Arbitrary order exactly divergence-free central discontinuous Galerkin methods for ideal MHD equations

Transcription

1 Arbitrary order exactly divergence-free central discontinuous Galerkin metods for ideal MHD equations Fengyan Li, Liwei Xu Department of Matematical Sciences, Rensselaer Polytecnic Institute, Troy, NY , United States Abstract Ideal magnetoydrodynamic (MHD equations consist of a set of nonlinear yperbolic conservation laws, wit a divergence-free constraint on te magnetic field. Neglecting tis constraint in te design of computational metods may lead to numerical instability or nonpysical features in solutions. In our recent work (Journal of Computational Pysics 230 ( , second and tird order exactly divergence-free central discontinuous Galerkin metods were proposed for ideal MHD equations. In tis paper, we furter develop suc metods wit iger order accuracy. Te novelty ere is tat te well-establised H(div-conforming finite element spaces are used in te constrained transport type framework, and te magnetic induction equations are extensively explored in order to extract sufficient information to uniquely reconstruct an exactly divergence-free magnetic field. Te overall algoritm is local, and it can be of arbitrary order of accuracy. Numerical examples are presented to demonstrate te performance of te proposed Corresponding autor. addresses: lif@rpi.edu (Fengyan Li, xul3@rpi.edu (Liwei Xu Preprint submitted to Journal of Computational Pysics January 10, 2012

2 metods especially wen tey are fourt order accurate. Keywords: Ideal magnetoydrodynamic (MHD equations, Exactly divergence-free, Central discontinuous Galerkin metods, Hig order accuracy, H(div-conforming finite element, BDM elements 1. Introduction In tis paper, we continue our recent development in [20] to devise exactly divergence-free numerical metods for ideal magnetoydrodynamic (MHD equations. Tese equations arise in many areas in pysics and engineering, and tey consist of a set of nonlinear yperbolic conservation laws, wit a divergence-free constraint on te magnetic field. Toug tis constraint olds for te exact solution as long as it does initially, neglecting tis condition in te design of computational algoritms may lead to numerical instability or nonpysical features of approximating solutions [17, 7, 15, 5, 30, 14, 19, 6, 21, 4]. In [20], exactly divergence-free central discontinuous Galerkin (DG metods wit second and tird order accuracy were developed for ideal MHD equations. Te metods are based on te central DG metods in [24, 25] and use a different discretization for magnetic induction equations. More specifically, wile oter conservative quantities are evolved wit central DG metods of [24], te magnetic field (or its two components in two dimensions is updated suc tat its normal component is first approximated by discretizing magnetic induction equations on interfaces of mes elements, and ten an element-by-element divergence-free reconstruction procedure follows wit compatible accuracy. Tis gives numerical metods wit exactly divergence- 2

3 free magnetic fields, and tese metods demonstrate good performance in bot accuracy and stability. On te oter and, suc reconstruction by matcing te pre-computed normal component of te magnetic field at mes interfaces is under-determined for iger order accuracy. In tis paper, a new strategy is proposed. In particular, we extensively explore te magnetic induction equations to extract sufficient information about te magnetic field, wit wic an element-by-element reconstruction is defined. We prove tat te reconstruction is uniquely determined, and te resulting magnetic field is exactly divergence-free wit arbitrary order of (formal accuracy. In addition, wen being second or tird order accurate, te proposed metod is te same as tat in [20]. Numerical experiments are carried out to illustrate te performance of te overall algoritms wen tey are fourt order accurate. Te metods are presented for two dimensions in tis paper, and tere is no difficulty to extend tem to tree dimensions. Our work is inspired by te development of H(div-conforming finite element spaces [8, 9] in classical finite element metods. A finite element space is said to be H(div-conforming if its function is piecewise smoot (suc as being piecewise polynomials and its normal component is continuous across mes element interfaces. Toug some of suc spaces ave been used in te design of exactly divergence-free numerical metods for ideal MHD equations [2, 21, 4, 22] (also see Section 5.2 for more discussion, to our best knowledge, te proposed metods are te first to utilize te establised results and analysis tecniques for H(div-conforming finite element spaces to systematically design exactly divergence-free numerical metods of any order of accuracy for MHD simulations. Not like in standard Galerkin type metods 3

4 using suc spaces, te proposed local procedure does not involve inverting a global mass matrix. Te proposed metods can be regarded as constrained transport type metods (see, e.g. [15, 5, 16, 21], wit central DG metods of [24] as te base metod and a new divergence-free reconstruction strategy. A basic idea of constrained transport metods is to work wit te magnetic induction system in its integral form along element boundaries. To acieve iger order accuracy in our proposed metods, we furter explore te magnetic induction equations in order to extract more information about te magnetic field. Central DG metods are a family of ig order numerical metods defined on overlapping meses wit two sets of numerical solutions. Compared wit standard DG metods [11, 19] wic are defined on a single mes wit one numerical solution and terefore are more efficient in memory usage, central DG metods do not explicitly use any numerical flux (wic is exact or approximate Riemann solver, and te time step allowed for linear stability is larger wen te accuracy of te metods is iger tan one [25, 20]. Moreover, no averaging or interpolation step is needed for our proposed metods to produce a single-valued electric field flux at grid-points. Suc step is inerent in our central framework on overlapping meses, and it is required explicitly for Godunov type constrained transport metods [16, 4] to define an exactly divergence-free reconstruction, and for tis, upwind mecanism needs to be incorporated due to te stability consideration especially for ig order accuracy. Te remainder of te paper is organized as follows. In Section 2, we describe te governing equations and introduce notations for meses and 4

5 discrete spaces. In Section 3, exactly divergence-free central DG metods are proposed for ideal MHD equations, and te main teorem is stated for a local reconstruction procedure, wit its proof given in Section 4. In Section 5, furter discussions are made on divergence-free discrete spaces, relations of te proposed metods to some oter divergence-free metods, ig order time discretizations, and nonlinear limiters. Numerical examples are reported in Section 6, followed by concluding remarks in Section 7. In Appendix A, te definition of te divergence-free reconstruction in [20] is given for te reference purpose. We include in Appendix B te formulas of te fourt order reconstruction proposed in tis paper. 2. Preliminaries: equations and notations Consider te ideal MHD equations, wic are a yperbolic system (ρu t ρ t + (ρu = 0, (1 + [ρuu + (p B 2 I BB ] = 0, (2 B t (u B = 0, (3 E t + [(E + p B 2 u B(u B] = 0, (4 wit a divergence-free constraint B = 0. (5 Here ρ is te density, p is te ydrodynamic pressure, u = (u x, u y, u z is te velocity field, and B = (B x, B y, B z is te magnetic field. Te total energy E is given by E = 1 2 ρ u B p γ 1 wit γ as te ratio of te

