A MULTIWAVE APPROXIMATE RIEMANN SOLVER FOR IDEAL MHD BASED ON RELAXATION II - NUMERICAL IMPLEMENTATION WITH 3 AND 5 WAVES

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1 A MULTIWAVE APPROXIMATE RIEMANN SOLVER FOR IDEAL MHD BASED ON RELAXATION II - NUMERICAL IMPLEMENTATION WITH 3 AND 5 WAVES FRANÇOIS BOUCHUT CHRISTIAN KLINGENBERG 2 KNUT WAAGAN 3 Abstract. In the first part of this work [5] we introduced an approimate Riemann solver for one-dimensional ideal MHD derived from a relaation system. We gave sufficient conditions for the solver to satisfy discrete entropy inequalities and to preserve positivity of density and internal energy. In this paper we consider the practical implementation and derive eplicit wave speed estimates satisfying the stability conditions of [5]. We present a 3-wave solver that well resolves fast waves and material contacts and a 5-wave solver that accurately resolves the cases when two eigenvalues coincide. A full 7-wave solver which is highly accurate on all types of waves will be described in a follow-up paper. We test the solvers on one-dimensional shock tube data and smooth shear waves... Introduction The equations for ideal MHD in one dimension are t + u = u t + u 2 + p + 2 B 2 2 B2 = 0 u t + uu B B = E t + [E + p + 2 B 2 2 B2 u B B u ] = 0 B t + B u B u = 0. The state variables are the mass density the pressure p the velocity split into its longitudinal and transverse components u and u and the magnetic field similarly into B and B. Hence u and B are two-dimensional vectors. Since the divergence of the magnetic field is zero at all times we take B constant for one-dimensional data but that restriction may be relaed. Finally there is the total energy E.6 E = 2 u2 + u 2 + e + 2 B2 + B 2 with e denoting the specific internal energy. The system is closed by an equation of state p = p e. Thermodynamical considerations leads to the assumption of eistence of a specific physical entropy s = s e that satisfies.7 de + p d = Tds for some temperature T e > 0. To ensure the hyperbolicity of.-.5 we assume that p.8 p > 0 Date: May Département de Mathématiques et Applications CNRS & Ecole Normale Supérieure 45 rue d Ulm Paris cede 05 - France. Francois.Bouchut@ens.fr. 2 Departement of Mathematics Würzburg University Am Hubland Würzburg Germany. klingenberg@mathematik.uni-wuerzburg.de. 3 High Altitude Observatory National Center for Atmospheric Research P.O. Bo 3000 Boulder Colorado USA. knut.waagan@cma.uio.no. s

2 2 FRANÇOIS BOUCHUT CHRISTIAN KLINGENBERG 2 KNUT WAAGAN 3 where the subscript s means that the partial derivative is taken with s constant. We shall also make the classical assumption that.9 s is a conve function of e. This is a follow-up paper to [5] where we proposed an approimate Riemann solver for.-.5. It is based on a relaation approimation which generalized the Suliciu relaation approach for the Euler equations see [2] and [3]. In section. of [5] we described Godunov schemes and the idea of approimate Riemann solvers. We then introduced discrete entropy inequalities as a stability constraint which can be seen also as a way to numerically impose the second law of thermodynamics. A second important stability criterion is the positivity of density and internal energy. In section.2 we discussed positivity and entropy inequalities in the contet of relaation systems. This was eemplified by the Suliciu relaation system for the Euler equations that our relaation system for.-.5 generalizes. In the net section we recall the results of [5]. In Section 3 we describe our approimate Riemann solvers with 3 and 5 waves that are special cases of our underlying approach. We give eplicit formulas for wave speeds that ensure entropy inequalities and positivity in both the 3-wave and the 5-wave case. These formulas are etensions of the eplicit signal speeds for the HLLC solver introduced in [3]. Finally we run some numerical tests. Our solvers are compared against each other and against the Roe solver the HLL solver and the 5-wave solver of []. At the end we summarize the results. In a follow-up paper [4] we treat the case of the full 7-wave solver. In multidimensional simulations one-dimensional solvers are commonly used as building blocks. In the case of ideal MHD the constraint that divb = 0 is an additional challenge. We describe in an appendi how Powell s idea of etending.-.5 see [2] can be easily incorporated into our relaation approach. Other methods used in the multidimensional finite volume setting should in principle be able to use our one-dimensional solver Relaation system and approimate Riemann solver In [5] we introduced the relaation system t + u = 0 u t + u 2 + π = 0 u t + uu + π = 0 E t + [E + πu + π u ] = 0 B t + B u B u = 0 with still.6 and where the relaation pressures π and π evolve according to π t + πu + b 2 + c 2 b u c a b u = 0 π t + π u c a b u + c 2 a u = 0. The parameters c a 0 c b 0 and b R 2 play the role of approimations of B p and signb B respectively. Indeed c a c b b are not taken constant but are evolved with 2.8 c a t + uc a = 0 c b t + uc b = 0 b t + ub = 0. The eigenvalues of the system are u u cs u ca and u c f where c 2 s = c 2 b 2 + c2 a + b 2 c 2 b + c2 a + b 2 2 4c 2 a c2 b 2.9 c 2 f = c 2 b 2 + c2 a + b 2 + c 2 b + c2 a + b 2 2 4c 2 a c2 b u having multiplicity 8. All are linearly degenerate. Note that c s c a c f c s c b c f and that the eigenvalues of equal the eigenvalues of.-.5 whenever c a = B c b = p and b = signb B. However in order to simplify we shall make here different

