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We present a well-balanced, second order, Godunov-type finite volume schemefor compressible Euler equations with gravity. By construction, the schemeadmits a discrete stationary solution which is a second order accurateapproximation to the exact stationary solution. Such a scheme is useful forproblems involving complex equations of state and/or hydrostatic solutionswhich are not known in closed form expression. No \'a priori knowledge of thehydrostatic solution is required to achieve the well-balanced property. Theperformance of the scheme is demonstrated on several test cases in terms ofpreservation of hydrostatic solution and computation of small perturbationsaround a hydrostatic solution.

A second-order, discretely well-balanced finite volume scheme for Euler equations with gravity