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On the role of Riemann solvers in Discontinuous Galerkin methods for magnetohydrodynamics

机译：关于黎曼求解器在磁流体动力学非连续Galerkin方法中的作用

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摘要

It has been claimed that the particular numerical flux used in Runge-Kutta Discontinuous Galerkin (RKDG) methods does not have a significant effect on the results of high-order simulations. We investigate this claim for the case of compressible ideal magnetohydrodynamics (MHD). We also address the role of limiting in RKDG methods. For smooth nonlinear solutions, we find that the use of a more accurate Riemann solver in third-order simulations results in lower errors and more rapid convergence. However, in the corresponding fourth-order simulations we find that varying the Riemann solver has a negligible effect on the solutions. In the vicinity of discontinuities, we find that high-order RKDG methods behave in a similar manner to the second-order method due to the use of a piecewise linear limiter. Thus, for solutions dominated by discontinuities, the choice of Riemann solver in a high-order method has similar significance to that in a second-order method. Our analysis of second-order methods indicates that the choice of Riemann solver is highly significant, with the more accurate Riemann solvers having the lowest computational effort required to obtain a given accuracy. This allows the error in fourth-order simulations of a discontinuous solution to be mitigated through the use of a more accurate Riemann solver. We demonstrate the minmod limiter is unsuitable for use in a high-order RKDG method. It tends to restrict the polynomial order of the trial space, and hence the order of accuracy of the method, even when this is not needed to maintain the TVD property of the scheme.
机译：据称，在Runge-Kutta间断Galerkin（RKDG）方法中使用的特定数值通量对高阶模拟的结果没有显着影响。我们针对可压缩理想磁流体动力学（MHD）的情况调查了这一主张。我们还解决了RKDG方法中限制的作用。对于平滑的非线性解决方案，我们发现在三阶仿真中使用更精确的Riemann求解器可降低误差并加快收敛速度​​。但是，在相应的四阶模拟中，我们发现改变Riemann解算器对解的影响可忽略不计。在不连续点附近，由于使用分段线性限制器，我们发现高阶RKDG方法的行为与二阶方法类似。因此，对于以不连续性为主的解，在高阶方法中选择黎曼求解器与在二阶方法中具有相似的意义。我们对二阶方法的分析表明，黎曼求解器的选择非常重要，更精确的黎曼求解器具有获得给定精度所需的最低计算量。这样可以通过使用更精确的Riemann解算器来减轻不连续解的四阶仿真中的误差。我们证明了minmod限制器不适合用于高阶RKDG方法。它倾向于限制试验空间的多项式阶数，从而限制方法精度的阶数，即使不需要它来保持方案的TVD属性也是如此。

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