3 APPROXIMATE RIEMANN SOLVER FOR IDEAL MHD 3 choices leading to a solver with 3 waves or 5 waves instead of 7 waves. The full motivation for the relaation system is given in Section 2 of [5]. The approimate Riemann solver associated to the above relaation system is a function R/t U l U r where U stands for the MHD variable U = u u E B. It is obtained by solving the Riemann problem for and dropping the etra components π π c a c b b. Initially this Riemann problem starts with the relaation pressures at equilibrium 2.0 π = p + 2 B 2 2 B2 and π = B B. The signal speeds c a c b b have to be specified initially on the left and on the right i.e. one has to give values for 2. c bl c br c al c ar b l b r. This choice is the key issue for stability and accuracy. This approimate Riemann solver is consistent with.-.5 and conservative whatever is the choice of these signal speeds see [3] for the precise meaning of this. Our relaation system generalizes the Suliciu relaation system for gas dynamics hence since the HLLC solver is associated to the Suliciu system our approimate Riemann solver is an etension of the HLLC solver to MHD. If the initial data U l U r consist of a single material contact discontinuity the approimate Riemann solver gives the eact solution to.-.5 π and π remaining at equilibrium. Isolated Alfven contact discontinuities are as well eactly resolved under some additional conditions specified in [5]. These additional conditions cannot be satisfied for the 3-wave or 5-wave solvers considered here. As usual an approimate Riemann solver leads to a conservative scheme 2.2 U n+ i = Ui n t F n i+ F n 2 i 2 where the inde i refers to the cell the inde n to time and 2.3 F n i+/2 = FUn i U n i+ with FU l U r the numerical flu. In our case since our solver R/t U l U r comes from the eact solution to which is conservative in U it is given by the eact flu of evaluated at /t = FU l U r = u u 2 + π uu + π E + πu + π u B u B u /t=0. The CFL-condition for this scheme is 2.5 ma u l c fl u r + c fr t C l r for some CFL-number C. A value C = /2 ensures that the waves emerging from the cell interfaces do not interact. However it is common in practice to use C = for the first-order scheme. Since all characteristic fields of are linearly degenerate the Riemann problem is much easier to solve than for the original MHD system. Indeed its solution consists of constant states we shall call them intermediate states in the sequel separated by discontinuities. In order to get the solution one has only to list the weak Riemann invariants associated to each eigenvalue and to write that each of them does not jump through the associated discontinuity. The aim of this work is to produce an accurate positive and entropy satisfying approimate Riemann solver for MHD. By entropy satisfying we mean that the scheme 2.2 satisfies discrete entropy inequalities 2.6 ηu n+ i ηui n t Gn i+ G n 2 i 2 for entropy flu pairs η G where G n = GU i+ i n Un i+ and GU l U r is a numerical entropy 2 flu satisfying GU U = GU. For ideal MHD we consider η = φs and G = ηu where φ is any decreasing and conve function. The assumption.9 ensures that η = φs is conve with respect to U. A stronger entropy inequality can indeed be formulated on the approimate Riemann solver itself see [3]. The positivity of density and internal energy for the approimate

4 4 FRANÇOIS BOUCHUT CHRISTIAN KLINGENBERG 2 KNUT WAAGAN 3 Riemann solver i.e. for its intermediate values is also retained as the useful formulation instead of the weaker one stating that the scheme 2.2 is positive. In [5] we derived stability conditions that ensure that the approimate Riemann solver is positive and entropy satisfying. These stability conditions must hold for each intermediate state of the approimate solver and involve also the initial states and the relaation parameters 2.. It is convenient to denote by a star any value corresponding to an intermediate state while the sub- or superscript l/r will be used to refer to the initial state on the same side of the central wave as the intermediate value considered. In [5] we proved the following result. Proposition 2.. The approimate Riemann solver defined by the relaation system is positive and satisfies the discrete entropy inequalities if for each intermediate state denoted by a star we have > 0 and 2.7 Bl/r where 2 p l/r is defined as 2 p l/r c 2 b B2 c 2 0 a B b 2 c a c 2 b 2 p l/r B2 c 2 a p l/r sup 2 p s l/r [ l/r ] and where c b c a b are evaluated locally i.e. they stand for c bl/r c al/r b l/r. 3. Riemann solvers with 3 and 5 waves The solvers considered in this paper are obtained with the choice b l = b r = 0. This makes the formulas for the intermediate states simpler resulting in fast codes and a relatively simple analysis. 3.. Intermediate states for the 3-wave solver. The most simple choice for the signal speeds is obtained by taking 3. b = 0 c a = c b c which means that we have only two parameters c l > 0 c r > 0. Then c s = c f = c thus we have only three eigenvalues for the system which are u c/ u u + c/. This leads to a solver with 3 waves and 2 intermediate states that will be denoted as l and r. The left and right waves have multiplicity 3. There are 8 strong Riemann invariants associated to the central wave i.e. quantities that lie in the kernel of t + u which are c a c b b and π c 2 B + B c 2 π e + B2 + B 2 π2 2 2c 2 π 2 2c 2. These quantities are thus weak Riemann invariants for the left and right waves. They must be completed with 3 weak Riemann invariants that are found to be π + cu π + cu for the left wave and π cu π cu for the right wave. For the central wave 6 weak Riemann invariants are u u π π. The wave speeds are therefore σ < σ 2 < σ σ = u l c l l σ 2 = u l = u r u σ 3 = u r + c r r. The values of c a c b b are the left values for the l state and the right values for the r state. The intermediate values for B e are deduced from the fact that the quantities in 3.2 do not jump through the left and right waves. It remains to determine the values u u π π which are common for the l and r states. They are determined by the relations 3.4 π + cu l = π + cu l π cu r = π cu r π + cu l = π + cu l π cu r = π cu r.

