Stability analysis of preconditioned approximations of the Euler equations on unstructuctured meshes

Abstract

This paper analyses the stability of a discretisation of the Euler equations on 3D unstructured grids using an edge-based data structure, first-order characteristic smoothing, a block-Jacobi preconditioner and Runge-Kutta time-marching. This is motivated by multigrid Navier-Stokes calculations in which this inviscid discretisation is the dominant component on coarse grids.

The analysis uses algebraic stability theory, which allows, at worst, a bounded linear growth in a suitably defined "perturbation energy" provided the range of values of the preconditioned spatial operator lies within the stability region of the Runge-Kutta algorithm. The analysis also includes consideration of the effect of solid wall boundary conditions, and the addition of a low Mach number preconditioner to accelerate compressible flows in which the Mach number is very low in a significant portion of the flow.

Numerical results for both inviscid and viscous applications confirm the effectiveness of the numerical algorithm, and show that the analysis provides accurate stability bounds.