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

TL;DR

This paper introduces a gradient-robust, well-balanced finite element/finite volume scheme for the compressible isothermal Stokes problem, improving accuracy in low Mach number flows and degenerating to an incompressible scheme at low Mach numbers.

Contribution

It presents a novel gradient-robust scheme that is well-balanced, asymptotic-preserving, and applicable to nonlinear, isothermal Stokes equations, with proven convergence and enhanced accuracy.

Findings

Numerical examples demonstrate increased accuracy in low Mach number flows.

The scheme degenerates to a pressure-robust incompressible Stokes discretization at low Mach numbers.

Extension to nonlinear barotropic equations appears feasible.

Abstract

A novel notion for constructing a well-balanced scheme - a gradient-robust scheme - is introduced and a showcase application for a steady compressible, isothermal Stokes equations is presented. Gradient-robustness means that arbitrary gradient fields in the momentum balance are well-balanced by the discrete pressure gradient - if there is enough mass in the system to compensate the force. The scheme is asymptotic-preserving in the sense that it degenerates 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 straight-forward extension to barotropic situations with nonlinear equations of state is feasible.