6 specific eats. We use te superscript to denote te vector transpose. In addition, I is te identity matrix, is te divergence operator, and is te curl operator. Equations (1, (2, and (4 are from te conservation of mass, momentum, and energy, and (3 is te magnetic induction equation system. In two dimensions wen all unknown functions depend only on spatial variables x and y, equations (1-(4 can be rewritten as U t B t + x F 1(U, B + y F 2(U, B = 0, (6 + E z (U, B = 0, (7 were U = (ρ, ρu x, ρu y, ρu z, B z, E, B = (B x, B y, and F 1 (U, B = (ρu x, ρu 2 x + p B 2 Bx, 2 ρu x u y B x B y, ρu x u z B x B z, u x B z u z B x, u x (E + p B 2 B x (u B, (8 F 2 (U, B = (ρu y, ρu y u x B y B x, ρu 2 y + p B 2 By, 2 ρu y u z B y B z, u y B z u z B y, u y (E + p B 2 B y (u B. (9 In addition, E z (U, B = u y B x u x B y, and it is te z-component of te electric field E = u B. We also use E z = ( Ez y, Ez x, wic gives te first two components of (0, 0, E z. Note tat B z does not contribute to B in two dimensions, terefore for convenience we call B te magnetic field and (7 te magnetic induction system from now on. Next, we introduce notations used in numerical scemes. Since only Cartesian grids are considered in tis paper, we assume te computational domain is Ω = (x min, x max (y min, y max R d, wit d = 2. and {y j } j Let {x i } i be partitions of (x min, x max and (y min, y max, respectively, and 6

7 x i+ 1 2 = 1 2 (x i + x i+1 and y j+ 1 2 = 1 2 (y j + y j+1. Ten T C = {C i,j, i, j} and T D = {D i,j, i, j} define two overlapping meses for Ω, wit C i,j = (x i, x i+1 (y j, y j+1 and D i,j = (x i 1 2, x i+ 1 2 (y j 1 2, y j+ 1. Tey are also 2 called primal and dual meses, respectively. Discrete spaces will be defined associated wit eac mes. In te numerical metods introduced in next section, different strategies are used to approximate U and B. For U, we use a piecewise polynomial vector space U,k as te discrete space, tat is, U,k = { v : v K [P k (K] 8 d, K T }, (10 were P k (K denotes te space of polynomials in K wit te total degree at most k, and [P k (K] r = {v = (v 1,, v r : v i P k (K, i = 1,, r} is its vector version wit any positive integer r. Here and below denotes C or D. For te magnetic field B, we want to approximate it by a piecewise polynomial vector field wic is exactly divergence-free. Suc function is caracterized by its being piecewise divergence-free, and its normal component being continuous across mes elements. Motivated by te development of classical finite elements on Cartesian meses, we take te following M,k as te discrete space for B, M,k = { v H(div 0 ; Ω : v K W k (K, K T }, (11 = { v : v K W k (K, v K = 0, K T, and te normal component of v is continuous across eac mes element interface}, wit W k (K being an augmented space of te polynomial vector space of 7

8 degree k, { W k (K = [P k (K] d span (x k+1 y, } (xy k+1 In fact, M,k, (12 = {v : v = u + a (x k+1 y + b (xy k+1, u [P k (K] d, a, b R}. is te divergence-free subspace of te following Brezzi-Douglas- Marini (BDM finite element space, BDM k = { v H(div; Ω : v K W k (K, K T }, (13 = { v : v K W k (K, K T, and te normal component of v is continuous across eac mes element interface}, wic is one of te widely used H(div-conforming finite element spaces and was introduced in [8] to solve second order elliptic problems in teir first order form. Moreover, M,k as optimal approximation properties for exactly divergence-free smoot functions on Cartesian meses wit respect to index k (see Lemma 2.1 in [8]. Toug te building block W k (K is an augmented space of a polynomial vector space, te added part, span{ (x k+1 y, (xy k+1 }, will not contribute to te divergence of a function in W k (K. Tis is stated below togeter wit anoter property of W k (K we will use later. Tey can be verified easily based on te definition of W k (K. Lemma 2.1 (Properties of functions in W k. Given C = [x L, x R ] [y L, y R ]. For any v = (v 1, v 2 W k (C, tere are Property 1: v 1 (x, y P k (y L, y R and v 2 (x, y P k (x L, x R for = L, R. Property 2: v P k 1 (C. 8

9 Tere are in fact oter ways to define te exactly divergence-free discrete space for B, and some examples are given in Section 5. For te numerical algoritm introduced in next section, we need one more discrete space, V,r = { v : v K [P r (K] d, K T }, (14 wic also consists of piecewise polynomial vector fields. Te role of tis space will become clear later. 3. Numerical scemes Toug te divergence-free constraint (5 seems to be redundant on te PDE level as it can be derived from te magnetic induction equations wit a compatible initial condition, suc constraint is not always satisfied by a numerical sceme, and tis may lead to nonpysical features of approximating solutions or numerical instability [7, 30, 19, 6]. In tis section, we propose central DG metods wit an exactly divergence-free magnetic field to solve te system (5-(7 and terefore (1-(5. Te proposed metods can be of arbitrary (at least formal order of accuracy and tey are completely local. In addition, wen te accuracy is second or tird order, te proposed metod is te same as te one developed in [20], and tis fact will be establised in Teorem 3.1. For simplicity, we present te scemes wit te forward Euler metod as te time discretization. In Section 5, we comment on iger order temporal accuracy wic can be acieved by using strong stability preserving (SSP ig order time discretizations [18]. Suc discretizations can also be needed for stability reason (see e.g. Table 6 in [25]. Te proposed metods evolve two copies of numerical solutions, wic are assumed to be available at t = t n, denoted as (U n, 9, Bn, U,k M,k

