Locally divergence-free spectral-dg methods for ideal magnetohydrodynamic equations on cylindrical coordinates

Transcription

1 Locally divergence-free spectral-dg methods for ideal magnetohydrodynamic equations on cylindrical coordinates Yong Liu, Qingyuan Liu, Yuan Liu 3, Chi-Wang Shu 4 and Mengping Zhang Abstract In this paper, we propose a class of high order locally divergence-free spectraldiscontinuous Galerkin (DG) methods for three dimensional (3D) ideal magnetohydrodynamic (MHD) equations on cylindrical geometry. Under the conventional cylindrical coordinates (r, ϕ, z), we adopt the Fourier spectral method in the ϕ-direction and discontinuous Galerkin (DG) approximation in the (r, z) plane, motivated by the structure of the particular physical flows of magnetically confined plasma. By a careful design of the DG approximation function space, our spectral-dg methods are divergence-free inside each element for the magnetic field. Numerical examples with third order strong-stability-preserving Runge-Kutta methods are provided to demonstrate the efficiency and performance of our proposed methods. Keywords: Discontinuous Galerkin method; Magnetohydrodynamics (MHD); Positivity-preserving; Divergence-free; Cylindrical coordinates. School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 3006, P.R. China. yong3@mail.ustc.edu.cn School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 3006, P.R. China. qyliu@ustc.edu.cn 3 Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 3976, USA. yliu@math.msstate.edu 4 Division of Applied Mathematics, Brown University, Providence, RI 09, USA. shu@dam.brown.edu. Research supported by ARO grant W9NF and NSF grant DMS School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 3006, P.R. China. mpzhang@ustc.edu.cn. Research supported by NSFC grant 4730.

2 Introduction The ideal magnetohydrodynamics (MHD) equations are fluid models of perfectly conducting quasi-neutral plasmas, which have been widely used in astrophysics, space physics and plasma applications. Mathematically, the ideal MHD equations consist of nonlinear hyperbolic conservation laws for the macroscopic quantities with an additional divergence-free restriction on the magnetic field, which take the following form ρ ρu ρu t E + ρu u + (p + B )I B B u(e + p + B ) B(u B) = 0, (.) B u B B u B = 0, (.) with E = p γ + ρ u + B. (.3) Here, ρ is density of mass, ρu is momentum, E is total energy, p is the hydrodynamic pressure, is used to denote the Euclidean vector norm and γ is the ideal gas constant. The ideal MHD equations (.) consist of eight coupled partial differential equations which are usually not solvable analytically. Directly simulating the full 3D MHD system is in general very difficult due to the high demand on computational resources. However, the computational cost can be reduced for certain problems when considering geometric symmetry or structures. MHD computations regarding the stable equilibrium of magnetically confined plasma usually take place on cylindrical or toroidal geometry [4, 6] and can be solved with cylindrical coordinates efficiently. It is known that for some physical problems such as the investigation of static equilibria and their perturbations, an important flow feature is toroidal symmetry or slow changes in the toroidal direction such as tokamak. In such cases, it is reasonable to assume that the radial component of the flow is smoother in the ϕ direction while the other components may involve large jumps [,, 39]. Starting with this knowledge, dimension decomposition or dimension reduction can be performed to simplify the 3D MHD computation [3, 3, 40, 7, ]. Here, we propose to use a similar approach to adopt the Fourier spectral method in the ϕ direction with a few modes and the DG approximation on the (r, z)-plane. By doing so, the 3D MHD computation is simplified to lower dimensions, and the computational cost is significantly reduced, which makes the numerical implementation more practical.

