A WELL-BALANCED SCHEME FOR EULER EQUATIONS WITH SINGULAR SOURCES^*

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 present challenges in 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 that the numerical scheme is 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 this difficulty, we propose a solution-structure based approximate Riemann solver, in which the structure of the 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.