Numerical Pseudo-Convergence for a MHD Model System Manuel Torrilhon Abstract. Recently, a strong non-uniform convergence of finite volume schemes for MHD has been demonstrated for special Riemann problems, see [17] and [18]. These Riemann problems have a unique solution but are close to initial conditions with non-unique solutions. This paper discusses the use of a model system in order to understand and improve the behavior of the schemes. The nonuniform convergence for the model system is demonstrated and the modified solver of Myong/Roe is investigated. Results for higher order schemes and adaptive meshes are mentioned. However, a numerical scheme that avoids pseudo-convergence has not been found so far. Contribution to Proc. 10th Intl. Conf. on Hyperbolic Problems, 2004
1. Full ideal MHD equations The equations of ideal magnetohydrodynamics couple the Euler equations with the induction equation, see e.g., [11]. The system consideres the variables U = {ρ, ρv, E, B}, i.e., density, momentum, energy and magnetic field. The 3d equations are given by ∂t ρ (1)
∂t ρv ∂t E ∂t B
+ div ρv
+ div ρvv + (p + 21 B2 )I − BBT + div (E + p + 12 B2 )v − BB · v T T T
+ div Bv − vB
=0 =0 =0 =0
1 together with the constitutive relation E = γ−1 p + 21 ρv2 + 21 B2 and divergence constraint div B = 0. The system forms a hyperbolic system of conservation laws. Numerical methods for the MHD system may be found e.g., in [2] or [5].
1.1. MHD Riemann problem In one dimension the vector of variables is written U = {ρ, ρvn , ρvt , E, Bn , Bt } with the normal and transverse components of velocity and magnetic field. In the
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Figure 1. Left: Solution of a MHD Riemann problem, Right: Convergence study following the representation Bt = Bt (cos ϕ, sin ϕ)T of the transverse field will be used frequently. The Riemann problem for MHD considers initial condition of the form U1 x ≤ 0 (2) U (x, t)|t=0 = U0 x ≥ 0 with constant values U0,1 . The left hand side of Fig. 1 shows the transverse magnetic field and the density of a typical MHD Riemann solution. Generally, the vector Bt changes its direction only in Alvenic waves, so-called rotational waves [11], while all other waves change only its amplitude. In the following we will set ϕ1 ≡ 0 and define α = ϕ0 − ϕ1 as the twist angle of the initial conditions. (0) (1) For α = π and vt = vt the MHD Riemann problem is known to be non-unique [4], [3]. The classical or regular solution contains a 180◦-rotational waves while the second solution exhibits a non-classical compound wave. The relevance and stability of these non-classical waves is subject of discussions, e.g., in [7], [8], [10], [13] and [19]. It is not the intention of this paper to contribute to this discussion. (1) (0) Important for the following is the fact that for vt = vt the Riemann problem is only non-unique if α = π, [16].
1.2. Non-uniform convergence We consider the Riemann problem (2) with left hand state U1 = 1, 0, 0, 1, (1, 0)T , 1 and right hand state U0 = 0.2, 0, 0, 1, (cos α, sin α)T , 0.2 with α 6= π which has a unique solution. In [17] a thorough convergence study of several standard MHD finite volume schemes has shown a distinct non-uniform convergence behavior if α is close to π for virtually any scheme. Solutions for different grid resolutions are shown
PSEUDO-CONVERGENCE FOR A MHD MODEL SYSTEM
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in the right hand side of Fig. 1 in the case of α = 3.0. The numerical solution seems to approximate the not admissible co-planar compound-wave solution on coarse grids. The behavior is quantified in the first row of Fig. 2 where error plots based on exact Riemann solutions [15] are shown with respect to the wrong c-solution (left) and the true solution (right). The initial pseudo-convergence to the wrong c-solution is clearly visible. In [17] the behavior is investigated and related to the results in [9] and the time dependent phanomenon in [20]. The crucial condition for the cancellation of the pseudo-convergence is the amount of Bz present in the profile of the rotational wave. If the integral Z ! (3) Az = Bz dBy > A(crit) z reaches the critical value the true wave patterns arise. The build-up of Bz depends on the numerical viscosity of the method. The question is how to develop numerical methods that do not show or reduce the initial non-uniform convergence.
1.3. Higher order and adaptivity Two possibilities to improve the performance of the numerical scheme are the use of higher order methods and specially adapted meshes. The middle row of Fig. 2 shows the error curves for the standard 2nd-order-scheme together with the results for 5thand 9th-order WENO methods, see [1]. Due to the reduced numerical viscosity the pseudo-convergences cancels at coarser meshes. However, the improvement is very small. This is due to the fact that all methods reduce to lower order around discontinuities and thus in the crucial region of our test problem. the extensive description of the higher order results are given in [18]. The crucial region of the 1dcomputation can easily be detected by searching for large variation in the angle ϕ of the transverse magnetic field Bt . This information can be used to successively refine this area in an adaptive computation. The lower row of Fig. 2 shows the error curves for the coarsest mesh of calculations with [6] using 4 and 5 grid levels cascaded in the crucial area. The adaptive mesh adds a fixed number of approximately 100 cells to the computation. Due to the time-step-wise restriction of the fine grid solution, the coarse mesh shows an improved convergence result. Essentially, the resolution in the highest level of the fine grids reaches the resolution needed for the cancellation of the pseudo-convergence. However, the recursive procedure to update the adaptive mesh reduces the speed of the computation considerably.