3 Another challenge in designing numerical methods for the ideal MHD equation is the divergence-free condition. That is, if the divergence of the initial magnetic field is zero, the divergence of the exact magnetic field at any future time is also zero. Honoring this divergence-free property for the numerical methods not only keeps consistency with the analytical conclusion but also avoids possible numerical instability that may cause break-down of the simulation. There are a lot of developments on divergence-free numerical methods for MHD equations in the literature, and those can be mainly categorized into four general approaches, i.e. the 8-wave formulation [, 34, 33, 9], the projection scheme [, 9], hyperbolic divergence-cleaning methods [3, ] and the constrained transport (CT) framework [, 4, 0, 4, 6, 8, 4, 30, 3, 3, 37, 4, 44, 43, 36]. In order to fulfill the divergence-free condition in our spectral-dg method, we will utilize the CT framework. The CT framework is to maintain a discrete version of the divergence-free condition. This framework was first introduced in [6] on staggered meshes for the magnetic field. Later in [43], together with the investigation and comparison of several variations of methods for ideal MHD equations, an unstaggered version in the CT framework is developed. Since then, tremendous efforts have been made within the CT framework both on staggered and unstaggered meshes. Among those works, the design of high-order and high-resolution numerical methods with shock-capturing capabilities has gained a lot of attention. Such methods include finite volume / finite difference essentially non-oscillatory (ENO) and weighted ENO (WENO) methods [3,, ], DG methods [8, 6] and central DG schemes [8, 4, 7]. The approach we use in the spectral-dg method is closely related to the locally divergence-free Runge-Kutta DG (RKDG) method which was originally developed in [6] for solving D ideal MHD equations with Cartesian meshes. The key step there is to design a special local divergence-free test function space for the magnetic field which can make the divergence-free condition automatically hold in each cell. This strategy has been further explored in the past years and many local and global divergence-free DG methods have been proposed, see [7, 8, 8, 4, 36, 9]. However, all the above mentioned works are on Cartesian meshes and for D MHD computation. In this paper, we will perform the nontrivial generalization to 3D MHD computation with cylindrical coordinates. The rest of the paper is organized as follows. In Section we will briefly review the MHD equation with cylindrical coordinates. In Section 3, we present our proposed high order spectral-dg methods which honor the locally divergence-free condition for 3D MHD simulation. The proposed schemes are implemented and tested on several 3

4 D and 3D numerical examples in Section 4. The conclusions are given in Section. MHD equation on cylindrical coordinates In the cylindrical coordinates (r, ϕ, z), the coordinate-free systems (.) with conservative variables can be expressed as [38] where U t + F r + G z + H r ϕ = r S r F (.) U = (ρ, ρu r, ρu z, ρu ϕ, B r, B z, B ϕ, E) T (.) F = ( ρu r, ρu r + p Br, ρu z u r B z B r, ρu ϕ u r B ϕ B r, 0, u r B z u z B r, u r B ϕ u ϕ B r, (.3) u r (p + E) B r (u r B r + u ϕ B ϕ + u z B z )) T, G = ( ρu z, ρu r u z B r B z, ρu z + p Bz, ρu ϕu z B ϕ B z, u z B r u r B z, 0, u z B ϕ u ϕ B z, (.4) u z (p + E) B z (u r B r + u ϕ B ϕ + u z B z )) T, H = ( ρu ϕ, ρu r u ϕ B r B ϕ, ρu z u ϕ B z B ϕ, ρu ϕ + p Bϕ, u ϕb r u r B ϕ, B z u ϕ u z B ϕ, 0, (.) u ϕ (p + E) B ϕ (u r B r + u ϕ B ϕ + u z B z )) T, S =(0, ρu ϕ + p Bϕ, 0, ρu ϕ u r + B r B ϕ, 0, 0, u r B ϕ u ϕ B r, 0) T. (.6) In particular, u Λ and B Λ denote the velocity and magnetic field in the Λ direction, with Λ = r, z, ϕ. p is defined as the total pressure p = p + (B r + B z + B ϕ ) (.7) and p is given as p = (γ )(E ρ(u r + u ϕ + u z) (B r + Bz + Bϕ)). (.8) Moreover, the divergence-free condition (.) is equivalent to under cylindrical coordinates. rb r r + rb z z + B ϕ ϕ = 0 (.9) 3 Numerical methods 3. The spectral-dg framework In this section, we present our spectral-dg scheme for solving 3D MHD equations on cylindrical geometry T = Ω [0, π]. Here, (r, z) Ω = [a, b] [c, d] and ϕ [0, π]. 4

