Multiple grid and Osher's scheme for the efficient solution of the steady Euler equations

Abstract

An iterative method is developed for the .solution of the steady Euler equations for inviscid flow. The system of hyperbolic conservation laws is discretized by a finite-volume Osher-discretization. The iterative method is a multiple grid (FAS) iteration with symmetric Gauss-Seidel (SGS) as a relaxation method. Initial estimates are obtained by full multigrid (FMG). In the pointwise relaxation the equations are kept in block-coupled form and local linearization of the equations and the boundary conditions is considered. The efficient formulation of Osher's discretization of the 2-D non-isentropic steady Euler equations and its linearization is presented. The efficiency of FAS-SGS iteration is shown for a transonic model problem. It appears that, for the problem considered, the rate of convergence is almost independent of the gridsize and that for all meshsizes the discrete system is solved up to truncation error accuracy in only a few (2 or 3) iteration cycles.

A new formulation of Kapila’s five-equation model for compressible two-fluid flow, and its numerical treatment

“…The Riemann invariants w 3 and w 4 show again that the mass fraction is convected with the flow, whereas the volume fraction is not. Along the wave path in solution space, with the subpaths in P-variant ordering [8], the Riemann invariants are distributed as shown in Fig. 2.…”

“…For the evaluation of the states at the cell faces an Osher-type approximate Riemann solver [16] is constructed, in the so-called P(hysical) variant [8]. The Riemann solver gets limited higher-order accurate left and right cell-face states as input (MUSCL approach).…”

Multigrid solution method for the steady RANS equations

“…For the second-order accurate fluxes, a limited upwind scheme is used [10,11]. It is known that the Minmod limiter is unsuitable for use with defect correction [9].…”

“…For the laminar Navier-Stokes equations, efficient multigrid techniques have been developed where the steady flow equations are solved directly with a combination of nonlinear multigrid and Gauss-Seidel smoothing; examples are found in the work of Hemker et al [10,11], of Dick et al [6,22], and of Trottenberg et al [7,23]. But due to the source terms in the turbulence model, the RANS equations cannot be solved with these techniques.…”

Low-Diffusion Rotated Upwind Schemes, Multigrid and Defect Correction for Steady, Multi-Dimensional Euler Flows

“…The numerical flux function to be applied should allow a good resolution of both oblique shock waves and oblique contact discontinuities, fixing choices to flux difference splitting schemes. Given the good experience with Osher's scheme [9] in combination with multigrid -(exact) Newton [4), here we will apply the latter flux difference splitting scheme.…”