 2 p ψ =0. Combing bot yields te mixed approac D(ψ) = 1 c 2 t ψ+ 1 ψ were ψ satisfies te telegrap c 2 p equation (3.2) 2 ttψ+ c2 c 2 t ψ c 2 ψ =0. (3.3) p

8 8 Te variable ψ can be seen as an abstract measurement for B. For te parabolic approac we obtain ψ(x,t) = c 2 p B(x,t), and for te yperbolic approac we obtain for ψ 0 (x) := B(x,0) ψ(x,t) = c 2 t 0 B(x,τ) dτ+ψ 0 (x). For te mixed approac an analytic solution is not possible. Based on te parabolic and yperbolic approac and by our assumption B(x,0) =0, we initialize ψ(x,0) =0. Te parameter c (0, ) influences te advection speed, at wic te error in te divergence is transported to te boundary of te domain. And te parameter c p (0, ) controls te smooting effect of te parabolic approac. For more details see [15] and for possible coices of c,c p see Altmann ( [1]). We coose in our simulations c = 7c v. To compute c v, we first compute te speeds c i of te fast waves arising on te boundaries of eac cell. Ten set c v = max i c i. Our aim is to coose c faster tan te maximum wave speed of system (2.1), to ave te divergence transported fast enoug to te boundary. To determine c p we follow te suggestion in [15] and take te ratio between yperbolic and parabolic effects to be constant c r = c2 p c (0, ). It is found by Dedner tat tis coice gives results wic are quite independent of te grid resolution and te sceme used. Furtermore tey suggested te value c r = 0.18, wic is a good compromise between damping and yperbolic transport. If more damping is forced, we lose te effect of te yperbolic transport of te error in te divergence (see [1]). For te one-dimensional MHD system and under some simplifying assumptions, Dedner proved tat B gets transported out of te domain wen te yperbolic approac is used. If te parabolic approac is used, we obtain a smooting on B. For te mixed approac bot properties are obtained. Tis metod is also very effective in more tan one space dimension. We solve te system (3.2) in an operator split approac. First, we evolve te magnetic equation B t + (vb T Bv T ) = 0 in (2.1). It follows by solving te system of Dedner s yperbolic approac B t + ψ =0, t ψ+c 2 B =0, (3.4) wit a DG sceme. We apply te DG formalism to (3.4) wit te Riemann solver suggested in [15]. In te computation of te Riemann solver for (3.4), we need te value of ψ

9 9 and B n at bot sides of te cell boundary. Terefore, te yperbolic divergence cleaning works on te global part of our measurement of te divergence. If we want to use te mixed approac, we ave to take te parabolic source term from te equation (3.3) into account. Again, in te operator splitting approac we only need to solve te additional ordinary differential equation t ψ = c2 c 2 ψ. p Tis can be done exactly ψ n+1 =exp ( t n c 2 c 2 p ) ψ n. (3.5) were ψ n is te solution to system (3.4). For te yperbolic approac, it can be sown (see [15]) tat B gets transported to te boundary. For te parabolic approac, te damping is forced on B, ence tis approac works on te local part of B. We see tat te mixed approac operates on te local and on te global part of te divergence. An overview of te implementation is sown in algoritm Locally divergence free basis To deal wit te local part, anoter possibility is to modify te test function space V to guarantee tat te numerical solution is locally divergence free. Hence use V := {v ( L 2 (Ω) ) ( 6 6 : v K P (K)) N (v4,v 5 ) K =0, K T } were v 4 and v 5 denote te x and y component of te two dimensional magnetic field respectively. For MHD suc an approac is made in [19]. To construct a basis of V we define te test-function space for te magnetic field ( 2 LDF N (K) := {p P (K)) N : p =0}. In te following we will sow, tat it is not easy to find a suitable basis of V. First we will concentrate on finding a basis of te space LDF N (K). Lemma 3.1. Te dimension of te space LDF N (K) is given by 1 2 (N+2)(N+3) 1. Furtermore a basis of LDF N (K) can be obtained by taking te curl of basis function of te space P N+1 (K).

