High-Order Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for Euler Equations with Gravitation on Unstructured Meshes
Year: 2022
Author: Weijie Zhang, Yulong Xing, Yinhua Xia, Yan Xu
Communications in Computational Physics, Vol. 31 (2022), Iss. 3 : pp. 771–815
Abstract
In this paper, we propose a high-order accurate discontinuous Galerkin (DG) method for the compressible Euler equations under gravitational fields on unstructured meshes. The scheme preserves a general hydrostatic equilibrium state and provably guarantees the positivity of density and pressure at the same time. Comparing with the work on the well-balanced scheme for Euler equations with gravitation on rectangular meshes, the extension to triangular meshes is conceptually plausible but highly nontrivial. We first introduce a special way to recover the equilibrium state and then design a group of novel variables at the interface of two adjacent cells, which plays an important role in the well-balanced and positivity-preserving properties. One main challenge is that the well-balanced schemes may not have the weak positivity property. In order to achieve the well-balanced and positivity-preserving properties simultaneously while maintaining high-order accuracy, we carefully design DG spatial discretization with well-balanced numerical fluxes and suitable source term approximation. For the ideal gas, we prove that the resulting well-balanced scheme, coupled with strong stability preserving time discretizations, satisfies a weak positivity property. A simple existing limiter can be applied to enforce the positivity-preserving property, without losing high-order accuracy and conservation. Extensive one- and two-dimensional numerical examples demonstrate the desired properties of the proposed scheme, as well as its high resolution and robustness.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.OA-2021-0126
Communications in Computational Physics, Vol. 31 (2022), Iss. 3 : pp. 771–815
Published online: 2022-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 45
Keywords: Discontinuous Galerkin methods Euler equations gravitational field positivity-preserving well-balanced triangular mesh.