5 To define our numerical method, we perform the spatial partition T = i,j,s K i,j ϕ s, where K i,j is a rectangular element in the (r, z) plane, K i,j = {(r, z) : r i for i N and j M, where and r i a = r c = z and z j = z j+ r r i+, z j < r3 < z3 z j < < r N+ < < z M+ = b z z j+ } = d. are denoted as the element sizes in the r r i = r i+ and z directions respectively. In the ϕ-direction, the interval [0, π] is discretized as 0 = ϕ 0 < ϕ <... < ϕ L = π. We firstly use the Fourier spectral method to perform discretization in the ϕ- direction, since the solution is periodic in this direction and also is smoother in this direction than in other directions for the particular flow problems we are interested in, as discussed in the introduction section. Denote u h (r, z, ϕ s, t) as the semi-discrete numerical solution at each grid point in ϕ s, the flux function H(u h ) can be expressed as H(u h ) L l= L where the discrete Fourier coefficients Ĥl takes the form Ĥ l = L L s= Ĥ l e ilϕ (3.) H(u h (r, z, ϕ s, t))e ilϕs, l = L,..., L. (3.) Direct calculation will give us the approximation of H ϕ as L H ϕ (ilĥl)e ilϕ. (3.3) l= L The system (.) can then be rewritten as U t + F r + G z = r S r F H r ϕ. (3.4)

6 where the ϕ-related term is treated as a source term. Notice that (3.4) is defined on the rectangular domain of the (r, z) plane, and we propose to numerically solve u h (r, z, ϕ s, t) by the DG methods. Define the finite element space of polynomial V k h = { v h : v h Ki,j {P k (K i,j )} 8, i N, j M } (3.) where P k (K i,j ) is the space of polynomials of degree at most k over the elements K i,j. Following the traditional RKDG formulation, we need to find u h Vh k, such that for an arbitrary test function v h Vh k u h K i,j t v h drdz = F v h K i,j r drdz + + zj+ z j ( ˆF i+ ri+ + ( Ĝj+ r i + ( K i,j r S r F r v h (r, z) + ˆF i+ i v h (r +, z)) dz i K i,j G v h z drdz v h (r, z ) + j+ Ĝj v h (r, z + dr j )) H ϕ ) v h drdz (3.6) where ˆF i+, Ĝ j+ are the so-called numerical fluxes. In this paper, the local Lax- Friedrichs flux is used for all the numerical implementation, ˆF i+ (U +, U ) = (F(U+ ) + F(U ) α i+ (U + U )) (3.7) where U ± are the values of U from both sides at the cell interface x i+ and α i+ is the local maximum (in magnitude) of the eigenvalues of the flux Jacobian in the normal direction of the cell boundary. After the spatial spectral-dg discretization, the semi-discrete ODE system is evolved by the third order Runge-Kutta method (u h ) t = L(u h, t) (3.8) u () h = u n h + tl(un h, tn ), u () h = 3 4 un h + 4 u() h + 4 tl(u() h, tn + t), (3.9) u n+ h = 3 un h + 3 u() h 3. The divergence-free condition + 3 tl(u() h, tn + t). It is easy to verify that the proposed spectral-dg framework does not necessarily satisfy the divergence-free condition (.9). In this subsection, we propose to modify 6