10 10 curl of monomials GL N basis condition of mass matrix polynomial approximation order polynomial approximation order Figure 1: Reciprocal condition number for te mass matrix for two different bases of LDF N (K) on a reference triangle. A proof can be found in [19] or [6]. Tis allows us to construct a basis, e.g. if we take te curl of te monomial basis of P N+1 (K). To avoid te poor condition of te resulting mass-matrix (Figure 1), we take te Gauss-Lobatto points into account, to opefully adapt te stability and approximation property of te Lagrange polynomials. Denote {p 1,...,p s } te Gauss-Lobatto points of order N+1 in te element K. Let gl i P N+1 (K) te unique functions satisfying gl i (p j ) = δ ij i,j =1,...,s. By Lemma 3.1, te set GL N (K) := { gl 1,..., gl s 1 } is a basis of LDF N (K). In our simulations we observed tat tis basis yields good approximation results and at te same time an acceptable condition for te mass matrix, see Figure 1. Note tat tis metod (we will refer to it as LDFB, see Figure 2) only controls te local part of te divergence. Tis is due to te fact, tat te basis functions are cosen independent from te neigbouring cells. In addition, it could be easy to implement a projection step into an existing code, tan to cange te underlying basis functions ( P N (K) ) 2. Terefore, we use a local L 2 projection from ( P N (K) ) 2 to LDF N (K) instead of canging te basis, wic can be done nearly witout additional numerical expense. Note tat te projection is decoupled from

11 11 te DG-discretization. Tus we can use te basis by applying Gram-Scmidt procedure to te monomial basis of LDF N (K). First let P K be te L 2 projection in te element K, ( 2 P K : P (K)) N LDF N (K) Now define te local operator PK ldf : ( P N (K) ) 8 ( P N (K) ) 8 troug q = P ldf K (p) : q i = p i, i =4,5and(q 4,q 5 ) = P K ((p 4,p 5 )) Te global operator P ldf : V V is now defined q = P ldf (p) : q K = P ldf K (p K), K T. In Figure 3 we compare two different approaces. In te first approac (LDFB), we adopt te test function space V in te DG-discretization. In te second approac (LDFP), we use V as test function space and applying P ldf every time before computing te Runge-Kutta step. Te results do not differ muc. We terefore suggest a ybridized metod ProjDED presented in Figure 2. Also a comparison between te different metods is provided in Figure 2. For te description of te oter scemes see Section 4. In te ybridized metod ProjDED, te projection is applied before te RK-step in order to work as a pre-smooter for te divergence. Dedner s mixed approac is used as a post-process to deal wit te global part of te divergence and te additional local error introduced by te RK-step. Since te operator P K is a projection between finite dimensional spaces te numerical for applying P ldf cost are negligible (see also Table 2). In te following lemma we sow tat te projection is compatible wit a positivity preserving metod, as long as we use positivity preserving limiter. Lemma 3.2. Let q = (ϱ,v,b,e) (L 2 (K)) 6 suc tat ϱ > 0 and p > 0 olds. Ten te operator P ldf defined above preserves positivity of te density and pressure in te means, meaning for q := ( ϱ,ṽ, B,Ẽ) := P ld f (q) we obtain p dx >0 and ϱ dx >0. K K Proof. Wit P K 1 we obtain B K B K. Hence, we get te positivity of te pressure

12 12 in te mean K p dx = = K K K (γ 1)(E 1 2 ϱv2 1 2 B 2 ) dx (γ 1)(E 1 2 ϱv2 1 2 B2 ) dx p dx >0. Meanwile, te density is not canged by P ldf, yielding ϱ = ϱ. Remark 3.1. Lemma 3.2 sows tat te projection can be included into an existing positivity preserving sceme, as long as a positivity preserving limiter is applied after eac projection. Here we refer te reader to te results obtained for MHD-DG given by Li ( [7]). For some positivity preserving limiter we refer to [31, 35]. 4 Details of te implementation Te aim of tis capter is to give some insigt into our implementation. Te step to compute te solution at te next time level is divided into tree levels: Te pre-process level, te DG-Runge-Kutta step and and a post-process level. In te pre- and post-process level we apply some metods to deal wit te error in te divergence. We only describe te ProjDED metod, since te oters can be described similar. Due to te operator splitting approac for DEDNER te value ψ is treated after we performed a DG-discretization to (2.1), so it beaves like a post-processing operation.

13 13 Initialize q 0 := ( ϱ 0 v 0 B 0 E) 0 V ψ 0 :=0 k :=0 wile Breaking condition is not satisfied do pre-process q k+1/3 := P ld f (q k ) DG-Runge-Kutta Determine t k and apply SSP-RK of order 4 wit initial data q k+1/3 to te DG discretization (LLF flux) of te MHD equations (2.1) and obtain q k+2/3 post-process Apply SSP-RK of order 4 wit initial data B k+2/3 and ψ k to te DG discretization (flux given in [15]) of te system B t + ψ =0 t ψ+c 2 B =0 to obtain B k+1 and ψ k+1/2. Now apply operator splitting for te source term coming from te parabolic approac ( ) ψ k+1 := exp c2 c 2 t k ψ k+1/2 p end ( Set q k+1 := ϱ k+2/3 Set k k+1 v k+2/3 B k+1 ) E k+2/3 Algoritm 1: ProjDED. For clarity we dropped te use of a limiter.

