论文标题
具有奇异源的Euler方程的平衡方案
A well-balanced scheme for Euler equations with singular sources
论文作者
论文摘要
本文讨论了具有单数源的Euler方程的数值方法。 奇异源引起的固定不连续性及其与流体对流的耦合给数值方法带来了挑战。 我们表明，分裂方案没有很好的平衡，并导致结果不正确。此外，由于来源的奇异性，在极端情况下，一些流行的均衡方案也提供了错误的解决方案。 为了解决此类困难，我们提出了一个基于解决方案结构的近似Riemann求解器，其中首先预测Riemann溶液的结构，然后给出其相应的近似求解器。 所提出的求解器可以应用于一般有限体积方法中数值通量的计算，这可以导致新的平衡方案。 数值测试表明，基于当前近似Riemann求解器的不连续的Galerkin方法具有准确捕获每个波的能力。
Numerical methods for the Euler equations with a singular source are discussed in this paper. The stationary discontinuity induced by the singular source and its coupling with the convection of fluid presents challenges to numerical methods. We show that the splitting scheme is not well-balanced and leads to incorrect results; in addition, some popular well-balanced schemes also give incorrect solutions in extreme cases due to the singularity of source. To fix such difficulties, we propose a solution-structure based approximate Riemann solver, in which the structure of Riemann solution is first predicted and then its corresponding approximate solver is given. The proposed solver can be applied to the calculation of numerical fluxes in a general finite volume method, which can lead to a new well-balanced scheme. Numerical tests show that the discontinuous Galerkin method based on the present approximate Riemann solver has the ability to capture each wave accurately.
