New Factorizable Discretizations for the Euler Equations

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 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 principal linearization of 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 reflect the physical anisotropies, be stable, and be easily solvable. This paper presents an approach to the derivation of discretization schemes for which TME can be achieved by multigrid solvers with distributed relaxation. In particular, discrete schemes for the nonconservative Euler equations possessing properties (1) and (2) have been derived and analyzed. The accuracy of these scheme has been tested for subsonic flow regimes and compared with accuracy of standard schemes. TME has been demonstrated in solving fully subsonic quasi--one-dimensional flow in a convergent/divergent channel.