New Factorizable Discretizations for the Compressible Euler Equations

Boris Diskin

ICASE, NASA Langley Research Center, Mail Stop 132C, Hampton, VA 23681


A multigrid method is defined as having textbook multigrid efficiency (TME) if solutions to the governing system of equations are attained in a computational work  that is a small (less than 10) multiple of the operation count in one target-grid residual evaluation. A way to achieve TME for the the Euler and Navier-Stokes  equations is to apply the distributed relaxation method thereby separating the elliptic and hyperbolic partitions of the equations. Design of a distributed relaxation scheme can be significantly simplified if the target discretization possesses two properties: (1) factorizability and (2) consistent approximations for the separate factors.  The first property implies that the discrete system determinant can be represented as a product of discrete factors, each of them approximating a corresponding factor of the determinant of the differential equations. The second property requires that  the discrete factors should reflect the physical anisotropies, be stable, and easily solvable. The majority of discrete schemes in current use are not factorizable. Even for factorizable schemes, the discretizations obtained for scalar factors of the determinant are not often stable or/and have wrong strong-anisotropy directions.   In this talk, I am going to introduce new collocated-grid discrete schemes for the compressible Euler equations possessing properties (1) and (2). The accuracy of these scheme has been tested for subsonic flow regimes and is comparable with the accuracy of standard schemes. TME has been demonstrated in solving quasi-one-dimensional flow in a convergent/divergent channel.