5 APPROXIMATE RIEMANN SOLVER FOR IDEAL MHD 5 Hence we get the intermediate values 3.5 Note that since σ = u l c l l u = c lu l + c r u r + π l π r c l + c r π = c rπ l + c l π r c l c r u r u l c l + c r u = c lu l + c ru r + πl πr c l + c r π = c rπ l + c lπ r c lc r u r ul c l + c r. and σ 3 = u r + cr positivity of is equivalent to σ < σ 2 < σ 3. r 3.2. Intermediate states for the 5-wave solver. A more general solver is obtained if we only set 3.6 b = 0 and keep arbitrary c a c b. Then we have the four parameters c bl c al c br c ar. We observe that 3.7 c s = minc a c b c f = mac a c b thus we have five eigenvalues for our system which are u c f / u c s / u u + c s / u + c f /. The eigenvalues u ± c a / have double multiplicity while u ± c b / are simple. We get a 5-wave solver with four intermediate states. There are 8 strong Riemann invariants associated to the central wave i.e. quantities that lie in the kernel of t + u which are c a c b b and + π B c 2 b + B c 2 π e + B2 + B 2 π2 a 2 2c 2 π b 2c 2. a These quantities are thus weak Riemann invariants for the other waves. Si weak Riemann invariants for the central wave are u u π π. They take the same value on the left and on the right of this central wave we shall denote theses values by u u π π. The remaining weak Riemann invariants for the left and right waves are found to be 3.9 u c b / : π + c b u π u u c a / : π + c a u π u u + c b / : π c b u π u u + c a / : π c a u π u. Therefore everything is as if the longitudinal part of the velocity-pressure u π were resolved independently of the transverse velocity-pressure u π the first jumping only through the c b waves and the second jumping only through the c a waves. We deduce the values u u π π by replacing c by c b or by c a in u = c blu l + c br u r + π l π r c bl + c br π = c brπ l + c bl π r c bl c br u r u l c bl + c br u = c alu l + c aru r + πl πr c al + c ar π = c arπ l + c alπ r c alc ar u r ul. c al + c ar We complete the values of u π u π in the noncentral intermediate states by setting them to either their star value or their l/r value depending on the ordering between c a and c b. Indeed if c b > c a then u π = u π u π = u l/r πl/r and if c b < c a then u π = u l/r π l/r u π = u π. The strong Riemann invariants 3.8 are constant on each side of the middle wave providing formulas for the intermediate values B and e. In particular does not jump through the c a waves.

6 6 FRANÇOIS BOUCHUT CHRISTIAN KLINGENBERG 2 KNUT WAAGAN 3 Finally we remark that u and π are given by u and π between the waves with speeds u l c bl l and u r + c br r while there takes the values l and r. The remaining left c a wave hence has speed u l c al l if c al c bl and u c al l c ar c br and u + car r π. otherwise. The remaining right c a wave has speed u r + car r if otherwise. Between these two c a waves u and π are given by u and 3.3. Previous 3-wave and 5-wave solvers. In [9] and [0] approimate Riemann solvers with 3 waves that resolve material contact discontinuities are given. No stability analysis is known in their case and indeed instabilities can occur in practice. Miyoshi and Kusano proposed in [] an approimate Riemann solver with 5 waves that also accurately resolves Alfven contact waves. Even if no analysis of stability is provided it is stable in practice Choice of signal speeds for the 3-wave solver. Here we derive eplicit values for the signal speeds that are sufficient for positivity and entropy inequalities. This is done with the use of Proposition 2.. Following the analysis of the classical gas dynamics case performed in [2] we make some natural assumptions on the pressure law that is 3. p > 0 s 3.2 p α p for some constant α > s where we recall that p is defined in.8. Notice that these assumptions only involve the pressure law p at fied s and that the first inequality is equivalent to p being conve with respect to / at fied s. For an ideal gas these conditions hold with α = 2 γ +. Considering now the 3-wave case we make the following a priori choice of the relaation speeds c l = l a 0 l + α l u l u r + + π r π l l a ql + r a qr c r = r a 0 r + α r u l u r + + π l π r + l a ql + r a qr Here a ql a qr denote the left and right fast MHD speeds 3.4 a 2 q = p + B2 + B 2 + p 2 + B2 and a 0 l and a 0 r need to be determined in such a way that 3.5 a 0 l a ql a 0 r a qr. + B p B2 This last restriction implies in particular that c l l a ql c r r a qr. We would like now to find a value for a 0 l such that l > 0 and that the conditions 2.7 of Proposition 2. are satisfied on the l state. From the first invariant of 3.2 we have that / l + π /c 2 l = / l + π l /c 2 l thus with the value of π given in Define now 3.7 X l = a ql l = l + c ru r u l + π l π r c l c l + c r l c ru l u r + c l c l + c r l u l u r + c l π r π l + c l c l + c r π r π l + c l l a ql + r a qr. u l u r + + π r π l + l a ql + r a qr so that by the definition 3.3 of c l and by 3.5 one has c l 3.8 a ql + αx l. l