10 wit B n, = (B n, x,, Bn, y,. Here and below denotes C or D. We will describe ow to obtain numerical solutions at t n+1 as (U n+1,, B n+1, U,k M,k wit B n+1, = (B n+1, x, = t n + t n, denoted, B n+1, y,. Due to similarity, we only present te procedure to update (U n+1,c, B n+1,c Updating U n+1,c To get U n+1,c, we apply to (6 te central DG metods of [24] (also see e.g. [20] for a brief review as te spatial discretization and te forward Euler metod as te time discretization. Tat is, to look for U n+1,c tat for any V U C,k Ci,j = [P k (C i,j ] 8 d wit any i, j, ( U n+1,c Vdx = θ n U n,d C i,j + t n ( C i,j C i,j ( F n,d 1 V x + Fn,D 2 V y U C,k suc + (1 θ n U n,c Vdx (15 ( dx n 1 F n,d 1 + n 2 F n,d 2 Vds C i,j Here θ n = t n /τ n [0, 1], wit τ n being te maximal time step allowed by te CFL restriction [24, 20]. (n 1, n 2 is te outward pointing unit normal vector along C i,j, and F n,d l = F l (U n,d, B n,d for l = 1, 2. Wit two sets of numerical solutions available at time t n, te metods do not explicitly use numerical fluxes, wic are exact or approximate Riemann solvers and are used in Godunov type metods suc as standard DG metods [11, 19] Updating te exactly divergence-free B n+1,c Te finite element space M C,k consists of exactly divergence-free vector fields. One can use tis space directly in te Galerkin framework to approximate te magnetic field to ensure its zero divergence, see e.g. [10], tis owever often needs to invert a large mass matrix for eac time step. 10

11 (or eac inner stage for multi-stage time discretizations. In tis paper, we propose a local strategy to obtain an exactly divergence free approximation B n+1,c = (B n+1,c x,, B n+1,c y, M C,k. Te algoritm is defined element by element and terefore no mass matrix inversion is involved. Note tat an exactly divergence-free vector field is caracterized by its continuous normal component across element interfaces and its divergencefree restriction inside eac mes element, and our metods start wit approximating te normal component of te magnetic field on te mes skeleton. For te Cartesian mes considered ere, tese are B n+1,c x, element interfaces and B n+1,c y, along y-direction along x-direction element interfaces. To tis end, we discretize two one-dimensional equations in (7 B x t = E z y, B y t = E z x (16 wit respect to te primal mes T C : for any i, j, one looks for bi,j x (y P k (y j, y j+1 and b i,j y (x P k (x i, x i+1, suc tat yj+1 yj+1 ( b i,j x (yµ(ydy = θ n B n,d x, (x i, y + (1 θ n B n,c x, (x i, y µ(ydy y j y ( j yj+1 + t n Ez n,d (x i, y µ(y dy En,D z,i,j+1 y µ(y j+1 + E n,d z,i,j µ(y j, (17 y j for any µ(y P k (y j, y j+1, and xi+1 b i,j x i xi+1 + t n ( y (xν(xdx = x i E n,d z xi+1 x i (x, y j ν(x x for any ν(x P k (x i, x i+1, were E n,d z and E n,d z,i,j ( θ n B n,d y, (x, y j + (1 θ n B n,c y, (x, y j ν(xdx dx + En,D z,i+1,j ν(x i+1 E n,d z,i,j ν(x i, (18 (x, y = E z (U n,d (x, y, B n,d (x, y = E n,d z (x i, y j for any i and j, and θ n is te same as before. 11

12 Here b i,j x and b i,j y approximate B x (x i, y for y (y j, y j+1 and B y (x, y j for x (x i, x i+1 at t n+1, respectively, and tey were used in [20] to reconstruct an exactly divergence-free magnetic field B n+1,c of te second order accuracy wen k = 1 and of te tird order accuracy wen k = 2. However for k > 2, since {b i,j x, b i,j y } i,j are defined only on mes skeleton, tey alone are insufficient to reconstruct a function in M C,k wic is defined on te wole domain Ω. In order to extract more information about B, we revisit (7 as a real d- dimensional system of equations instead of a set of one dimensional problems on te mes skeleton, and discretize it using te standard central DG metod of relatively lower order accuracy. More specifically, for k 2, we look for B V C,k 2 any i, j, C i,j B vdx = suc tat for any v = (v 1, v 2 in V C,k 2 Ci,j = [P k 2 (C i,j ] d wit C i,j ( θ n B n,d t n ( C i,j + (1 θ n B n,c vdx (19 Ez n,d ( v 2 x v 1 y dx + Ez n,d C i,j (n 2 v 1 n 1 v 2 ds were (n 1, n 2 is te outward pointing unit normal vector along C i,j. It is assumed ere and later [P r (K] d = {0} for any negative integer r. In fact if te trial and test spaces above are replaced wit V C,k, te sceme wit (15 and te modified (19 is just te (k + 1st order central DG metod in [24] applied to te ideal MHD system. In our algoritm, only te P k 2 part of tat numerical magnetic field is needed. Wit {b i,j x, b i,j y } i,j from (17- (18 and B from (19, we are now ready to introduce an element-by-element reconstruction wic defines B n+1,c k 0. = (B n+1,c x,, B n+1,c y, for an arbitrary index, 12

13 Reconstruction. Given i and j, reconstruct B n+1,c Ci,j W k (C i,j suc tat B n+1,c = (B n+1,c x,, B n+1,c y, satisfying (i B n+1,c x, (x l, y = b l,j x (y for l = i, i + 1 and y (y j, y j+1, (ii B n+1,c y, (x, y l = b i,l y (x for l = j, j + 1 and x (x i, x i+1, (iii For k 2, tere is also C i,j ( B n+1,c B vdx = 0, v [P k 2 (C i,j ] d. One can see tat te normal component of te reconstructed magnetic field B n+1,c across element interfaces is given eiter by b i,j x index i and j. Te remaining degrees of freedom of B n+1,c or b i,j y determined by B in suc a way tat te L 2 projection of B n+1,c wit some wen k 2 are onto V C,k 2 is exactly B. Te main results for te reconstruction are summarized in next Teorem, wit its proof given in Section 4. Teorem 3.1. For any k 0, (R1 B n+1,c Ci,j W k (C i,j is uniquely determined. (R2 B n+1,c Ci,j = 0, and terefore B n+1,c M C,k and it is exactly divergence-free. (R3 Wen k = 1, 2, te reconstructions are te same as tose in [20]. Remark 3.2. (1 To define te reconstruction, we ave used Property 1 in Lemma 2.1. In fact, (i and (ii are equivalent to (i For l = i, i + 1, ( y j+1 y j P k (y j, y j+1, B n+1,c x, 13 (x l, y b l,j x (y w(ydy = 0, w

14 (ii For l = j, j +1, ( x i+1 x i P k (x i, x i+1. B n+1,c y, (x, y l b i,l y (x w(xdx = 0, w Tese equivalent formulations were used in [8] wen BDM elements were introduced. (2 Teorem 3.1 implies tat te local reconstruction procedure defined above produces an exactly divergence-free magnetic field B n+1,c. Te index k 0 can be arbitrary. Not like in oter divergence-free reconstruction strategies, suc as in [4, 22, 20] were te reconstructed magnetic field being divergence-free is part of te definition of te reconstruction, in our reconstruction, tis is a derived property. 4. Proof of Teorem 3.1 Te following Lemma provides a key relation of {b i,j x } i,j, {b i,j y } i,j and B. It is related to K V nwds = K Vwdx + K V wdx, (20 an equality derived from te divergence teorem, and it ensures te reconstructed magnetic field to be divergence-free. Lemma 4.1 (Relation of {b i,j x } i,j, {b i,j y } i,j and B. For any w P k 1 (C i,j, tere is C Θ nwds = i,j B wdx, k > 1, C i,j 0, k = 1. (21 14

