An Element-Based Spectrally Optimized Approximate Inverse Preconditioner for the Euler Equations

We introduce a method for constructing an element-by-element sparse approximate inverse (SAI) preconditioner designed to be effective in a massively-parallel spectral element modeling environment involving nonsymmetric systems. This new preconditioning approach is based on a spectral optimization of a low-resolution preconditioned system matrix (PSM). We show that the local preconditioning matrices obtained via this element-based, spectrum-optimized (EBSO) approach may be applied to arbitrarily high-resolution versions of the same system matrix without appreciable loss of preconditioner performance. We demonstrate the performance of the EBSO preconditioning approach using 2-D spectral element method (SEM) formulations for a simple linear conservation law and for the fully-compressible 2-D Euler equations with various boundary conditions. For the latter model running at sufficiently large Courant number, the EBSO preconditioner significantly reduces both iteration count and wall-clock time regardless of whether a generalized minimum residual (GMRES) or a stabilized biconjugate gradient (BICGSTAB) iterative scheme is employed. To assess the value added by this new preconditioning approach, we compare its performance against two other equally-parallel SAI preconditioning methods: low-order Chebyshev generalized least-squares polynomials and an element-based variant of the well-known Frobenius norm optimization preconditioner which we also develop herein. The EBSO preconditioner significantly out-performs both the Chebyshev polynomials and the element-based Frobenius-norm-optimized (EBFO) preconditioner regardless of whether the GMRES or BICGSTAB iterative scheme is employed. Moreover, when the EBSO preconditioner is combined with the Chebyshev polynomial method dramatic reductions in iterations per time-step can be achieved while still achieving a significant reduction in wall-clock time.