7 APPROXIMATE RIEMANN SOLVER FOR IDEAL MHD 7 Using this in 3.6 gives 3.9 hence l > 0 l X l l + αx l < l l/ l with 3.2 l = X l + αx l ] α α. The following Lemma generalizes the analysis performed for the Euler equations in [3]. Lemma 3.. Consider a pressure law p satisfying Let = X/ + αx for some X 0. Then for all > p p + αx. Proof. The assumptions imply that 3.23 and thus that d α p d r r p r /α p r /α. Taking r = α gives 3.25 p p α thus in order to conclude it only remains to prove that α + αx or equivalently that + αx /α. This can be written 3.26 X + αx + αx /α 0. Defining θ = / + αx [0 ] this reduces to 3.27 θ α θ/α 0 which holds true for all θ [0 ] and α. Applying the lemma with = l gives l l 3.28 p s l l p l s l + αx l. l l According to the monotonicity of p stated in 3. and to 3.20 this allows to estimate the supremum of the speeds in p l l p l + αx l with p l = p l s l. Now in order to satisfy 2.7 it is sufficient to have 3.30 B l 2 c 2 l 2 l p l + αx l 2 l B2 l c 2 l

8 8 FRANÇOIS BOUCHUT CHRISTIAN KLINGENBERG 2 KNUT WAAGAN 3 provided that the two factors on the right-hand side are nonnegative. In order to simplify this we write that c l = l a 0 l + αa qlx l thus c 2 l = 2 l a 0 l 2 + 2αa 0 l a ql X l + αa ql X l l a 0 l 2 + 2αa 2 ql X l + αa ql X l 2. Therefore we get 3.32 c 2 l 2 l p l + αx l 2 2 l a 0 l 2 p l + 2αX l a 2 ql p l + αx l 2 a 2 ql p l 2 l a 0 l 2 p l. Therefore for 3.30 to hold it is enough that 3.33 B l 2 2 l a 0 l 2 p l l B2 l 2 l a0 l 2 provided that the two factors on the right-hand side are nonnegative. Multiplying this by a 0 l 2 we get a second degree inequality and a 0 l 2 must be larger than its largest root. Therefore we can take for a 0 l 2 this largest root 3.34 a 0 l 2 = 2 p l + B2 + Bl 2 l l + p l + B2 + B l 2 2 4p l l l B 2. l l Notice that this formula differs only by the appearance of l from the definition of a ql. Since l the condition a 0 l a ql in 3.5 holds true. The same analysis is valid on the right with 3.35 X r = a qr 3.36 a 0 r 2 = 2 We have proved the following: u l u r + + π l π r + l a ql + r a qr p r + B2 + B r 2 r r + r = X r + αx r p r + B2 + B r 2 2 4p r r r B 2. r r Proposition 3.2. If the pressure law satisfies the 3-wave solver is positive and entropy satisfying for the choice of c l c r given by 3.3 with The above formulas are sharp in the sense that they give the true fast MHD speeds for constant initial data. Indeed it is enough that u l = u r and π l = π r for getting c l = l a ql c r = r a qr. This shows that fast waves are well resolved with this solver. Furthermore one observes that with our choice of c l c r the maimum speed involved in the CFL condition 2.5 remains bounded when the speeds of the left and the right states are bounded u +a q l/r C. This is true in particular if for eample l approaches 0 with r fied but under the previous left/right bounds as it was required in order to treat the vacuum in the Euler case see [3] Choice of signal speeds for the 5-wave solver. We here provide formulas for the signal speeds c bl c br c al c ar for the 5-wave solver that enable to get positivity and entropy inequalities. We still make the assumptions In order to apply Proposition 2. we need that 3.37 B l/r 2 c 2 b 2 p l/r B2 c 2 a with both factors on the right-hand side being nonnegative. If 3.37 is to hold sharply a decrease in c a implies an increase in c b and vice versa. In the 3-wave case we had c b = c a and clearly that choice gives the least restrictive CFL-condition 2.5 where c f = mac a c b is involved. Another criterion for choosing c a and c b is to minimize the effect of the diffusion obtained from the

9 APPROXIMATE RIEMANN SOLVER FOR IDEAL MHD 9 Chapman-Enskog epansion. We shall make this choice here. The computation of the Chapman- Enskog epansion was performed in [5] and one can check that minimizing the largest eigenvalue of the diffusion matri means minimizing c 2 b + c2 a. If we assume that 3.37 holds sharply and for smooth solutions in the Chapman-Enskog contet minimum is achieved when 3.38 c 2 a = B2 + B B and c 2 b = 2 p + B 2 + B B. Then for this choice the largest eigenvalue of the diffusion matri is 2 B B /. We notice that it implies that the diffusion matri vanishes identically when B = 0 or B = 0. The case when B = 0 or B = 0 means that two eigenvalues of the MHD system coincide the system has at most 5 waves instead of 7. Therefore it means that whenever the MHD system has at most 5 waves the 5-wave solver with the choice 3.38 becomes fully accurate it has only the residual viscosity due to averaging on the cells or equivalently it has the same viscosity as the eact Godunov scheme. We would like now to make a choice of c al c bl c ar c br for our solver in such a way that for constant data we obtain the values In order to analyze the stability conditions we notice that the intermediate densities are given by a formula similar to the 3-wave case 3.39 l = + c bru r u l + π l π r l c bl c bl + c br r = + c blu r u l + π r π l. r c br c bl + c br This suggests to take c bl c br similarly as in 3.3 c bl = l a 0 l + α l u l u r + + π r π l l a bql + r a bqr c br = r a 0 r + α r u l u r + + π l π r + l a bql + r a bqr but now with a bql a bqr defined by 3.4 a 2 bq = p + B 2 + B B. We look for values a 0 l a0 r satisfying 3.42 a 0 l a bql a 0 r a bqr. This restriction implies in particular that c bl l a bql c br r a bqr. Then we set 3.43 X l = u l u r + + π r π l + a bql l a bql + r a bqr so that by 3.40 and 3.42 one has 3.44 c bl l a bql + αx l. Estimating the density l given in 3.39 similarly as in 3.6 yields that < l l / l with 3.46 l = X l + αx l ] α α. Now we observe that we have two intermediate states on the left but however for any of them its density is equal either to l or to l. Thus > 0 and 3.47 sup l. [ l ] l Therefore according to the monotonicity of p stated in 3. the maimal speed in 2.8 can be estimated as p l l l p s l l p l l + αx l l