15 Here C i,j Θ nwds := yj+1 y j + x i+1 x i b i+1,j x (yw(x i+1, ydy y j+1 y j b i,j x (yw(x i, ydy b i,j+1 y (xw(x, y j+1 dx x i+1 x i b i,j y (xw(x, y j dx.(22 Proof. For any w P k 1 (C i,j wit k 1, by taking te test function in (17-(18 to be w, we ave = = C i,j Θ nwds (23 yj+1 + y j xi+1 x i yj+1 y j b i+1,j x (yw(x i+1, y b i,j x (yw(x i, ydy b i,j+1 y (xw(x, y j+1 b i,j y (xw(x, y j dx ( θ n B n,d x, (x i+1, y + (1 θ n B n,c x, (x i+1, y w(x i+1, ydy + t n ( yj+1 yj+1 y j t n ( yj+1 xi+1 + x i t n ( xi+1 xi+1 x i Ez n,d (x i+1, y w(x i+1, y y j y ( θ n B n,d x, (x i, y + (1 θ n B n,c x, (x i, y Ez n,d (x i, y w(x i, y y j y ( θ n B n,d y, (x, y j+1 + (1 θ n B n,c y, (x, y j+1 x i + t n ( xi+1 E n,d z dy E n,d z,i+1,j+1 w i+1,j+1 + E n,d z,i+1,j w i+1,j w(x i, ydy dy E n,d z,i,j+1 w i,j+1 + E n,d z,i,j w i,j w(x, y j+1 dx (x, y j+1 w(x, y j+1 dx E n,d z,i+1,j+1 x w i+1,j+1 + E n,d z,i,j+1 w i,j+1 ( θ n B n,d y, (x, y j + (1 θ n B n,c y, (x, y j x i E n,d z (x, y j w(x, y j x w(x, y j dx dx E n,d z,i+1,j w i+1,j + E n,d z,i,j w i,j All terms containing Ez n,d w at vertices are perfectly canceled. Wit furter 15.

16 simplification, one gets Θ nwds C i,j = = = C i,j C i,j + C i,j ( θ n B n,d ( θ n B n,d C i,j ( θ n B n,d ( θ n B n,d + (1 θ n B n,c nwds t n + (1 θ n B n,c wdx + (1 θ n B n,c wdx t n + (1 θ n B n,c wdx t n For te last step, we use te fact tat bot B n,c divergence-free. Ez n,d C i,j Ez n,d C i,j Ez n,d C i,j and B n,d ( w x n 2 w y n 1ds ( w x n 2 w y n 1ds ( w x n 2 w y n 1ds. are exactly Wen k = 1, w is constant on C i,j and its gradient is zero. Terefore C i,j Θ nwds = 0, and tis gives (21 for k = 1. For k > 1, we furter take v = w [P k 2 (C i,j ] d in (19, ten ( B wdx = θ n B n,d + (1 θ n B n,c wdx C i,j C i,j t n ( w x n 2 w y n 1ds, Ez n,d C i,j and tis is exactly te same as C i,j Θ nwds, ence (21 olds for k > 1. Remark 4.2. (1 Te proof of (21 relies on te use of te same θ n in (17, (18 and (19. Tis θ n can be different from te one in (15, wic is not directly related to te reconstruction. (2 Wit w 1, (21 becomes yj+1 y j b i+1,j x (ydy yj+1 y j b i,j x (ydy+ 16 xi+1 x i xi+1 b i,j+1 y (xdx b i,j y (xdx = 0, x i

17 and tis is te compatible condition in [20]. (3 Te electric field flux {E n,d z,i,j } i,j, used to discretize te induction equations (17 and (18, are evaluated based on te numerical solution on te dual mes, tey are single-valued at te grid-points of te primal mes and terefore all relevant terms are canceled out. For Godunov type metods [5, 16, 4] under te constrained transport framework, additional interpolation or averaging procedure is required to produce a single-valued electric magnetic flux at te grid-points. Certain upwind mecanism also needs to be incorporated for stability consideration especially for ig order scemes. We are now ready to prove Teorem 3.1 for te proposed reconstruction. Proof of Teorem 3.1. Step 1. For (R1, we first prove te result wen k 1 by following [8]. Note tat for two dimensions wit d = 2, te reconstruction of B n+1,c Ci,j involves 4(k dim([p k 2 ] d = 4(k d k(k 1 2 = k 2 + 3k + 4 conditions, wic equals to te dimension of W k (C i,j. Terefore to get (R1, one only needs to sow tat wit te zero data, namely, b i,j x (y = b i+1,j x (y = 0, b i,j y (x = b i,j+1 y (x = 0, and if k > 1 tere is also B(x, y Ci,j = 0, te reconstructed B n+1,c Ci,j as to be zero. To tis end, we denote B n+1,c = ( B n+1,c x, = B n+1,c y, ( w1 w 2 ( ( x k+1 (k + 1xy k + a 1 + a (k + 1x k 2 y y k+1 (24 in C i,j wit w l (x, y = 0 r+s k a(l r,sx r y s. According to (i in te definition 17

18 of te reconstruction, and b i,j x B n+1,c x, (x, y = 0 r+s k s k and b i+1,j x being zero, tere is a (1 r,sx r y s + a 1 x k+1 + (a (1 0,k + a 2(k + 1xy k = 0 at x = x i and x i+1. Since te last term is te only one containing te monomial y k, it as to be zero and terefore a 2 = 0. Wit a similar argument to B n+1,c y,, one can furter sow a 1 = 0. Now (24 becomes (w 1, w 2 [P k (C i,j ] 2 were w 1 vanises at x i, x i+1 and w 2 vanises at y j, y j+1. Tis indicates, if k = 1, (w 1, w 2 as to be zero and so does B n+1,c Ci,j. if k > 1, tere is (w 1, w 2 = ( (x x i (x x i+1 w 1, (y y j (y y j+1 w 2 wit some ( w 1, w 2 [P k 2 (C i,j ] 2. Now we can take v = ( w 1, w 2 in (iii of te reconstruction. Wit te assumption B(x, y Ci,j gets ( w 1, w 2 = (0, 0, and ence B n+1,c Ci,j = 0. = 0, one Next we want to prove (R1 wen k = 0. In tis case, ( ( B n+1,c a1 x = + c y in C i,j for some constants a 1, a 2 and c, and te reconstruction becomes a 2 (25 a 1 + cx i+1 = b i+1,j x, a 1 + cx i = b i,j x, a 2 cy j+1 = b i,j+1 y On te oter and, Lemma 4.1 wit w 1 for k = 0 gives (y j+1 y j (b i+1,j x b i,j x + (x i+1 x i (b i,j+1 y b i,j y = 0,, a 2 cy j = b i,j y. (26 wic ensures tat te system (26 is uniquely solvable, in particular wit c = bi+1,j x b i,j x x i+1 x i = bi,j+1 y b i,j y y j+1 y j, a 1 = b i,j x cx i, a 2 = b i,j y + cy j, (27 18

