Well-balanced finite volume schemes for nearly steady adiabatic flows

We present well-balanced finite volume schemes designed to approximate the Euler equations with gravitation. They are based on a novel local steady state reconstruction. The schemes preserve a discrete equivalent of steady adiabatic flow, which includes non-hydrostatic equilibria. The proposed method works in Cartesian, cylindrical and spherical coordinates. The scheme is not tied to any specific numerical flux and can be combined with any consistent numerical flux for the Euler equations, which provides great flexibility and simplifies the integration into any standard finite volume algorithm. Furthermore, the schemes can cope with general convex equations of state, which is particularly important in astrophysical applications. Both first- and second-order accurate versions of the schemes and their extension to several space dimensions are presented. The superior performance of the well-balanced schemes compared to standard schemes is demonstrated in a variety of numerical experiments. The chosen numerical experiments include simple one-dimensional problems in both Cartesian and spherical geometry, as well as two-dimensional simulations of stellar accretion in cylindrical geometry with a complex multi-physics equation of state

ARBITRARY ORDER FINITE VOLUME WELL-BALANCED SCHEMES FOR THE EULER EQUATIONS WITH GRAVITY

This work presents arbitrary high order well balanced finite volume schemes for the Euler equations with a prescribed gravitational field. It is assumed that the desired equilibrium solution is known, and we construct a scheme which is exactly well balanced for that particular equilibrium. The scheme is based on high order reconstructions of the fluctuations from equilibrium of density, velocity, and pressure, and on a well-balanced integration of the source terms, while no assumptions are needed on the numerical flux, beside consistency. This technique also allows one to construct well-balanced methods for a class of moving equilibria. Several numerical tests demonstrate the performance of the scheme on different scenarios, from equilibrium solutions to nonsteady problems involving shocks. The numerical tests are carried out with methods up to fifth order in one dimension, and third order accuracy in two dimensions