Acceleration techniques for explicit Euler codes.

Abstract

In this thesis we study two steps in the acceleration of Euler computations to steady solutions: (1) Using full multi-grid to march from an arbitrary initial guess to within the range of attraction of the steady solution; (2) Using vector-sequencing to converge to the steady solution from a nearby state. Regarding the first step, in order to design schemes that combine well with multi-grid acceleration, a method has been developed for designing optimally smoothing multi-stage time-marching schemes, given any spatial-differencing operator. The analysis has been extended to the Euler and Navier-Stokes equations in one space-dimension by use of characteristic time-stepping. Convergence rates independent of the number of cells in the finest grid have been achieved with these optimal schemes, for transonic flow with and without a shock. Besides characteristic time-stepping, local time-stepping has been tested with these schemes. While the analysis is only truly applicable with characteristic time-stepping, good convergence has still been obtained with local time-stepping. Finally, the analysis has been extended to scalar, Burgers, and Euler equations in two space dimensions. The successful application to multi-dimensional scalar equations turns out to depend on the possibility to damp numerical signals that move normal to the physical transport direction. We, therefore, have tested several techniques that do this. The best way found is to add some artificial cross-diffusion, but this tends to deteriorate the accuracy of the solution. Still needed is a general technique of making the cross-diffusion term vanish in the steady state. Regarding the second step, two vector-sequencing strategies (GMRES and MPE), which can quickly converge to the steady solution from a nearby state, have been explored and applied to linear and nonlinear problems. The results obtained with GMRES and MPE in nested iterations suggest that there is an advantage in the combination of the multi-grid strategy with vector-sequencing ideas.