19 and terefore te reconstruction is well-defined for k = 0. Step 2. For (R2, we first sow tat te reconstructed B n+1,c is divergencefree on C i,j. For k = 0, tis comes directly wit te explicit formula (25 and (27 of te reconstruction. For k 1, consider any w P k 1 (C i,j, tere is B n+1,c wdx = B n+1,c wdx + (B n+1,c nwds C i.j C i.j C i.j = B wdx + (Θ nwds = 0. (28 C i.j C i.j Te last two equalities are due to te definition of te reconstruction and Lemma 4.1. Moreover, Lemma 2.1 implies B n+1,c tis, one can take w = B n+1,c P k 1 (C i,j. Wit in (28 and gets B n+1,c Ci,j = 0. Now B n+1,c Ci,j W k (C i,j, it is divergence-free in eac mes element and as continuous normal component across element interfaces, terefore B n+1,c M C,k and it is exactly divergence-free. Step 3. Finally, we want to prove (R3. Tat is, for k = 1 or k = 2, our proposed reconstruction and te one in [20] (also given in Appendix A are te same. First note tat (i and (ii in our current reconstruction are te same as (i and (ii in Appendix A. Suppose B n+1,c is te reconstructed magnetic field on C i,j based on te current definition. Wit (R2 proved above, B n+1,c Ci,j = 0, and terefore tis B n+1,c satisfies (i -(iii in Appendix A. Suppose B n+1,c is te reconstructed magnetic field on C i,j based on te definition in Appendix A (and also [20]. We only need to sow tat wen k = 2, tis B n+1,c satisfies (iii of te current reconstruction. For any v = ( v1 v 2 [P k 2 (C i,j ] d, one can write it as v = w wit w = v 1 x + v 2 y 19

20 P k 1 (C i,j. Based on (21, tere is C i,j Θ nwds = C i,j B wdx, were C i,j Θ nwds is defined in (22. Wit tis ( B n+1,c B ( vdx = B n+1,c B wdx C i,j C i,j = B n+1,c wdx + B n+1,c nwds B wdx C i,j C i,j C i,j = B n+1,c wdx + Θ nwds B wdx C i,j C i,j C i,j = B n+1,c wdx = 0. C i,j Te last equality uses tat te reconstructed magnetic field B n+1,c is divergencefree on C i,j due to (iii in Appendix A. Remark 4.3. Te relation (21 is only used to sow te reconstructed magnetic field being divergence-free in (R2, and it is not needed for te unique solvability in (R1 of te proposed reconstruction. 5. Furter discussions 5.1. Divergence-free discrete spaces For te numerical metods proposed in Section 3, te divergence-free subspace of a H(div-conforming discrete space, te BDM finite element space, is used to approximate te magnetic field B. Witin te current framework, one can also use some oter H(div-conforming finite element spaces, suc as te Brezzi-Douglas-Fortin-Marini (BDFM [9] or Raviart- Tomas (RT [28] finite element spaces. Of all tree, te BDM finite element space is te smallest in order to acieve te same order of accuracy in L 2 norm. 20

21 5.2. Relation to some oter divergence-free metods Our metods are closely related to [2, 4] and [21, 22] among various exactly divergence-free numerical metods in te constrained transport framework for MHD simulations. In all tese metods (including ours, one first approximates te normal component B n (or B n in tree dimensions of te magnetic field on mes interfaces, and ten applies a divergencefree reconstruction procedure, wit te reconstructed magnetic field aving te same normal component on mes interfaces as obtained in te previous step. Besides te difference in te base metods, suc as being finite volume [2, 4, 21, 22] or finite element [20] (and tis work metods, being Godunov [2, 4] or central [21, 22, 20] type, tese metods are also different or related in two more aspects: one is in te coice of te discrete spaces te reconstructed magnetic field belongs to; te oter is te additional conditions wic may be needed to uniquely determine te reconstruction after one matces te normal component on mes interfaces. In particular, our reconstruction uses te BDM finite element space, it is te same as te one in [2] for k = 1, and for k = 2 it is te same as te one in [21]. Wen k = 3, bot our metod and [22] employ te same discrete space BDM k. Wile being divergence-free is part of te definition of te reconstruction in [22], it is a derived property of te reconstructed magnetic field in our approac. After one matces te normal component of te magnetic field on mes interfaces, te additional conditions to ensure te unique solvability of te reconstruction are also quite different. Te strategy proposed in our current work relies on extensive use of te magnetic induction equations in order to extract sufficient information about te magnetic 21

22 field, it provides reconstructions for any integer index k 0 and results in exactly divergence-free numerical metods of arbitrary order of accuracy. Te actual performance of te metods wit iger k certainly needs furter investigation. Te additional conditions in [22] on te oter and are motivated eiter by using more compact stencil in reconstruction or by better resolving planar, grid-aligned flows. In [4], te divergence-free reconstruction uses different discrete spaces for te magnetic field: wen k = 2, te space is BDFM k+1 [9], and wen k = 3, a space between BDM k and BDFM k+1 is used. In bot cases, te additional conditions to ensure te unique solvability of te reconstruction are motivated by minimizing te magnetic energy. Te reconstructions in [4] are given for tree dimensions, and te two-dimensional version can be obtained by neglecting te spatial variable z. It is not easy to see wic strategy of [22], [4] and our current work is better in terms of accuracy and stability wen uniquely determining te reconstruction of exactly divergence-free ig order accurate magnetic fields in MHD simulations. Tis is partially due to te difference in te base metods. Our proposed strategy seems to be a promising candidate as it relies on extensive use of te magnetic induction equations. More analysis is needed to better understand our approac and to evaluate te strengt of different strategies Hig order time discretizations To acieve better accuracy in time, strong stability preserving (SSP ig order time discretizations [18] can be used. Suc discretization can be written as a convex combination of te forward Euler metod, and terefore te full sceme wit a ig order SSP time discretization still produces an exactly 22

