A High-Order Discontinuous Galerkin Multigrid Solver for Aerodynamic Applications

Krzysztof Fidkowski, Massachusetts Institute of Technology

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We present the results from the development of a high-order discontinuous Galerkin finite element solver using p-multigrid with line Jacobi smoothing. The line smoothing algorithm and p-multigrid have been developed for the nonlinear Euler equations of gas dynamics on unstructured meshes. Analysis of 2-D advection shows the improved performance of the line implicit relaxation method versus common point implicit methods. The results of a mesh refinement study demonstrate that the accuracy of the discretization is the optimal O(h^(p+1)) for smooth problems in 2-D and 3-D. In terms of performance, the multigrid convergence rate is found to be independent of the polynomial order but weakly dependent on the grid size. Timing studies for each problem indicate that higher order is advantageous over grid refinement when high accuracy is required. Finally, the ease of parallelization of the method is outlined and scalability results are presented.

The application of this research is in the area of robust aerodynamic flow computations where high accuracy is required. Our work with the Euler equations shows the accuracy and performance benefit of using a discontinuous Galerkin higher-order finite element method over current finite volume schemes. The end goal is to extend this work to the full Reynolds-averaged Navier Stokes equations, and to attain similar improvements.

Abstract Author(s): Krzysztof Fidkowski and David Darmofal