14 14 LDFB LDFP DEDNER ProjDED Pre-Process Apply operator Pre-Process Apply operator P ldf P ldf DG-RK-step Use V as testfunction space DG-RK-step Use V as testfunction space DG-RK-step Use V as testfunction space DG-RK-step Use V as testfunction space Post-Process Apply Dedner div. cleaning Post-Process Apply Dedner div. cleaning Figure 2: Overview over te different scemes used. We separate eac DG-RK step into a pre-process step, te actual DG-RK-step and a post-process step. 5 Numerical experiments Te scemes we are using in te numerical tests are te following. Te BASE sceme is just te DG sceme wit test function space V. Te LDFB (locally divergence free basis) sceme is te DG sceme wit te test function space V. If we use instead a locally divergence free projection as a pre-process step for te BASE sceme, we call te sceme LDFP. In te DEDNER sceme, te mixed approac of Dedner is applied. Te implementation is given similar as in algoritm 1. Hence we interpret it as a post-process metod. And ProjDED is te combined metod of LDFP and DEDNER described above (see figure 2 and algoritm 1). Unless oterwise stated te local Lax-Friedrics (LLF) flux is used. To sow te impact of te divergence, we compute te numerical convergence rates for te genuinely two dimensional vortex problem given in [19, Section 3.2.1]. We only give te results for te density in Table 1, since for te oter components te results look te same. We see tat te error in te divergence decreases significantly te numerical order of convergence. It can be seen tat our ybrid metod is not destroying te numerical convergence order. Furtermore we encounter a strange beavior of our convergence order. For even polynomial order we get order N+1/2 but for uneven polynomial order we obtain te optimal rate of N+1. In te ydrodynamic case we do not discover suc a beavior. Te different beavior of te L 2 -rate of convergence for even and uneven polynomial approximations was also discovered in [17] and [23]. If we take a look at te computational time needed for te local projection metod we define one complete time-step in te 4t order SSP-RK witout any modifications to be 100%. In Table 2 we compare te time needed for one projection step compared wit one

15 15 ϱ N=1 sceme BASE LDFB LDFP DEDNER ProjDED /0 L 2 -error order L 2 -error order L 2 -error order L 2 -error order L 2 -error order e e e e e e e e e e e e e e e e e e e e N= e e e e e e e e e e e e e e e e e e e e N= e e e e e e e e e e e e e e e e e e e e N= e e e e e e e e e e e e e e e e e e e e Table 1: Numerical order of convergence for te density of te vortex problem

16 16 grid size 8x8 16x16 32x32 64x64 128x128 N=1 3.7% 1.8% 1.1% 0.4% 0.2% N=2 3.2% 1.6% 1.0% 0.3% 0.2% N=3 2.0% 1.5% 0.7% 0.2% 0.2% N=4 2.0% 1.5% 0.5% 0.2% 0.2% N=5 2.0% 1.4% 0.5% 0.2% 0.3% N=6 2.1% 1.4% 0.4% 0.3% 0.3% N=7 2.0% 1.3% 0.3% 0.3% 0.3% N=8 2.0% 1.1% 0.3% 0.3% 0.3% Table 2: Computational time needed for one projection step in relation to one time-step wit 4t order SSP-RK for different mes sizes and polynomial orders. step in te SSP-RK. We furtermore test our numerical sceme wit 3 different test problems: Orszag- Tang ( [24]), blast problem [2] and te magnetic rotor ( [33]). We observed tat ProjDED runs stable and te results are in good agreement wit results given in te literature. In Figure 4-6, we compare te divergence B div for different numerical metods for tis problems. Te combined metod ProjDED gives good result in tese tests. In te computation, triangles are used. To sow te difference between te sceme DEDNER and ProjDED, we also plot te relative error div(dedner) div(projded), (5.1) div(dedner) were div(dedner) denotes B div wen te algoritm DEDNER is used. Te results are given in figure 7 and we see tat ProjDED is about 25% more efficient tan te Dedner sceme in our test simulations. Note tat bot scemes ave almost te same numerical effort. We also tested te 3-wave solver as a numerical flux specially designed for MHD by Klingenberg ( [3, 4]). Tis flux takes te divergence constraint into account. We expect for te combination of te 3-wave solver and our ybrid metod ProjDED an improvement in our numerical sceme bot in resolution of details for long time simulations and a decrease in te divergence compared to LLF. Bot results are confirmed - see Figure 8. 6 Conclusion We ave presented four metods to deal wit te error of te divergence of te magnetic field and compared teir efficiency on several test problems. Te advantage of LDFB is tat te resulting magnetic field is locally divergence free for all time. Tis is guaranteed by te sceme itself, we do not need any pre- or post-process operators.

