Well-Balanced Schemes for the Euler Equations with Gravitation: Conservative Formulation Using Global Fluxes

摘要

We develop a second-order well-balanced central-upwind scheme for thecompressible Euler equations with gravitational source term. Here, we advocatea new paradigm based on a purely conservative reformulation of the equationsusing global fluxes. The proposed scheme is capable of exactly preservingsteady-state solutions expressed in terms of a nonlocal equilibrium variable. Acrucial step in the construction of the second-order scheme is a well-balancedpiecewise linear reconstruction of equilibrium variables combined with awell-balanced central-upwind evolution in time, which is adapted to reduce theamount of numerical viscosity when the flow is at (near) steady-state regime.We show the performance of our newly developed central-upwind scheme anddemonstrate importance of perfect balance between the fluxes and gravitationalforces in a series of one- and two-dimensional examples.