10 0 FRANÇOIS BOUCHUT CHRISTIAN KLINGENBERG 2 KNUT WAAGAN 3 the latter inequality resulting from Lemma 3.. Thus in order to get 3.37 it is enough that 3.49 B l 2 c 2 bl 2 l p l + αx l 2 l B2 l c 2 al provided that the two factors on the right-hand side are nonnegative. Net we write c bl = l a 0 l + αa bqlx l and the same estimates as in give 3.50 c 2 bl 2 l p l + αx l 2 2 l a 0 l 2 p l. Therefore for 3.49 to hold it is sufficient to have 3.5 B l 2 2 l a 0 l 2 p l l B2 l c 2 al with the two factors on the right-hand side being nonnegative. Our choice is then to take 3.52 a 0 l 2 = p l + Bl 2 + B B l l l 3.53 c 2 al = l l B 2 + B B l that gives equality in 3.5. Since l the condition that a 0 l a bql in 3.42 is satisfied. Notice again that 3.52 differs only by the factor / l from the definition of a bql. A similar analysis on the right leads to 3.54 X r = a bqr u l u r + + π l π r + l a bql + r a bqr r = X r + αx r 3.55 a 0 r 2 = p r + Br 2 + B B r r r c 2 ar = r r B 2 + B B r. We have proved the following: Proposition 3.3. If the pressure law satisfies the 5-wave solver is positive and entropy satisfying for the choice of c bl c br c al c ar given by As it was chosen for constant data the values of c a c b reduce to 3.38 hence the solver becomes fully accurate whenever B or B vanishes or is sufficiently small. Thus an interest of this 5-wave solver is to be able to use it as a patch for other solvers using 7 waves that may have singularities when two eigenvalues become too close. However a weak point in our 5-wave solver is that its maimal wave speed c f / = mac a / c b / involved in the CFL condition 2.5 eceeds the maimal speed of the MHD system which is a q in 3.4. This ecess can be evaluated by the ratios c a /a q and c b /a q where c a c b are given in For the first one studying the variations with respect to p one can easily prove that c a a q = c b a q = B 2 + B B a 2 q For the second it is not so easy but with the same method we get the same upper bound p a 2 q + B 2 + B B a 2 q This implies that the overall computational cost can be only of 0% in the worst case. In practice the 5-wave solver uses timesteps close to what the 3-wave solver uses.

11 APPROXIMATE RIEMANN SOLVER FOR IDEAL MHD Test u v w p B B y B z γ 4 Dai-Woodward left π 4π 4π 3 Dai-Woodward right π 2 4π Brio-Wu I left Brio-Wu I right Brio-Wu II left Brio-Wu II right Slow rarefaction left Slow rarefaction right Vacuum problem left Vacuum problem right Epansion problem left Epansion problem right Epansion problem II left Epansion problem II right Figure 4.. Table of initial data for shock tube tests 4. Numerical tests In this section we investigate the accuracy and robustness in numerical computations of the relaation-based Riemann solvers with 3 and 5 waves. We compare them with each other and with other methods. In all cases we use the solvers in the first-order Godunov scheme 2.2. In order to get reference solutions we performed high resolution computations with the entropy satisfying 3-wave solver. The CFL-number was 0.9 in all tests. We summarize the Riemann initial data taken as test cases in Table 4.. Note that we use the notation u = v w and B = B y B z. For comparison purposes we consider the Roe solver of [] the HLL solver and the 5-wave solver of [] denoted by MK5. As signal speeds for HLL and MK5 we used the speeds for the 3-wave solver from Section 3. For the HLL solver this choice ensures positivity and entropy inequalities. This is because if we consider the conserved quantities the intermediate value of the HLL solver is equal to the spatial average between the intermediate states of the 3-wave solver. Hence positivity of density and internal energy is inherited by conve combination and entropy inequalities are obtained via Jensen s inequality. 4.. Shock tube of Dai and Woodward. This shock tube test was introduced in [7]. The initial data are given as Dai-Woodward in Table 4.. The solution consists of shocks and contact discontinuities for all characteristic fields. On this test we compare our solvers against the Roe solver of []. This particular Roe solver based on entropy variables seems to give very sharp resolution. For the Roe solver we used a CFL number of 0.45 since it is unstable for values too close to. It gives the epected high accuracy on this fairly mild test case see Figure 4.2. We also note that the 3-wave solver gives a much better resolution than HLL especially of the density. The 5-wave solver performs similarly to the 3-wave solver ecept that it slightly improves the resolution of the left-going Alfven wave. In figure 4.3 we compare our 3-wave solver to the MK5 solver. As signal speeds in the MK5 solver we used our speeds for the 3-wave solver. For that reason it is not surprising that the performance on fast waves differ very little. At the left-going Alfven and slow waves MK5 is sharper than with our 3- and 5-wave solvers Brio-Wu shock tube I. Net we consider the shock tube tests of [6] denoted by Brio-Wu I. Figure 4.4 shows the resulting and B y. The solution consists of from left to right a fast rarefaction a compound wave a contact discontinuity and a slow shock. The compound wave is a discontinuity attached to a slow rarefaction it can be attributed to the non strict hyperbolicity of the system.-.5. There is also a small Alfven wave and a fast rarefaction going to the