7 the DG approximation of B r and B z in order to construct the locally divergence-free spectral-dg method. This will be realized with the help of a special solution set and test function space for the magnetic field. Specifically, in the spectral-dg method (3.6), when using the DG scheme to approximate B r and B z, we propose to use the solution function set Ṽk h defined as Ṽ k h = {v h : v h Ki,j {P k (K i,j )} 8, rb r r k and the test function space Ṽh defined as k Ṽh {v = h : v h Ki,j {P k (K i,j )} 8, rb r r + rb z z + rb z z = B } ϕ ϕ, i N, j M, (3.0) } = 0, i N, j M. (3.) Notice that Ṽk h is not a linear space but a function set, but functions in it can be expressed as a particular polynomial function satisfying the constraint plus general k functions in the linear space Ṽh. Clearly, the locally divergence-free condition holds automatically for the magnetic field B in Ṽk h. Here, we call Ṽk h the divergence-free set k and Ṽh the divergence-free space. Obviously, Ṽh is a subspace of the standard DG test function space Vh k, and the dimensionality decreases due to the extra constraints. To clearly illustrate how we perform the numerical implementation based on the k divergence-free set and space Ṽk h and Ṽ h, we present the details of the P case in the following with the understanding that the method can be easily generalized to the arbitrary P k case in the same fashion. On each DG element K i,j = [r i, r i+ ] [z j, z j+ ], we let ξ = r r i and η = z z j, then the standard orthogonal basis function r z space is V (K i,j ) = {φ, φ, φ 3, φ 4, φ, φ 6 }, where φ 4 = k φ =, φ = ξ, φ 3 = η, (3.) (ξ ), φ = ξη, φ 6 = (η ), with ξ and η. For simplicity, we denote r i and z j as r and z in the following section when K i,j is the target cell. 3.. The axisymmetric case In the case of axial symmetry under cylindrical coordinates, the divergence-free condition is rb r r + rb z z 7 = 0. (3.3)

8 Denote 6 6 B r = Bs r φ s, B z = Bs z φ s (3.4) s= s= where Bs r and Bz s are the DG coefficients obtained by (3.6). Plug into (3.3) and utilize the orthogonality of basis functions, we obtain B r + r i ( r Br + z Bz 3 ) + r( r Br 4 + z Bz ) = 0, B r + r i ( r Br 4 + z Bz ) + r z Bz 3 = 0, B3 r + r i ( r Br + z Bz 6) = 0, 3B r 4 + r z Bz = 0, B r + z Bz 6 = 0, B r 6 = 0. (3.) It is easy to see the original degrees of freedom for B r and B z decrease to 6 due to the extra constraints (3.), and it can help us to form a new test function space with degree 6 for B r and B z. Specifically, we choose {B r, B z, B z, B z 4, B z, B z 6} as the independent unknown DG coefficients. Introducing the notation [ ] [ ] a a a 3 a 4 a a 6 a φ = + a φ + a 3 φ 3 + a 4 φ 4 + a φ + a 6 φ 6, b b b 3 b 4 b b 6 b φ + b φ + b 3 φ 3 + b 4 φ 4 + b φ + b 6 φ 6 we can directly write out the basis functions satisfying (3.) as {ˆφ, ˆφ, ˆφ 3, ˆφ 4, ˆφ, ˆφ 6 } where ˆφ = [ ] [ ] ˆφ = [ ] ˆφ 3 = ri ˆφ 4 = β r 0 0 z r 8

9 ri r 0 0 r ˆφ = β 3 r z z ri r ˆφ 6 = β ri r 0 0 z z in which β, β and β 3 are positive constants such that the basis functions are normalized. Since {φ s } 6 s= are orthonormal, it is easy to check that {ˆφ s } 6 s= are orthogonal except that < ˆφ 4, ˆφ > 0. By a further orthogonalization process, we derive the divergence-free orthonormal basis for B r and B z as and Ṽ (K i,j ) = { φ, φ, φ 3, φ 4, φ, φ 6 } (3.6) φ = ˆφ, φ = ˆφ, φ3 = ˆφ 3, [ e r,4 e φ r,4 ] = 0 0 e z, φ = [ e r, e r, 0 e r, ] e z, 3 0 e z, 0 (3.7) φ 6 = [ 0 0 e r,6 3 0 e r,6 ] e z,6 6 wherein and e r,4 = β ri r, e r, = θ (β r i r 3 r z er,4 = β, e z,4 3 = β z r, θ e r,4 ), er, e r, 4 = θ β r 3 z, ez, 3 = θ ( e r,6 3 = β 3 ri z, er,6 = β 3 r z, = θ θ e r,4, ri 3 r β θ e z,4 3 ), ez, = θ β, ez,6 6 = β 3, θ =< ˆφ 4, ˆφ >, θ =. θ (3.8) 9

