We develop high order discontinuous Galerkin (DG) methods with Lax-Friedrich fluxes for Euler equations under gravitational fields. Such problems may yield steady-state solutions and the density and pressure are positive. There were plenty of previous works developing either well-balanced (WB) schemes to preserve the steady states or positivity-preserving (PP) schemes to get positive density and pressure. However, it is rather difficult to construct WB PP schemes with Lax-Friedrich fluxes, due to the penalty term in the flux. In fact, for general PP DG methods, the penalty coefficient must be sufficiently large, while the WB scheme requires that to be zero. This contradiction can hardly be fixed following the original design of the PP technique. In this talk, we reformulate the source term such that it balances with the flux term when the steady state has reached. To obtain positive numerical density and pressure, we choose a special penalty coefficient in the Lax-Friedrich flux, which is zero at the steady state. The technique works for general steady-state solutions with zero velocities. Numerical experiments are given to show the performance of the proposed methods.
报告人简介：杜洁，华东师范大学紫江青年学者。2015年于中国科学技术大学获得理学博士学位。读博期间多次前往香港大学担任研究助理，并作为联合培养博士研究生前往布朗大学学习。2015年进入香港中文大学进行博士后的工作。2017年就职于清华大学。2023年就职于华东师范大学。主要从事偏微分方程高精度数值算法研究工作，于数值计算及其应用方向的主流杂志上已发表20余篇学术论文，其中包括应用数学类著名杂志SIAM Journal on Scientific Computing和Journal of Computational Physics以及工程类顶级期刊Transportation Research Part B等。现为科技部重点研发计划的课题负责人。