Well-balanced schemes for the Euler equations with gravitation: Conservative formulation using global fluxes

Highlights: • Systems of balance laws can be rewritten in a conservative form with global fluxes. • Development of well-balanced methods is an important and challenging task. • Reconstruction of equilibrium variables is crucial to design well-balanced schemes. • Central-upwind time evolution is an important step to ensure well-balanced property. We develop a second-order well-balanced central-upwind scheme for the compressible Euler equations with gravitational source term. Here, we advocate a new paradigm based on a purely conservative reformulation of the equations using global fluxes. The proposed scheme is capable of exactly preserving steady-state solutions expressed in terms of a nonlocal equilibrium variable. A crucial step in the construction of the second-order scheme is a well-balanced piecewise linear reconstruction of equilibrium variables combined with a well-balanced central-upwind evolution in time, which is adapted to reduce the amount of numerical viscosity when the flow is at (near) steady-state regime. We show the performance of our newly developed central-upwind scheme and demonstrate importance of perfect balance between the fluxes and gravitational forces in a series of one- and two-dimensional examples.