10 By the design of the divergence-free test function space, it is clear that our spectral- DG method is locally divergence-free. However, in the actual implementation, instead of using the divergence-free space Ṽk h (K i,j) to approximate the magnetic field separately, we perform the standard DG discretization (3.6) for all the quantities and then modify the DG coefficients of B r and B z in order to obtain the divergence-free magnetic field. Let B = (B r, B z ) T, through projection, we can get B = 6 bs φs. (3.9) s= where b s =< B, φ s >. We then examine the integral formulation of the basis functions { φ s } 6 s= (3.7), we obtain (B) t φ s dξdη. By the (B) t φ dξdη = R z, (B) t φ dξdη = R z, (B) t φ 3 dξdη = R z 4, (B) t φ 4 dξdη = e r,4 R r + e r,4 R r + e z,4 3 R z 3, (B) t φ dξdη = e r, R r + e r, R r + e r, 4 R r 4 + e z, 3 R z 3 + e z, R z, (B) t φ 6 dξdη = e r,6 3 Rr 3 + er,6 Rr + ez,6 6 Rz 6. (3.0) Here, R Λ s = and Λ is taken as the variable r or z. On the other hand, it is easy to see that (B Λ ) t φ s dξdη, s =,,..., 6, (3.) (B) t φ s dξdη = d dt b s, s =,..., 6 (3.) 0

11 because of the orthonormal property of the basis functions. Let (3.0) equal to (3.), d = R dt b z, d = R dt b z, d 3 = R4 dt b z, (3.3) d 4 = e dt b r,4 R r + e r,4 R r + e z,4 3 R3, z d = e dt b r, R r + e r, R r + e r, 4 R4 r + e z, 3 R3 z + e z, R, z d 6 = e dt b r,6 3 R3 r + e r,6 R r + e z,6 6 R6. z In order for (3.3) to hold for arbitrary t, we propose to modify the DG approximation of B r and B z as follows, where and B r = e r,4 B 4 r = er, 4 b4 + e r, B r = 6 B s r φ s, B z = s= b, Br = e r,4 b, Br = e r,6 b4 + e r, 6 B s z φ s (3.4) s= b, Br 3 = e r,6 3 b6, Br 6 = 0. B z = b, Bz = b, Bz 3 = e z,4 3 B z 4 = b 3, Bz = e z, b, Bz 6 = e z,6 6 b4 + e z, 3 b6. b6, b, (3.) (3.6) Next, we will show that the magnet field B obtained through our modification (3.4) satisfies the locally divergence-free condition. Theorem 3.. Coupled with third order Runge-Kutta time discretization (3.9), the magnetic field B obtained by the spectral-dg scheme (3.6) and (3.4)-(3.6) is locally divergence-free. Proof: We only need to show that the DG approximation with (3.4) is equivalent to that obtained by the DG scheme with the divergence-free test function space. Our proof is carried out with the forward Euler method, which also applies to the Runge- Kutta scheme (3.9) since it is a convex combination of the forward Euler methods. From (3.3), the discretization can be written as bn+ s b n s = tr z s, s =,, 3. bn+ 4 b n 4 = t(er,4 Rr + er,4 R r + ez,4 3 Rz 3 ), bn+ b n = t(e r, R r + e r, R r + e r, 4 R r 4 + e z, 3 R z 3 + e z, R z ), bn+ 6 b n 6 = t(er,6 3 R r 3 + er,6 R r + ez,6 6 R z 6 ). (3.7)

