A non-linear multigrid method for the steady Euler equations

Higher-order accurate Euler-flow solutions are presented for some airfoil test cases. Second-order accurate solutions are computed by an Iterative Defect Correction process. For two test cases even higher accuracy is obtained by the additional use of a ~xtrapolation technique. Finite volume Osher-type discretizations are applied throughout. Two interpolation schemes (one with and one w~hout a flux limiter) are used for the computation of the second-order defect. In each Defect Correction cycle, the solution is computed by non-linear mu~igrid iteration, in which Collective Symmetric Gauss-Seidel relaxation is used as the smoothing procedure. The computational method does not require tuning of parameters. The solutions show a good resolution of discontinuities, and they are obtained at low computational costs. The rate of convergence seems to be grid-independent

A fitted numerical method for singularly perturbed parabolic reaction-diffusion problems

This paper treats a time-dependent singularly perturbed reaction-diffusion problem. We semidiscretize the problem in time by means of the classical backward Euler method. We develop a fitted operator finite difference method (FOFDM) to solve the resulting set of linear problems (one at each time level). We prove that the underlying fitted operator satisfies the maximum principle. This result is then used in the error analysis of the FOFDM. The method is shown to be first order convergent in time and second order convergent in space, uniformly with respect to the perturbation parameter. We test the method on several numerical examples to confirm our theoretical findings.Web of Scienc

3D Multigrid on Partially Ordered Sets of Grids

. In this paper we discuss different possibilities of using partially ordered sets of grids in multigrid algorithms. Because, for a classical sequence of regular grids the number of degrees of freedom grows much faster with the refinement level for 3D than for 2D, it is more difficult to find sufficiently effective relaxation procedures. Therefore, we study the possibility of using different families of (regular rectangular) grids. Semi-coarsening is one technique in which a partially ordered set of grids is used. In this case still a unique discrete fine-grid problem is solved. On the other hand, sparse grid techniques are more efficient if we compare the accuracy obtained with the number of degrees of freedom used. However, in the latter case it is not always straightforward to identify an appropriate discrete equation that should be solved. The different approaches are compared. The relation between the different approaches is described by looking at hierarchical bases and by consid..

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

Two simple, multi-dimensional upwind discretizations for the steady Euler equations are derived, with the emphasis Iying on bath a good accuracy and a good solvability. The multi-dimensional upwinding consists of applying a one-dimensional Riemann solver with a locally rotated left and right state, the rotation angle depending on the local flow solution. First, a scheme is derived for which smoothing analysis of point Gauss-Seidel relaxation shows that despite its rather low numerical diffusion, it still enables a good acceleration by multigrid. Next, a scheme is derived which has not any numerical diffusion in cross wind direction, and of which convergence analysis shows that its corresponding discretized equations can be solved efficiently by means of defect correction iteration with in the inner multigrid iteration the first scheme. For the steady, two-dimensional Euler equations, numerical experiments are performed for same supersonic test cases with an oblique contact discontinuity. The numerical results are in good agreement with the theoreticaI predictions. Comparisons are made with results obtained by a standard, grid-aligned up wind scheme. The grid-decoupled results obtained are promising

Efficient multigrid computation of steady hypersonic flows

In steady hypersonic flow computations, Newton iteration as a local relaxation procedure and nonlinear multigrid iteration as an acceleration procedure may both easily fail. In the present chapter, same remedies are presented for overcoming these problems. The equations considered are the steady, two-dimensional Navier-Stokes equations. The equations are discretized by an upwind finite volume method. Collective point Gauss-Seidel relaxation is applied as the standard smoothing technique. In hypersonics this technique easily diverges. First, collective line Gauss-Seidel relaxation is applied as an alternative smoothing technique. Though promising, it also fails in hypersonics. Next, collective point Gauss-Seidel relaxation is reconsidered and improved; a divergence monitor is introduced and in case of divergence a switch is made to a local explicit time stepping technique. Satisfactory singlegrid convergence results are shown for the computation of a hypersonic reentry flow around a blunt forebody with canopy. Unfortunately, with this improved smoothing technique, standard nonlinear multigrid iteration still fails in hypersonics. The robustness improvements made therefore to the standard nonlinear multigrid method are a local damping of the restricted defect, a global upwind prolongation of the correction and a gIobal upwind restriction of the defect. Satisfactory multigrid convergence results are shown for the computation of a hypersonic launch and reentry flow around a blunt forebody with canopy. For the test cases considered, it appears that the improved multigrid method performs significantly better than a standard nonlinear multigrid method. For the test cases considered it appears that the most significant improvement comes from the upwind prolongation, rather than from the upwind restriction and the defect damping