12 2 FRANÇOIS BOUCHUT CHRISTIAN KLINGENBERG 2 KNUT WAAGAN Roe HLL B y.3.2. Roe HLL Figure 4.2. and B y for Dai-Woodward shock tube at time t = 0.2 with resolution = 0.0. The reference solution is a 3-wave simulation with = right that are not shown in Figure 4.4. We first compare our 3-wave solver with the HLL solver. As epected the 3-wave solver has much better resolution of the contact discontinuity. Figure 4.4 shows that the 3-wave solver also strongly improves the sharpness of the slow shock and the compound wave. The fast waves are well resolved by both solvers. We also see that the 5-wave solver gives good results on these waves. Figure 4.5 shows that the 5-wave solver improves the resolution of the compound wave compared to the 3-wave solver. This has to do with B being locally small Brio-Wu shock tube II. The second test from [6] is a case with high fast magnetosonic Mach number. Since B = 0 and u = 0 the 3-5-wave and MK5 solvers are identical and a full 7-wave solver cannot be epected to peform better. The results are shown in Figure 4.6. In this case we do not gain much compared to HLL from eactly resolving the contact wave since it moves much faster than the magnetosonic speeds. The smearing of the contact wave is mostly due to numerical diffusion inherent in the Godunov scheme. Both fast waves are reasonably well resolved.

13 APPROXIMATE RIEMANN SOLVER FOR IDEAL MHD MK B y.3.2 MK Figure 4.3. and B y for Dai-Woodward shock tube at time t = 0.2 with resolution = 0.0. The reference solution is a 3-wave simulation with = Shear waves. This eample illustrates the numerical accuracy on Alfven waves of the different solvers. We choose the initial data =.0 p =.0 u =.0 B y = sin2π B z = cos2π v = sin2π w = cos2π B =.0 γ = 5/3. This specifies a stationary left Alfven wave. We epect that the 5-wave solver will give a better approimation than the 3-wave and this is confirmed by the plot of B y in Figure 4.7. We also made the same computation with p = 00.0 which still gives a stationary Alfven wave. In this case the 5-wave solver is superior as epected as Figure 4.8 shows. We also note in Figures that the 3-wave solver is a clear improvement with respect to the HLL solver Slow sonic rarefaction. This test is a slow switch-on rarefaction suggested in [8] and also used in []. The data are given in table 4. There is a sonic point in this rarefaction that is a point where slow magnetosonic speed equals the fluid velocity. A linearized solver will typically produce an unphysical shock at the sonic point unless additional numerical diffusion is added.

14 4 FRANÇOIS BOUCHUT CHRISTIAN KLINGENBERG 2 KNUT WAAGAN HLL B y HLL Figure 4.4. and B y for Brio-Wu shock tube I at time t = 0.2 with resolution = 0.0. The reference solution is a 3-wave simulation with = Figure 4.9 shows that the 3-wave solver is able to handle this problem in a stable manner with good accuracy. It gives an improvement with respect to the HLL solver. The structures in the density profile behind the rarefaction can be referred to as start-up errors and these are basically unavoidable for certain kinds of Riemann problems. The 5-wave solver gives slightly better results than the 3-wave solver for the magnetic field B y otherwise they are practically the same Vacuum problem. An interesting test for the stability of a scheme is how it handles a Riemann problem where one initial state is vacuum see [2] and [3]. In the case of ideal MHD we have to take zero magnetic field in the vacuum region to keep the characteristic speeds finite. This implies that B = 0 everywhere. Since also u = 0 in our case the 3- and 5-wave solvers are the same. Consider a vacuum left state with l = 0 B l = 0 and a right state with r > 0 p r > 0. Then c l c r and the intermediate states are well defined. We assume in order to get a value for c l / l that B l / l = 0. Note that no quantities jump across the left-going wave. Therefore the quantity c l / l is only used in calculating the CFL condition. Furthermore its contribution to the CFL condition is not strictly needed. In the case when γ = 2 i.e. p = e and B z = 0 we can write the solution as B y = Br y r together with a pure Euler system written in terms of u and an auiliary internal energy ẽ =

15 APPROXIMATE RIEMANN SOLVER FOR IDEAL MHD wave B y wave Figure 4.5. and B y for Brio-Wu shock tube I at time t = 0.2 with resolution = 0.0. The reference solution is a 3-wave simulation with = Note the different -ais ranges. e + 2 B r y 2. r Then since a rarefaction wave is isentropic we get the classical solution 4. ut = 2 ma 3 t = r + 2 ẽt = t ẽr r. 2 t 2 ut 2ẽr + 2 2ẽ r 2ẽ r The figures show that our solver handles this vacuum test case well. The MK5 solver with our 3-wave speeds is identical to the 3-wave solver in this case u = 0 B = 0 and hence would give the same result Epansion problem. This test is from []. It consists of two rarefactions separating a low density region which is difficult to compute especially for linearized solvers. Since B = 0 and u = 0 the 3-5-wave and MK5 solvers are the same as in subsection 4.3. HLL with the same