2. The model system In [12] (see also [9]) a model system is investigated which mimics the slow, alvenic and fast wave families of the full MHD system. It consideres three variables U = {u, v, w} which may be related to {ρ, By , Bz } of the MHD system. The governing
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Figure 2. Error curves for a standard 2nd-order-scheme with different values of α (upper) and, in the case of α = π, for higher order schemes (middle) and for adaptive meshes (lower)
equations are given by (4)
∂t u ∂t v ∂t w
+ ∂x cu2 + v 2 + w2 + ∂x (2 u v) + ∂x (2 u w)
=0 =0 =0
in the 1d case. The system has one linearly degenerated wave with speed λ = 2u, which shows the rotational character of the Alven wave. The other two wave families are genuinely non-linear and are related to the slow and fast MHD waves. In the following we discuss how the model system can be used to guide the way to improved numerical methods for MHD.
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Figure 3. Non-unique solutions of the model system (left), solutions in phase space (right)
2.1. Non-unique Riemann solutions The Riemann solutions of the model system correspond to the three MHD waves to the left or to the right of the contact surface. The solution for the initial conditions (1.2, 1.65, 0.0)T x≤0 (5) U (x, t)|t=0 = (0.3, 0.85 cos α, 0.85 sin α)T x ≥ 0 with α = π is non-unique. Both solutions are shown in the left hand side of Fig. 3. The structure is analogous to the co-planar MHD Riemann problem described in Sec. 1.1. The right hand side of Fig. 3 shows the solutions in phase space (u, v, w) which is rotationally symmetric around the u-axis. The Sc -wave curve for the compound wave can only connect co-planar states, while the rotational waves connect to a non-planar or co-planar plane. This shows directly that the Riemann problem of the model system is only non-unique, if state 0 is in the same plane as state 1, i.e., α = π.
2.2. Convergence study We consider the Riemann problems of the model system (5) for α = 2.9, 3.0, 3.1 which have a unique solution and conduct the same study of convergence as in the MHD case using a classical 2nd-order Roe-scheme. The error curves with respect to the (wrong) co-planar solution and the true solution are shown in the first row of Fig. 4. The show exactly the same pseudo-convergent behavior as in the full MHD case, see Fig. 2. Hence, it is worthwhile to investigate numerical methods for the simpler model system in order to find a way to avoid the pseudo-convergence. The hope is that these results may then be extended to the more complicated full MHD system.
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Figure 4. Error curves for the model system for various α (upper), results for the solver of Myong/Roe (middle), results with modified initial conditions (lower)
2.3. Modified solver and initial conditions In [14] Myong and Roe proposed a modified Riemann solver for the model system in order to improve the representation of the rotational waves. The ordinary Roeaverage for the model system uses v¯ = 21 (vR + vL ) and analogous expressions for w. ¯ The authors of [14] realize that (¯ v , w) ¯ will almost vanish for the almost co-planar case. They used (v, w) = r (cos ϕ, sin ϕ) and introduced the average L rR + r L cos ϕR +ϕ v¯ 2 = . (6) L w ¯ sin ϕR +ϕ 2 2 Furthermore, they solve the planar equations at each cell interface and find an apropriately rotated flux afterwards. Note that this can not been done in the MHD case due to strong coupling. While the new scheme is successfull for stationary rotational waves [14], there is no improvement for the pseudo-convergence of the
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present test problem. The corresponding error curves are shown in the middle of Fig. 4. As described in Sec. 1.2 the main problem is the slow build-up of Bz , that is, w in the rotational wave. This is especially problematic since w is a conservative quantity. The above modification tries to incorporate higher values of w into the solver, but apearently not enough. To show the relevance of the amount of w we (1) (0) and analogous expressions for modify the initial conditions. Using ∆ϕ = ϕ N−ϕ +1 ∆r and ∆u with a fixed integer N we define u(0) + n ∆u (7) Un = (r(0) + n ∆r) cos(α(0) + n ∆α) , for n = 0, 1, ..., N + 1 (r(0) + n ∆r) sin(α(0) + n ∆α) and modify the initial conditions by (1) x/∆x ≤ −N/2 U Un N/2 − n ≤ x/∆x ≤ N/2 − n + 1, n = 1, 2, ...N . (8) U (x, t)|t=0 = (0) U x/∆x ≥ N/2
This smoothes U (x, 0) in a narrow zone (−N/2 ∆x, N/2 ∆x) around the origin which is filled with a rotational distribution of (v, w). The error curves obtained from solutions with N = 1, 3 and 5 are shown in the lower row of Fig. 4. The pseudo-convergence indeed vanishes. However, this modifications incorporates a priori knowledge about the w-build-up and is not applicable to more complicated, e.g., 2d, situations.
3. Conclusions The MHD Riemann problem in Sec. 1.1 with α = 3.0 has a unique solution consisting of rarefaction fans, Lax-shocks and linear waves. A numerical method for ideal MHD should be able to approximate this solution with a reasonable convergence rate. This paper and the work in [17] and [18] shows that this is not the case for a huge class of 2nd and higher order methods. While this paper did not present a method that works, it investigated a promising model system which might help to design improved numerical methods. The solver proposed by Myong/Roe in [14] does not show any improvement for the convergence, though it increases the amount of Bz in the solver. A modification of the initial conditions shows that this is the key in circumventing the pseuod-convergence.
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[email protected]
