Stability Analysis of Preconditioned Approximations of the Euler Equations on Unstructured Meshes

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 timemarching. 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.

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