A gradient-robust well-balanced scheme for the compressible isothermal Stokes problem

Abstract

A novel notion for constructing a well-balanced scheme – a gradient-robust scheme – is introduced and a showcase application for the steady compressible, isothermal Stokes equations in a nearly-hydrostatic situation is presented. Gradient-robustness means that gradient fields in the momentum balance are well-balanced by the discrete pressure gradient — which is possible on arbitrary, unstructured grids. The scheme is asymptotic-preserving in the sense that it reduces for low Mach numbers to a recent inf–sup stable and pressure-robust discretization for the incompressible Stokes equations. The convergence of the coupled FEM–FVM scheme for the nonlinear, isothermal Stokes equations is proved by compactness arguments. Numerical examples illustrate the numerical analysis, and show that the novel approach can lead to a dramatically increased accuracy in nearly-hydrostatic low Mach number flows. Numerical examples also suggest that a straightforward extension to barotropic situations with nonlinear equations of state is feasible.