Efficient Algorithms for High-Order Discretizations of the Euler and Navier-Stokes Equations

Higher order discretizations have not been widely successful in industrial applications to compressible flow simulation. Among several reasons for this, one may identify the lack of tailor-suited, best-practice relaxation techniques that compare favorably to highly tuned lower order methods, such as finite-volume schemes. In this paper we investigate efficient Spectral Difference discretization methods for the Euler and Navier-Stokes equations, and present solution algorithms using such techniques as h/p-Multigrid. We present a novel hybrid multilevel relaxation method that combines matrix-free implicit relaxation techniques with explicit time-stepping using geometric multigrid.