TY - JOUR
T1 - High-Order Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for Euler Equations with Gravitation on Unstructured Meshes
AU - Zhang , Weijie
AU - Xing , Yulong
AU - Xia , Yinhua
AU - Xu , Yan
JO - Communications in Computational Physics
VL - 3
SP - 771
EP - 815
PY - 2022
DA - 2022/03
SN - 31
DO - http://doi.org/10.4208/cicp.OA-2021-0126
UR - https://global-sci.org/intro/article_detail/cicp/20298.html
KW - Discontinuous Galerkin methods, Euler equations, gravitational field, positivity-preserving, well-balanced, triangular mesh.
AB - <p style="text-align: justify;">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.</p>