12 When s =, using (3.) we can further obtain that bn+ b n = tr z = B z,n+ B z,n. (3.8) Here Bs Λ,n+ represents the numerical approximation of the DG coefficients Bs Λ and before we perform the divergence-free modification. Since B,n z = b n, we have Similarly, for s =, 3, we get at tn+ bn+ = B z,n+. (3.9) bn+ = B z,n+, bn+ 3 = B z,n+ 4. (3.30) For s = 4, plug in the discretization of R Λ s will result in bn+ 4 b n 4 = e r,4 (B r,n+ B r,n ) + e r,4 (B r,n+ B r,n ) + e z,4 3 (B z,n+ 3 B z,n 3 ). (3.3) Since e r,4 B r,n + e r,4 B r,n + e z,4 3 B z,n 3 = ((e r,4 ) + (e r,4 ) + (e z,4 3 ) ) b n 4 + (er,4 e r, + e r,4 e r, + e z,4 3 e z, 3 ) b n = b n 4, (3.3) we can derive that bn+ 4 = e r,4 B r,n+ + e r,4 B r,n+ + e z,4 3 B z,n+ 3. (3.33) Similar derivations can be made to b and b 6, which will give us bn+ = e r, Br,n+ + e r, Br,n+ + e r, 4 Br,n+ 4 + e z, bn+ 6 = e r,6 3 B r,n+ 3 + e r,6 B r,n+ + e z,6 6 B z,n Bz,n+ 3 + e z, Bz,n+, It is easy to see that B n+ is divergence-free, which completes our proof. (3.34) 3.. The non-axisymmetric case In the case of non-symmetry, i.e. cylindrical coordinates is equivalent to B ϕ ϕ 0, the divergence-free condition under rb r r + rb z z = B ϕ ϕ (3.3) Since both terms on the left hand side of equation (3.3) are in P polynomial space throughout our DG discretization, we project the right hand side term Bϕ ϕ,

13 obtained from the Fourier spectral approximation, into the P space. This computation will explicitly give us with B ϕ ϕ 6 c s φ s (3.36) s= Then the divergence-free condition (3.3) is equivalent to c s =< B ϕ ϕ, φ s >. (3.37) (rb r ) r + (rb z) z = 6 c s φ s. (3.38) s= in the numerical simulation. We assume that the solution of the non-homogeneous equation (3.38) is in the form of B = q(t) + 6 bs φs. (3.39) Here, q(t) is a special solution of the nonhomogeneous equation (3.38). It is easy to verify that q(t) can be chosen as q(t) = s= [ ] g r 0 g3 r g4 r g r g6 r 0 0 g3 z. (3.40) where g Λ s are given as g r = c g r 3 = c 3 ri ( ri ) c r 3 r c 4 3 c 4, ri r c, g4 r = 3 c 4, g r = c, g r 6(t) = c 6, g z 3 = z r ( c 3 ri ) r c 4. (3.4) Similar to the axis-symmetric case, we examine the integral Using the solution form (3.39), we have (B) t φ s dξdη. (B) t φ s dξdη = d dt b s + q t φ s dξdη, s =,..., 6. (3.4) 3

14 have Let (3.4) equal to (3.0), together with the known form of q(t) (3.40), we will d = R dt b, z d = R dt b z, d 3 = R4 dt b z, d ) ( b4 + g r er,4 + g3 z ez,4 3 dt d dt d dt ( b + e r, gr + er, 4 gr 4 + ez, 3 gz 3 ) ( b6 + e r,6 3 gr 3 + er,6 gr = e r,4 Rr + er,4 Rr + ez,4 ) 3 Rz 3, = e r, Rr + er, Rr + er, 4 Rr 4 + ez, 3 Rz 3 + ez, Rz, = e r,6 3 Rr 3 + er,6 Rr + ez,6 6 Rz 6. (3.43) Similarly, in order that (3.43) holds throughout the numerical evolution, we propose to modify the DG approximation of B r and B z as B r = 6 s= B r s φ s, B z = 6 s= B z s φ s (3.44) where B r = er,4 b 4 + e r, b + g r, Br = e r,4 b 4 + e r, b, B r 4 = e r, 4 B r 3 = er,6 3 b 6 + g r 3 b + g r 4, Br = e r,6 b 6 + g r, Br 6 = 0. (3.4) and B z = b, Bz = b, Bz 3 = e z,4 3 b 4 + e z, 3 b + g z 3, B z 4 = b 3, Bz = e z, b, Bz 6 = e z,6 6 b6. (3.46) This completes our construction of the DG approximation of B r and B z. Theorem 3.. Coupled with the third order Runge-Kutta time discretization, the magnetic field B obtained by (3.44) has the divergence-free property. The proof is similar to that for Theorem 3. and we omit the details here. 4 Numerical examples In this section, we perform numerical simulations with our spectral-dg methods in 3D space to demonstrate their accuracy and efficiency. For the simulation of extreme cases, where the density or pressure is very small, negative numerical density or 4

