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. 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 two-dimensional spectral element method formulations for a simple linear conservation law and for the fully compressible two-dimensional 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 regar...