WELL-BALANCED SCHEME FOR EULER EQUATIONS WITH SINGULAR SOURCES
科研大讨论系列报告
报告题目(Title)：WELL-BALANCED SCHEME FOR EULER EQUATIONS WITH SINGULAR SOURCES
报告人(Speaker)：刘铁钢（北京航空航天大学）
地点(Place)：后主楼1223
时间(Time)：2023年10月20日15:00-16:00
邀请人(Inviter)：潘亮
报告摘要
In this talk, numerical methods for the Euler equations with a singular source are discussed. The stationary discontinuity induced by the singular source and its coupling with the convection of fluid present challenges to numerical computation. We introduce a definition of the well-balanced property of the numerical scheme for the singular source of interest, which is necessary for the numerical solution to be correct. We theoretically show that the splitting scheme is always not well-balanced and leads to incorrect results. For the unsplitting scheme, we present a consistency condition of the numerical fluxes for singular sources, which ensures the numerical scheme to be well-balanced. However, it can be shown that the well-balanced property of a scheme cannot guarantee the correct numerical solutions in extreme cases. 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 solution 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.