16 6 FRANÇOIS BOUCHUT CHRISTIAN KLINGENBERG 2 KNUT WAAGAN B y Figure 4.6. and B y for Brio-Wu shock tube II at time t = 0.02 with resolution = 0.0. The reference solution is a 3-wave simulation with = The 3- and 5-wave solvers are the same in this case. signal speeds as the 3-wave solver does an equally good job as the 3-wave solver. Figure 4.2 shows computed with the 3-wave solver. This is a good result as one would epect Low thermal pressure. Taking B nonzero in the above eample causes p B to become 2 small in the center region in the wake of two strong slow rarefactions. This is an additional difficulty to the low density and pressure. The 3-wave and 5-wave solvers both handle this case well and give almost identical results. They resolve the density better than HLL as seen from Figure 4.3. The magnetic field B y is shown in Figure 4.4 and for that quantity there is less difference between the codes. We also plotted β = 2 p B see Figure 4.5. The low values make this an interesting test case. 2 In Figure 4.6 we compare our solvers with the MK5 solver. We notice that the latter gives similar results to the 3-wave solver while our 5-wave solver computes the thermal pressure more accurately. The specific internal energy reaches its maimum e ma at = 0.5 in all cases. With our 5-wave solver we get e ma = while the MK5 gives e ma =.2 which is a significant difference. The reference solution has e ma = Note that the quantity e is proportional to the temperature for an ideal gas.

17 APPROXIMATE RIEMANN SOLVER FOR IDEAL MHD Eact 5 wave HLL 0.2 y Figure 4.7. B y for a smooth Alfven wave with resolution = 0.0 at time t = Eact 5 wave HLL 0.2 y Figure 4.8. The same as in Figure 4.7 but with pressure 00 times higher 4.9. Conclusion. We summarize the results from this section in the following points. The 3-wave solver resolves slow moving contacts much better than HLL. The 3-wave solver also improves the resolution of Alfven waves and slow waves compared to HLL. The 3-and 5-wave solvers can handle rarefactions into low density and low β = 2 p B. 2 The 5-wave solver can in contrast to the 3-wave solver sharply resolve all waves when B or B vanishes. The 5-wave solver is significantly sharper on Alfven waves in certain regimes. In particular when Alfven speed is smaller than sound speed or when B is smaller than B. This appears from the numerical tests to also be the case for the slow modes in certain cases. The 3- and 5-wave solvers are ready to be applied on physical problems. They provide good accuracy and ecellent stability properties with simple and eplicit formulas. In most cases the 3-wave solver should perform better because it runs faster. It also allows larger timesteps

18 8 FRANÇOIS BOUCHUT CHRISTIAN KLINGENBERG 2 KNUT WAAGAN HLL B y HLL Figure 4.9. Slow rarefaction at time t = 0.2 with resolution = 0.0. The reference solution is a 3-wave simulation with = Eact Figure 4.0. Vacuum problem at time t = 0.2 with resolution = The 3- and 5-wave solvers are the same in this case.

19 APPROXIMATE RIEMANN SOLVER FOR IDEAL MHD Eact Figure 4.. Vacuum problem at time t = 0.2 with resolution = The 3- and 5-wave solvers are the same in this case Figure 4.2. Epansion problem at time t = 0.2 with resolution = Note the logarithmic y-ais. The reference solution is a 3-wave simulation with = The 3- and 5-wave solvers are the same in this case. although the difference is not really significant. At high β though the 5-wave solver is much less diffusive. This is also the case if one has a strong grid-aligned magnetic field. 5. Appendi: Powell s system In this appendi we propose a method to numerically deal with nonconstant B. This can be usefull when dealing with multidimensional MHD and the divb = 0 constraint. We shall denote here B = B B and u = u u. Following Powell [2] the so called Powell system in three dimensions indeed one of its versions is obtained by removing the constraint divb = 0 in the MHD system and by adding a term u divb

20 20 FRANÇOIS BOUCHUT CHRISTIAN KLINGENBERG 2 KNUT WAAGAN HLL Figure 4.3. Epansion problem with B =.0 at time t = 0.5 with resolution = Note the logarithmic y-ais. The reference solution is a 3-wave simulation with = B y Figure 4.4. Epansion problem with B =.0 at time t = 0.5 with resolution = on the induction equation leading to 5. t + divu = 0 u t + divu u + p + 2 B 2 Id B B = 0 E t + div[e + p + 2 B 2 u B ub] = 0 B t + divb u u B + u divb = 0 where E is still given by.6. This system is a classical quasilinear system and only the induction equation is not in conservative form. Some of the main properties of Powell s system are:. Powell s system is hyperbolic its eigenvalues being given by the same formulas as for the MHD system the only difference being that u has now multiplicity 2

21 APPROXIMATE RIEMANN SOLVER FOR IDEAL MHD p/B HLL Figure 4.5. Epansion problem with B =.0 at time t = 0.5 with resolution = Note the logarithmic y-ais. The figure shows the ratio of thermal and magnetic pressure p/B wave MK Figure 4.6. Epansion problem with B =.0 at time t = 0.5 with resolution = Note the logarithmic y-ais. The reference solution is a 3-wave simulation with = Under the assumptions Powell s system has the same entropy inequalities as the classical MHD system 5.2 φs t + divφsu 0 for all nonincreasing conve φ 3. Solutions to Powell s system such that divb 0 are solutions to the MHD system 4. Solutions to Powell s system such that divb0 0 satisfy also divbt 0 for all times t because one has 5.3 divb t + divu divb = 0. Thus such solutions are also solutions to the MHD system.-.5.