23 divergence-free approximation for te magnetic field. For multi-stage time discretizations suc as SSP Runge-Kutta metods, one needs to apply te divergence-free reconstruction for eac inner stage Nonlinear limiters Wen central DG metods are applied to nonlinear problems suc as te ideal MHD system, nonlinear limiters are often needed. In tis paper we use te minmod TVB slope limiter in [29, 12], wic is simple yet involves a nonnegative parameter M. Tis limiter can be implemented componentwisely or in local caracteristic fields. We take M = 10 in all numerical simulations in Section 6 wenever te limiter is applied. Different values of M or oter limiters [27, 23] may produce better results for eac individual example, and tis will not be explored in tis paper. As in [20], wen nonlinear limiter tecnique is used, it is only applied to U, not to B, B or (b i,j x, b i,j y i,j. Tis is to ensure tat te property (21 in Lemma 4.1 still olds. Suc a strategy may not always be able to effectively control te oscillation in te magnetic field and terefore may affect te numerical stability of te algoritm. Numerical experiments for k = 3 in tis paper and for k = 1, 2 in [20] owever indicate tat te proposed metods wit suc limiting strategy perform well wen tey are applied to many commonly used two-dimensional MHD examples, wic include some examples involving a discontinuous magnetic field, suc as te Orszag-Tang vortex problem, te rotor problem, and te blast problem. 23

24 6. Numerical examples In tis section, we report a set of numerical experiments to illustrate te accuracy and stability of te proposed metods. Except for te first tree examples, we only present te numerical results of te fourt order metod wit k = 3. Te formulation of te local reconstruction used in tis case is given in Appendix B. In all simulations, te primal mes is Cartesian and uniform wit messizes x and y. Te tird order TVD Runge-Kutta metod [12] is used as te time discretization, wit te time step dynamically determined by Here c x f and cy f C cfl ( max ( ux +c. (29 x f + max ( uy +cy f x y are te fast speed in x and y directions, respectively (see [26] for te definition, and te CFL number C cfl is taken as 1.0 for k = 1, 0.6 for k = 2, and 0.3 for k = 3 unless oterwise specified. We use θ n = 1 in (15 and (17-(19. Wen demonstrating te accuracy order for k = 3, one needs to reduce te CFL number by a factor c 0 as te mes is refined in order to matc te tird order temporal accuracy wit te fourt order spatial accuracy. Te value of c 0 will be specified below for individual example. All reported results are based on te numerical solutions on te primal mes. Te minmod TVB slope limiter wit M = 10 is applied to non-smoot examples in order to enance te numerical stability. One can refer to [20] for more discussion on initialization and boundary conditions. 24

25 6.1. Accuracy test Smoot Alfvén wave Te first example describes a circularly polarized smoot Alfvén wave ([30, 21, 20]. Following [21], te initial conditions are taken as ρ = 1, u = 0, u = 0.1 sin(2πβ, u z = 0.1 cos(2πβ, B = 1, B = u, B z = u z, p = 0.1, on Ω = [0, 1/ cos α] [0, 1/ sin α]. Here α = π/4 is te angle wit respect to te x-axis at wic te wave propagates, and β = x cos α + y sin α. Te subscripts and denote te directions parallel and perpendicular to te wave propagation direction, respectively. Te boundary conditions are periodic and γ = 5/3. Te Alfvén wave travels at a constant Alfvén speed B / ρ = 1 and te solution returns to its initial configuration wen t is an integer. In te simulation wit k = 3, te CFL number C cfl is reduced by a factor c 0 = 0.5 wen te mes is refined from N N to N/2 N/2, wit te initial C cfl as 0.4. No nonlinear limiter is applied. In Table 1, we report L 2 errors and orders of accuracy for u x, u z, B x and B z at t = 2. Te results sow tat te proposed metods are (k + 1-st order accurate wen k = 1, 2, 3, and tey are optimal. We also want to use tis example to illustrate some advantage of ig order metods. In our simulations, a tird order TVD Runge-Kutta metod is applied as te time discretization for k = 1, 2, 3. On a given mes, wen k increases, smaller CFL number (ence smaller time step needs to be taken due to stability consideration, wile te computational complexity for one time step increases. It turns out tat te increase in te overall computational cost on a given mes by using iger order metods is paid back by 25

26 muc smaller errors and terefore better solution resolution, and tis can be seen from Table 1. In addition, we also report te computational time in te last column of Table 1, and te computational time (orizontal axis versus L 2 error in Figure 1 for u x (left and u z (rigt. Te results sow tat in order to reduce te errors of approximating solutions to a given tresold especially wen tis tresold is small, it is more cost efficient to work wit ig order metods. Tis is consistent to many oter studies in literature. Our results are obtained from simulations performed on a computer wit 2.27 GHz Intel Core 2 Duo processor and 12GB DDR3 memory Figure 1: Computational time (orizontal axis versus L 2 error in u x (left and u z (rigt for te smoot Alfvén wave problem at t = 2. From top to bottom: k = 1 (solid star in blue, k = 2 (circle in red, k = 3 (diamond in black Smoot vortex problem Te second example, introduced in [3], describes a smoot vortex propagating at te speed (1, 1 in te two-dimensional domain. Te initial condi- 26

27 Table 1: L 2 errors, orders of accuracy, and computational time (in seconds for te smoot Alfvén wave problem at t = 2. C cfl is 1.0 for k = 1 and 0.6 for k = 2. For k = 3, C cfl is 0.4 on te initial mes wit a reduction factor c 0 = 0.5. u x u z B x B z N L 2 error Order L 2 error Order L 2 error Order L 2 error Order Time k = E E E E E E E E E E E E E E E E k = E E E E E E E E E E E E E E E E k = E E E E E E E E E E E E E E E E tions are (ρ, u x, u y, u z, B x, B y, B z, p = (1, 1 + δu x, 1 + δu y, 0, δb x, δb y, 0, 1 + δp 27