15 pressure may be observed. In order to maintain the positivity of those two quantities and avoid numerical instability, we apply the positivity-preserving limiter for the DG scheme introduced in [8, 46]. We note that a more restrictive CFL condition is used with the application of the positivity-preserving limiter. Example 4.. Consider the 3D non-axisymmetric problem on the domain (r, ϕ, z) [π, 4π] [0, π] [0, π] with periodic boundary condition. The initial conditions are taken as ρ 0 = + sin(r + z), (u r, u ϕ, u z ) = (,, ), B r = sin(r) sin(z) sin(ϕ), B B z = cos(r) cos(z) sin(ϕ), p The constant γ = /3. ϕ = sin(r) sin(z) cos(ϕ), = + cos(r + z). (4.) It is easy to find that the exact solution has the form ρ 0 = + sin(r + z t), (u r, u ϕ, u z ) = (,, ), B r = sin(r) sin(z t) sin(ϕ), B ϕ = sin(r) sin(z t) cos(ϕ), (4.) B z = cos(r) cos(z t) sin(ϕ), p = + cos(r + z t), and the initial magnetic field B = (B r, B ϕ, B z ) satisfies the divergence-free condition (.9). We use the proposed spectral-dg method to simulate this problem, in which the partition in the ϕ-direction is fixed as L = 4. In Table 4., errors with different norms are presented at T = 0. and third order optimal accuracy is attained. Example 4.. We consider the Orszag-Tang vortex problem which is a widely used test example in MHD simulations and the numerical results with Cartesian coordinates have been well demonstrated in the literature [0, 4]. We take the initial conditions [0] ρ =, u = ( sin(πz), sin(πr), 0), 36π p = π, B = ( sin(πz), sin(4πr), 0) 4π The computational domain is taken to be (r, z) [, ] [0, ] with periodic boundary conditions on all sides, the constant γ = /3.

16 Table 4.: Error table for B r and B z, CFL = 0., T = 0. N component L -err L -ord L -err L -ord L -err L -ord 0 B r.6e-0.3e-0 3.3E-03 0 B z.94e-0.74e-0 4.7E-03 0 B r.30e e E B z 3.03E E E B r 3.47E E E B z 4.38E E E B r 4.60E E E B z.9e E E (a) (b) Figure 4.: Distributions of (a) density and (b) the divergence of the magnetic field in each side of cell in the Orszag-Tang vortex problem. The smooth initial conditions evolve into a more complex flow with many discontinuities. We use uniform meshes to implement the schemes. The solution at t = 0. is shown in Fig. 4.a on the grid of the size The divergence of the magnetic field in each side of elements, which is defined in [6], is shown in Fig. 4.b. Note that our method is locally divergence-free but there is divergence across element boundaries since the normal magnetic field is discontinuous across these boundaries, and we can see that such divergence is larger near the shocks and other discontinuities of the flow. We also note that the result is not symmetric, different from the case in Cartesian coordinates. Since we have just used the (r, z) variables to replace the (x, y) in the initial conditions, and the MHD equations in cylindrical coordinates are not symmetric about the variable r, the nonsymmetric result is expected. The positivity-preserving limiter is used in this example for the positivity of the density and pressure. 6