17 Base LDFP LDFB Figure 3: Measurement of te divergence for Orszag-Tang vortex problem. Te results of LFDB and LDFP do not differ muc BASE LDFP DEDNER ProjDED Figure 4: Divergence error for different scemes for te Orszag-Tang vortex problem

18 BASE LDFP DEDNER ProjDED Figure 5: Divergence error for different scemes for te blast problem BASE LDFP DEDNER ProjDED Figure 6: Divergence error for different scemes for te magnetic rotor problem

19 Figure 7: Relative error (5.1) for different test problems. Top: Orszag-Tang vortex problem, Middle: Blast problem, Bottom: Magnetic rotor

20 LLF 3-wave Figure 8: Demonstration of te impact of different numerical fluxes bot on te level of resolution for long time computations and on te divergence. Top: k Bkdiv for te LLF-flux and for te 3-wave solver. Bottom: Grayscale image of te density of Orszag-Tang vortex problem at time t = 6 wit different numerical flux functions. Left: local Lax-Friedrics, Rigt: 3-wave solver. Bot computations are done on triangles wit linear polynomials, TVB-limiter and LDFP.

21 21 In LDFP, we project te magnetic field into te subspace of locally divergence free vector fields. Te results do not differ from tose obtained by LDFB and we ave decoupled te control of te divergence from te RK-step. In fact, if we use te projection before eac stage in te RK-step, bot scemes yield almost te same results. Note tat LDFB and LDFP only control te local part of te divergence. Tis is due to te fact tat te basis functions are cosen independently from te neigbouring cells. A very effective way of controlling te global error in te divergence is te approac of Dedner et al. (DEDNER), were we ave two parameters to control bot te advection speed and te smooting factor of te divergence, but coosing tese parameters is not trivial. To control bot te local and te global part of te divergence, we developed te sceme ProjDED as a combination of te local projection as a pre-process step, and te mixed approac of Dedner et al. as a post-process step. If te numerical metod used is positivity preserving, te combination of te projection is still positivity preserving. Tus, te projection does not introduce additional instabilities regarding positivity. We see tat all of tese metods stabilize te DG metod in te sense tat te error in te divergence stays small or at least is significant smaller tan te solution of BASE. As a result, we avoid non-pysical oscillations leading to a more stable sceme. If we look at te relation of computational costs and efficiency, LDFP clearly wins. But since we are interested in getting B close to zero, it is wort to implement DEDNER. As already mentioned, te numerical costs of LDFP are negligible. ProjDED is as costly as te Dedner approac. As a result we conclude tat ProjDED is a suitable and robust metod to deal wit te negative influence of a non-vanising divergence of te magnetic field. Acknowledgments Researc of Y. Xia is supported by te NSFC grants No , No References [1] C. Altmann. Explicit Discontinuous Galerkin Metods for Magnetoydrodynamics. PD tesis, University of Suttgart, [2] D.S. Balsara and D.S. Spicer. A staggered mes algoritm using ig order godunov fluxes to ensure solenoidal magnetic fields in magnetoydrodynamic simulations. Journal of Computational Pysics, 149: , [3] F. Boucut, C. Klingenberg, and K. Waagan. A multiwave approximate rieman solver for ideal md based on relaxation. i: teoretical framework. Numerisce Matematik, 108:7 42, 2007.

