In this paper, we develop a high order structure-preserving local discontinuous Galerkin (DG) scheme for the compressible self-gravitating Euler equations, which pose great challenges due to the presence of time-dependent gravitational potential. The designed scheme is well-balanced for general polytropic equilibrium state and total energy conserving for multiple spatial dimensions without an assumption of spherical symmetry. The well-balanced property is achieved by decomposing the gravitational potential into equilibrium and perturbation parts, employing a modified Harten-Lax-van Leer-contact flux and a modification of the discretization for the source term. Conservation of total energy is particularly challenging in the presence of self-gravity, especially when aiming for high order accuracy. To address this, we rewrite the energy equation into a conservative form, and carefully design an energy flux with the aid of weak formulation from the DG method to maintain conservation as well as high order accuracy. The resulting scheme can be extended to high order in time discretizations. Numerical examples for two and three dimensional problems are provided to verify the desired properties of our proposed scheme, including shock-capturing, high order accuracy, well balance, and total energy conservation.
Pan et al. (Sun,) studied this question.
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