17 Example 4.3. In this example, we consider the rotor problem which was originally introduced in [4]. Following the setup in [44], we take the computational domain (r, z) [, ] [0, ] with periodic boundary conditions on all sides, the constant γ =.4 and initial condition is given as follows. (0, u 0 ( (z /)), u 0 (r 3/)) for R < r 0 r 0 r 0 (ρ, u r, u z ) = ( + 9f, fu 0 R ( (z /)), fu 0 R (r 3/)), f = r R for r 0 R < r r r 0 (, 0, 0) for r R (4.3) where R = (r 3/) + (z /), r 0 = 0., r = 0. and u 0 =. The rest of the quantities are constants and are given by u ϕ = 0, p =.0, B = (B r, B ϕ, B z ) = 4π (, 0, 0) We perform the numerical simulation to T = 0.. In the ϕ direction, we use L = 4 and mesh partition is used for DG approximation on the (r, z) plane. The numerical density ρ and pressure p are reported in Fig. 4.a and Fig. 4.b. The contours of the Mach number u /c s with the sound speed c s = γp/ρ are presented in Fig. 4.3a, from which we can clearly see that our methods can capture the structures of the physical quantities well. Besides, we can observe that the divergence of the magnetic field is successfully controlled (except near the shocks and other discontinuities of the flow) as shown in Fig. 4.3b. Here, we note that, different from the Cartesian coordinate case, our numerical results is not symmetric. This is because the initial condition is not symmetric under the cylindrical coordinates. Example 4.4. In this last example, we consider 3D jet in cylindrical coordinates. We take γ = /3 and initialize the jet interior region where r r jet =. and z z jet = 3 with density ρ jet = 00 and axial velocity v z,jet =. In the region outside the jet, the medium density ρ ext = 00 and axial velocity v z,jet = 0. The magnetic field configuration is such that the magnetic energy is contained inside the jet and respect the solenoid nature of a magnetic field. We set the magnetic field as B r (r, z) = B 0 rjet (z/z jet ) 3 tanh((z/z jet ) 4 ) tanh((r/r jet ) ) rz jet cosh((z/z jet ) 4 ) B 0 B z (r, z) = cosh((z/z jet ) 4 ) cosh ((r/r jet ) ) (r/r jet ) B ϕ (r, z) = B cosh((z/z jet ) 4 ) cosh ((r/r jet ) ) (4.4) 7

18 (a) (b) Figure 4.: Contour plots of (a) density and (b) pressure in the rotor test. (a) (b) Figure 4.3: Contour lines of (a) Mach numbers and (b) the divergence of the magnetic field in each side of cell in the rotor test. 8

19 (a) (b) Figure 4.4: Contour plot of (a) Log 0 ρ and (b) B ϕ in the 3D jet where B 0 is the poloidal jet magnetic field strength measured at the axis and B controls the field helicity. The value of B 0 and B will also control the geometry of the terminal shock. We set B 0 = 0, B = to lead to a perpendicular shock. We set up the simulation on a cylindrical domain of radius r = 0 and extended to z = 40. The boundary conditions are open boundaries for r = 0, z = 40 and also at the base of the jet when r > r jet. Boundary conditions are frozen to the initial values at the base of the jet where r < r jet. The jet axis is treated as usual with a combination of symmetric and antisymmetric conditions. In Fig. 4.4, we display the logarithmic contours of density and divergence of the magnetic field at time t = 60 with mesh at ϕ = 0. We can clearly observe that the jet propagates through the medium creating a cocoon cavity that isolates the jet from the external medium. The cocoon is created by jet material expelled from the front shock located at the head of the jet. The numerical results we obtained is in good agreement with the solutions in [7]. Conclusion In this paper, we have proposed a class of high order locally divergence-free spectral- DG method for solving 3D MHD equations. Since the variable in the ϕ direction 9

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