28 wit (δu x, δu y = ξ 2π exp{0.5(1 r 2 }, (δb x, δb y = η 2π exp{0.5(1 r 2 }, and δp = η2 (1 r 2 ξ 2 8π 2 exp(1 r 2. Here r = x 2 + y 2, ξ = 1, η = 1, and γ = 5/3. In te simulation wit k = 3, te CFL number C cfl is reduced by a factor c 0 = (1/2 1/3 wen te mes is refined from N N to N/2 N/2, wit te initial C cfl as 0.3. Te computational domain is taken as [ 10, 10] [ 10, 10] wit periodic boundary conditions. Suc boundary condition treatment introduces an error wit te magnitude O(e = O(10 22, wic is muc smaller tan te errors we report and terefore will not affect te accuracy study. Note tat for k = 1, 2, a smaller computational domain [ 5, 5] [ 5, 5] wit periodic boundary conditions is sufficient and tis was used in [20]. In Table 2, we present L 2 errors and orders of accuracy for representative variables ρ, u x, B x and p at t = 20. Te results confirm te optimal accuracy orders of our proposed metods. No nonlinear limiter is applied in te computation Numerical dissipation and long-term decay of Alfvén waves We next investigate te numerical dissipation of te proposed metods by examining te long-term decay beavior of torsional Alfvén waves, wic propagate at a small angle to te y-axis. Tis problem was tested in [3, 22] wit various numerical scemes or metods wit different accuracy orders. We use te following initial conditions ρ = 1, u x = 0.2n y cos φ, u y = 0.2n x cos φ, u z = 0.2 sin φ, B x = n x + 0.2n y cos φ, B y = n y 0.2n x cos φ, B z = u z, p = 1. 4π 4π 28

29 Table 2: L 2 errors and orders of accuracy for te smoot vortex example on [ 10, 10] [ 10, 10] at t = 20. C cfl is 1.0 for k = 1 and 0.6 for k = 2. For k = 3, C cfl is 0.3 on te initial mes wit a reduction factor c 0 = (1/2 1/3. ρ u x B x p N L 2 error Order L 2 error Order L 2 error Order L 2 error Order k = E E E E E E E E E E E E E E E E k = E E E E E E E E E E E E E E E E k = E E E E E E E E E E E E E E E E Here (n x, n y = (, r r 2 +1 r is te wave propagation direction, and φ = π n y (n x x+n y y is te initial pase of te wave. Te simulation is performed in te domain Ω = [ r/2, r/2] [ r/2, r/2] wit r = 6. Boundary conditions 29

30 are periodic and γ = 5/ k=1 k=2 k= Figure 2: Te temporal evolution of max Ω u z up to t = 120 on a mes x60 90x90 120x x60 90x90 120x Figure 3: Te temporal evolution of max Ω u z up to t = 120 wit k = 2 (left and k = 3 (rigt on various meses. For tis example, te maximum values of u z and B z are constant in time for te exact solution, yet tey will decay in simulations due to numerical dissipation. In Figure 2, we present te temporal evolution of te maximum value of u z, max Ω u z, up to t = 120 for k = 1, 2, 3, and te mes is 30

31 Only te results for u z is presented as it as a similar decay rate as B z. Apparently, te fourt order sceme (k = 3 as te smallest numerical dissipation wile te second order sceme (k = 1 as te largest. In particular, for te fourt order sceme, te absolute cange in max Ω u z is max Ω u z t=0 max Ω u z t=120 = O(10 5 over suc a long time simulation. Moreover, tere is significant improvement from te second order sceme to iger tan second order scemes, indicating te advantages of using ig order metods in controlling numerical dissipation. Tis as been observed or analyzed for many oter ig order metods, suc as te divergence-free central finite volume metods in [22], and discontinuous Galerkin metods in [11, 1]. In Figure 3, we also present te results for k = 2 (left and k = 3 (rigt on different meses: 60 60, and Wile te numerical dissipation is smaller for finer grids, it as muc less dependence on te messizes for iger order metods Orszag-Tang vortex problem In tis subsection, we consider te Orszag-Tang vortex problem wic is a widely used test example in MHD simulations. Te initial conditions are taken as [19, 20] ρ = γ 2, u x = sin y, u y = sin x, u z = 0, B x = sin y, B y = sin 2x, B z = 0, p = γ, wit γ = 5/3. Te computational domain is [0, 2π] [0, 2π] wit periodic boundary conditions. Te solution involves formation and interaction of multiple socks as te nonlinear system evolves, and tis can be seen from 31

32 te time evolution of density ρ in Figure 4. Te computation is carried out wit k = 3 on a mes. Numerical evidence in [19, 22] suggests tat insufficient treatment of te divergence-free condition may affect te numerical stability for tis example. For instance, te DG metod wit te P 2 approximation breaks down in te simulation, and tis is partially overcome wen te locally divergencefree DG metod wit te same accuracy is used [19]. Wit our exactly divergence-free central DG metods, te numerical simulation can continue stably for k = 1, 2 [20] and for k = 3 up to t = 30 (te maximum time we run, and te simulation can still go on. We furter perform a qualitative convergence study for tis example. In Figure 5, we plot pressure p (left wit y = at t = 2 using meses and , and k = 3. Wit socks developed in te solution at tis time, convergence can be observed, and te pressure slices are almost te same as te second and tird order exactly divergence-free central DG approximations in [20] and te locally divergence-free DG approximations in [19]. Te reported results are computed wit te nonlinear limiter implemented in te local caracteristic fields. Toug tis limiter is applied only to U, tere is no significant oscillation in numerical solutions, see Figure 5 (rigt for te numerical B x wit x = π at t = 3. Finally, we want to point out tat no negative pressure is encountered trougout te simulation Rotor problem Next we consider te rotor problem wic was explained in greater details in [5]. Te problem describes a dense disk of fluid rapidly spinning in a ligt 32

33 Figure 4: Density ρ in Orszag-Tang vortex problem at t = 0.5 (top left, t = 2 (top rigt, t = 3 (bottom left, and t = 4 (bottom rigt, wit k = 3 on a mes. Fifteen equally spaced contours wit ranges [2.11, 5.84], [0.62, 6.25], [1.25, 6.03], and [1.28, 5.71], respectively. ambient fluid. Following [30, 20], te starting setup is given as (u z, B x, B y, B z, p = (0, 2.5/ 4π, 0, 0, 0.5, and (ρ, u x, u y = (10, (y 0.5/r 0, (x 0.5/r 0 if r < r 0 (1 + 9λ, λ(y 0.5/r, λ(x 0.5/r if r 0 < r < r 1 (1, 0, 0 if r > r 1 33

34 Figure 5: Orszag-Tang vortex problem wit k = 3 on (circle and (solid line meses. Left: p wit y = at t = 2; Rigt: B x wit x = π at t = 3. wit r = (x (y 0.5 2, r 0 = 0.1, r 1 = and λ = (r 1 r/(r 1 r 0. Te simulation is implemented in te domain [0, 1] [0, 1], and periodic boundary conditions are used wit γ = 5/3. Te nonlinear limiter is applied in local caracteristic fields. It turns out tat nonlinear limiters are necessary in tis example wen k = 3 for numerical stability, toug our metods witout limiters produce satisfactory results for k = 1 and k = 2, see [20]. Similar to [5, 30, 19, 20], in Figure 6 we examine te performance of te metods by zooming in te central part of te Mac number u /c (wit c = γp/ρ as te sound speed at t = Note tat tere is no distortion in te numerical solutions, and suc distortion was reported in [30, 19] and it was attributed to te divergence error in te magnetic field. We furter perform a qualitative convergence study for our metod. In Figure 7, Mac number is presented wit x = and k = 3 on , and meses. Wit several socks developed in te solution, te 34

