Adaptive least squares pseudospectral element methods for the two dimensional Euler equations

Abstract

The present dissertation deals with the application of an adaptive mesh hp-refinement method to the time-dependent compressible Euler equations describing the flow of an inviscid fluid. The physical simulation space is a two-dimensional channel with bump through which an ideal gas flows. A big advantage of the Euler equations is the applicability to different Mach regimes. Thus, smooth solutions and those containing discontinuities can be simulated. As the underlying numerical technique, a pseudo-spectral element method using least squares is used. The grid consists of rectangles that have to be transformed into curved deformed elements in the area of the bump. The h-refinement is realized by means of a quadtree and the p-refinement by means of a stepwise increase of the polynomial degree by 2, whereby hanging nodes can arise. Compared to previous research, the Euler equations are examined with respect to four different refinement criteria in combination with the hp-adaptive pseudo-spectral element method. The refinement criteria are the gradient criterion, two spectral criteria and an edge detection method. For a first determination of the suitability of the criteria for the h-refinement and for a validation of the pseudo-spectral element method, these are used for the solution of three steady state convection equations for which the exact solutions are known. Finally, the full method is applied to three test problems for the Euler equations. The flow through the channel is realized by a smooth subsonic flow, a transonic flow with a shock, and a supersonic flow that contains reflective shocks. Since no analytical solution is known, we compare our results to other works that consider stationary grids for these test cases. It turns out that the applied adaptive method refines the grid in the regions of the stagnation points and those that have discontinuities. Thus prior knowledge about the location of the problem areas is not necessary. A smaller number of elements and locally higher polynomial degrees reduce the sizes of the systems of equations to be solved.