A gradient-robust well-balanced scheme for the compressible isothermal Stokes problem
- Akbas, Mine
- Gallouët, Thierry
- Gaßmann, Almut
- Linke, Alexander
- Merdon, Christian
2010 Mathematics Subject Classification
- 76D07 65N30 65N12
2010 Physics and Astronomy Classification Scheme
- 47.10.ad 47.11.Fg
- compressible Stokes equations, finite element method, finite volume method, well-balanced scheme, upwind, convergence
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.
- Comput. Methods Appl. Mech. Engrg., 367 (2020), 113069, DOI 10.1016/j.cma.2020.113069 .