35 convergence of te metod is observed, and te solutions slices are almost te same as te second and tird order exactly divergence-free central DG approximations in [20] and te locally divergence-free DG approximations in [19]. Toug te nonlinear limiter is applied only to U in te simulation, tere is no significant oscillation in numerical solutions, see Figure 8 for B x wit x = 0.25 and B y at y = 0.5. Tese slices are cosen from te region were te magnetic field displays more interesting features. It was reported in [30] tat many one step TVD based scemes failed for tis problem due to te negative pressure, in our simulation, tere is no negative pressure observed Figure 6: Zoom-in central part of Mac number in rotor problem at t = wit k = 3. Tirty equally spaced contours on a mes wit te range of [0, 2.703] (left, and on a mes wit te range of [0, 3.005] (rigt Blast problem Te blast wave problem was introduced in [5], and te solution involves strong magnetosonic socks. We employ te same initial condition as in [5, 21, 20], tat is, (ρ, u x, u y, u z, B x, B y, B z = (1, 0, 0, 0, 100/ 4π, 0, 0. Te pressure is taken as p = 1000 if r R, and p = 0.1 oterwise, were r = 35

36 Figure 7: Mac number in rotor problem at t = wit x = and k = 3. Left: (circle and (solid line meses; Rigt: (circle and (solid line meses Figure 8: Te magnetic field in rotor problem at t = wit k = 3 on (circle and (solid line meses. Left: B x wit x = 0.25; Rigt: B y wit y =

37 x2 + y 2 and R = 0.1. Wit tis setup, te fluid in te region outside te initial pressure pulse as very small plasma β ( = p (B 2 x+b 2 y/2 = 2.513E 04. Te simulation is implemented in te domain [ 0.5, 0.5] [ 0.5, 0.5] wit a mes and k = 3. Outgoing boundary conditions are used, and γ = Figure 9: Te blast problem on a mes at t = 0.01 wit k = 3. Forty equally spaced contours are used. Top left: density ρ [0.206, 4.602]; Top rigt: pressure p [ 3.225, ]; Bottom left: square of total velocity u 2 x + u 2 y [0, ] ; Bottom rigt: magnetic pressure Bx 2 + By 2 [ , ]. In Figure 9, we present te numerical results at t = 0.01 for density ρ, pressure p, square of te total velocity u 2 x + u 2 y, and te magnetic pressure 37

38 Bx 2 + By. 2 As pointed out in [5, 21], tis is a stringent problem to solve. Negative pressure is generated near te sock front in our simulation, and tis is also observed in many oter metods wic are not positivity preserving (see e.g. [21]. In Figure 10 we plot te negative part of pressure min(0, p. Note tat at t = 0.01, te minimum value of pressure for k = 3 is wic is sligtly smaller tan 1.633, te minimum value of pressure of te tird order approximation (wit k = 2 wen M = 1 is used in te minmod TVB slope limiter in [20]. In fact, in all cases wit k = 1, 2, 3, te magnitude of te negative pressure is fairly small for tis low plasma β example, and tis illustrates te good performance of te proposed metods. We furter report solution slices for ρ and u 2 x + u 2 y along y = 0.0 in Figure 11, and for B x along x = 0.0 and B y along y = 0.25 in Figure 12 on and meses. Tese slices are cosen from te region were te considered field displays more interesting features. Te sceme performs well in convergence along wit mes refinement. In addition, tere is no significant oscillation observed toug te nonlinear limiter is only applied to U. For tis example, te componentwise TVB minmod limiter is applied. Since we only explicitly use pressure to determine te time step from (29, wen negative pressure occurs, max(p, ɛ wit ɛ = 10 5 is used to replace p in order to estimate te maximum wave speed in (29. Tis simple fix results in stable simulation during te time of our interest. Wen te limiters are implemented in local caracteristic fields, te stability of te simulations suffers from te numerical pressure being negative. It is expected tat positivity preserving tecniques will be very important for tis example to ensure te numerical stability, and tis is currently under investigation in 38

39 a separate project Figure 10: Negative part of te pressure min(0, p in te blast problem wit k = 3 on a mes at t = Figure 11: Te blast problem at t = 0.01 wit k = 3 and y = 0.0 on (circle and (solid line meses. Left: ρ; Rigt: u 2 x + u 2 y Cloud-sock interaction Te last example we consider is te cloud-sock interaction problem wic involves strong MHD socks interacting wit a dense cloud. We define 39

40 Figure 12: Te blast problem at t = 0.01 wit k = 3 on (circle and (solid line meses. Left: B x wit x = 0.0; Rigt: B y wit y = tree sets of data for (ρ, u x, u y, u z, B x, B y, B z, p, U 1 = ( , 0, 0, , 1, 0, , , U 2 = (1, , 0, 0, 1, 0, 1, 0.04, U 3 = (5, , 0, 0, 1, 0, 1, Te computational domain [0, 2] [0, 1] is divided into tree regions: te postsock region Ω 1 = {(x, y : 0 x 1.2, 0 y 1}, te pre-sock region Ω 2 = {(x, y : 1.2 < x 2, 0 y 1, (x (y }, and te cloud region Ω 3 = {(x, y : (x (y < 0.18}, were te solutions are initialized as U 1, U 2, and U 3, respectively. Note tat te cloud is five times denser tan its surrounding. Outgoing boundary conditions are used, and γ = 5/3. Figure 13 sows te gray-scale images of density ρ, te magnetic field B x and B y, and pressure p at t = 0.6 on te mes wit k = 3. Te darker area represents relatively smaller value. In Figure 14, we also plot te density ρ along y = 0.6 and B y along y = 0.59 on and

41 meses. Note te main features in Figure 13 and Figure 14 (left are te same as te tird order approximations in [20], yet wit some difference in local details. Tere is no significant oscillation in te solution, given tat te limiter is only applied to U. For tis example, te nonlinear limiter is implemented in local caracteristic fields. We would like to mention tat tis cloud-sock example is not te same as te one in [13] as different scalings are used in te MHD system for te magnetic field (see our equations (1-(4 and teirs on page 486 in [13]. Figure 13: Gray-scaled images in te cloud-sock interaction problem wit k = 3 on a mes at t = 0.6. Top left: ρ [1.853, ]; Top rigt: B x [ 1.750, 4.131]; Bottom left: B y [ 2.723, 2.723]; Bottom rigt: p [6.536, ]. 41