22 22 FRANÇOIS BOUCHUT CHRISTIAN KLINGENBERG 2 KNUT WAAGAN Powell s system in one dimension. In one space dimension Powell system reads 5.4 t + u = 0 u t + u 2 + p + 2 B 2 2 B2 = 0 u t + uu B B = 0 E t + [E + p + 2 B 2 2 B2 u B B u ] = 0 B t + ub = 0 B t + B u B u + u B = 0. Instead of being constant previously B is now advected at velocity u. Note that the nonconservative products in the induction equations do not induce any difficulty concerning definitions because B jumps only through a material contact discontinuity where u and u do not jump Relaation system associated with Powell s system. In order to approimate 5.4 by relaation we just add the nonconservative part u divb of the Powell system to the magnetic equations of the relaaton system and obtain 5.5 t + u = 0 u t + u 2 + π = 0 u t + uu + π = 0 E t + [E + πu + π u ] = 0 B t + ub = 0 B t + B u B u + u B = 0 with still.6 and no change in the other equations 5.6 π t + πu + b 2 + c 2 b u c a b u = 0 π t + π u c a b u + c 2 au = c a t + uc a = 0 c b t + uc b = 0 b t + ub = 0. The studies made previously in [5] and in the present paper for the case B = cst are valid in the more general case B non constant the only difference being that now B has to be understood as evaluated locally as it was the case for c a c b b Numerical flues. Computing the solution to the Riemann problem associated to the relaation system gives an approimate Riemann solver for the Powell system. Then one wants to compute the numerical flues associated to it. For the conservative quantities u u E they are given by the corresponding components of 2.4 with the additional information that B only jumps at the middle wave. For updating the magnetic quantities B and B one has to be more careful since we are solving nonconservative equations. In the spirit of approimate Riemann solvers one has to write that the new value B n+ i is obtained as the average over the cell of the approimate solution. The approimate solution satisfies 5.8 B t + Bu B u + ub = 0 where u = u u. Denote by u the value of u through the material contact. Then one has B l 5.9 B = if t < u B r if t > u. But since 5.0 B = B r Bl δ tu ub = u B r Bl δ tu where u is the value of u through the material contact one has 5. [Bu B u] =0 + u B r B l I u =0 = 0 where [...] =0 denotes the jump through the line = 0.

23 APPROXIMATE RIEMANN SOLVER FOR IDEAL MHD 23 Now integrate 5.8 over 0 t 0. We get B/ td B l + t Bu B u 0 Bu B u l + t u B r Bl I u <0 = 0. Net integrate 5.8 over 0 t 0. We get 5.3 B/ td B r + t 0 Bu B u r Bu B u 0+ + t u B r BI l u >0 = 0. Denote Fl B = Bu B 5.4 u 0 + u B r BI l u <0 Fr B = Bu B u 0+ u B r Bl I u >0. We end up with B n i + t According to 5. a formula for the numerical flues is { F 5.6 If u B 0 then 5.5 B n+ i 5.7 If u 0 then In all cases one has F B l i+/2 F B r i /2 = 0. l = Bu B u 0 Fr B = Bu B u 0 u B r B l { F B l = Bu B u 0+ + u B r Bl F B r = Bu B u Fr B FB l = u B r Bl which implies a consistent discretization of the nonconservative term in 5.8. Notice that the above derivation of the left and right numerical flues Fl B Fr B involved in 5.5 does not really use the relaation system but merely only 5.8 which is indeed the eact induction equation of Powell s system. Thus the formulas are true also for the eact Riemann solver for eample. This relaation solver is positive and entropy satisfying under the same conditions that are derived in [5]. The only modification is that B has always to be understood as evaluated locally according to 5.9. For the 3- and 5-wave solvers Propositions 3.2 and 3.3 remain valid with this interpretation as well as the formulas for the intermediate states. s [] T. J. Barth. Numerical methods for gasdynamic systems on unstructured meshes. In Dietmar Kröner Mario Ohlberger and Christian Rohde editors An introduction to recent developments in theory and numerics for conservation laws : proceedings of the International School on Theory and Numerics and Conservation Laws Freiburg/Littenweiler October pages Springer-Verlag Inc [2] François. Bouchut. Entropy satisfying flu vector splittings and kinetic BGK models. Numer. Math. 944: [3] François Bouchut. Nonlinear stability of finite volume methods for hyperbolic conservation laws and wellbalanced schemes for sources. Frontiers in Mathematics. Basel: Birkhäuser. viii 35 p [4] François Bouchut Christian Klingenberg and Knut Waagan. A multiwave approimate riemann solver for ideal mhd based on relaation III - numerical implementation with 7 waves. To appear. [5] François Bouchut Christian Klingenberg and Knut Waagan. A multiwave approimate riemann solver for ideal mhd based on relaation I - theoretical framework. Numerische Mathematik 08: [6] M. Brio and C.C. Wu. An upwind differencing scheme for the equations of ideal magnetohydrodynamics. J. Comput. Phys. 752: [7] Wenlong Dai and Paul R. Woodward. An approimate Riemann solver for ideal magnetohydrodynamics. J. Comput. Phys. 2: [8] S. A. E. G. Falle S. S. Komissarov and P. Joarder. A multidimensional upwind scheme for magnetohydrodynamics. Monthly Notices of the Royal Astronomical Society 297: [9] K.F. Gurski. An HLLC-type approimate Riemann solver for ideal magnetohydrodynamics. SIAM J. Sci. Comput. 256: [0] Shengtai Li. An HLLC Riemann solver for magneto-hydrodynamics. J. Comput. Phys. 203:

24 24 FRANÇOIS BOUCHUT CHRISTIAN KLINGENBERG 2 KNUT WAAGAN 3 [] Takahiro Miyoshi and Kanya Kusano. A multi-state HLL approimate Riemann solver for ideal magnetohydrodynamics. J. Comput. Phys. 208: [2] Kenneth G. Powell. An approimate Riemann solver for magnetohydrodynamics that works in more than one dimension. Technical report Institute for Computer Applications in Science and Engineering ICASE 994.