A Nonlinear Flux Vector Split Defect Correction Scheme for Fast Solutions of the Euler and Navier-Stokes Equations

Abstract

A defect correction scheme is suggested which makes use of embedding the high order accurate total numerical residuals of the dicretized EULER or NAVIER-STOKES (NS) equations into a simple Flux Vector Splitting (FVS) scheme formulated in the classic first order mode. The update of the flow equations (EULER or NS) is performed using a matrix free approach which even does not require the calculation of search directions as in Krylov subspace methods. It rather works like a point Gauss-Seidel method with immediate first order flux updates performed with the newest available flow quantities. By this concept the expensive calculation of the matrix for Newton type classic implicit solvers is avoided, and the storage requirements are those of an explicit scheme. The investigations undertaken with the present kind of implicit solution strategy revealed that a defect correction scheme should use diffusive first order accurate numerical fluxes for providing sufficient smoothing of the increments of the flow variables. For this purpose a novel non-polynomial flux vector splitting is used which exhibits extraordinary simplicity and still retains acceptable accuracy if used in the context of higher order reconstructions of the flow variables.