22 22 [4] F. Boucut, C. Klingenberg, and K. Waagan. A multiwave approximate rieman solver for ideal md based on relaxation. ii: numerical implementation wit 3 and 5 waves. Numerisce Matematik, 115: , [5] J. U. Brackbill and D.C. Barnes. Te effect of nonzero b on te numerical solution of te magnetoydrodynamic equations. Journal of Computational Pysics, 35: , [6] F. Brezzi, J. Douglas, and L.D. Marini. Two families of mixed finite elements for second order elliptic problems. Numerical Matematics, 47: , [7] Y. Ceng, F. Li, J. Qiu, and L. Xu. Positivity-preserving dg and central dg metods for ideal md equations. Journal of Computational Pysics, 238: , [8] B. Cockburn, F. Li, and C.-W. Su. Locally divergence-free discontinuous galerkin metods for te maxwell equations. Journal of Computational Pysics, 194: , [9] B. Cockburn and C.-W. Su. Tvb runge-kutta local projection discontinuous galerkin finite element metod for conservation laws ii: general framework. Matematics of Computation, 52: , [10] B. Cockburn and C.-W. Su. Tvb runge-kutta local projection discontinuous galerkin finite element metod for conservation laws iii: One-dimensional systems. Journal of Computational Pysics, 84:90 113, [11] B. Cockburn and C.-W. Su. Te runge-kutta local projection discontinuous galerkin finite element metod for conservation laws iv: Te multidimensional case. Matematics of Computation, 54: , [12] B. Cockburn and C.-W. Su. Te runge-kutta local projection p 1 -discontinuous-galerkin finite element metod for scalar conservation laws. Matematical Modelling and Numerical Analysis, 25: , [13] B. Cockburn and C.-W. Su. Te runge-kutta discontinuous galerkin metod for conservation laws v: Multidimensional systems. Journal of Computational Pysics, 141: , [14] B. Cockburn and C.-W. Su. Runge-kutta discontinuous galerkin metods for convectiondominated problems. Journal of Scientific Computing, 16: , [15] A. Dedner, F. Kemm, D. Kroener, T. Scnitzer C.-D. Munz, and M. Wesenberg. Hyperbolic divergence cleaning for te md equations. Journal of Computational Pysics, 175: , [16] S. Gottlieb, C.-W. Su, and E. Tadmor. Strong stability preserving ig order time discretization metod. SIAM Rev., 43:89 112, [17] J. Guzmán and B. Rivière. Sub-optimal convergence of non-symmetric discontinuous galerkin metods for odd polynomial approximations. J. Sci. Comp., 40: , [18] J. S. Hestaven and T. Warburton. Nodal Discontinuous Galerkin Metods. Springer Verlag, [19] F. Li and C.-W. Su. Locally divergence-free discontinuous galerkin metods for md equations. Journal of Scientific Computing, 22-23: , [20] F. Li and L. Xu. Arbitrary order exactly divergence-free central discontinuous galerkin metods for ideal md equations. Journal of Computational Pysics, 231: , [21] F. Li, L. Xu, and S. Yakovlev. Central discontinuous galerkin metods for ideal md equations wit te exactly divergence-free magnetic field. Journal of Computational Pysics, 230: , [22] J.C. Nédélec. A new family of mixed finite elements in r3. Numerical Matematics, 50:57 81, [23] J. T. Oden, I. Babuska, and C. E. Baumann. A discontinuous p finite element metod for diffusion problems. Journal of Computational Pysics, 146: , 1998.

23 [24] S. Orszag and C.-M. Tang. Small-scale structure of two-dimensional magnetoydrodynamic turbulence. Journal of Fluid Mecanics, 90: , [25] K. G. Powell. An approximate riemann solver for magnetoydrodynamics (tat works in more tan one dimension). ICASE-Report, 94-24, [26] K. G. Powell, P. L. Row, R. S. Myong, T. Gombosi, and D. De Zeeuw. Te symmetric form of magnetoydrodynamics equation. Numer. Metods Mec. Contin. Media, 1:26, [27] K. G. Powell, P. L. Row, R. S. Myong, T. Gombosi, and D. De Zeeuw. An upwind sceme for magnetoydrodynamics. Numerical Metods for Fluid Dynamics, V:163, [28] W. H. Reed and T. R. Hill. Triangular mes metods for te neutron transport equation. Tec. Report LA-UR , Los Alamos Scientific Laboratory, [29] C.-W. Su. Tvb uniformly ig-order scemes for conservation laws. Matematics of Computation, 49: , [30] C.-W. Su and S. Oser. Efficient implementation of essentially non-oscillatory soccapturing scemes. Journal of Computational Pysics, 77: , [31] C.-W. Su and X. Zang. On positivity-preserving ig order discontinuous galerkin scemes for compressible euler equations on rectangular meses. Journal of Computational Pysics, 229: , [32] M. Torrilon. Zur Numerik der idealen Magnetoydrodynamik. PD tesis, Eidgenssisce Tecnisce Hocscule Zric, [33] G. Tót. Te b = 0 constraint in sock-capturing magnetoydrodynamics codes. Journal of Computational Pysics, 161:605, [34] K. Waagan, C. Klingenberg, and C. Federrat. A robust numerical sceme for igly compressible magnetoydrodynamics: Nonlinear stability, implementation and tests. Journal of Computational Pysics, 230: , [35] X. Zang, Y. Xia, and C.-W. Su. Maximum-principle-satisfying and positivity-preserving ig order discontinuous galerkin scemes for conservation laws on triangular meses. Journal of Scientific Computing, 50(1):29 62,