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A high-order well-balanced scheme for the Euler equations with gravitation ispresented. The scheme is able to preserve a spatially high-order accuratediscrete representation of a large class of hydrostatic equilibria. It is basedon a novel local hydrostatic reconstruction, which, in combination with anystandard high-order accurate reconstruction procedure, achieves genuinehigh-order accuracy for smooth solutions close or away from equilibrium. Theresulting scheme is very simple and can be implemented into any existing finitevolume code with minimal effort. Moreover, the scheme is not tied to anyparticular form of the equation of state, which is crucial for example inastrophysical applications. Several numerical experiments demonstrate therobustness and high-order accuracy of the scheme nearby and out of hydrostaticequilibrium.

High-order well-balanced finite volume schemes for the Euler equations with gravitation