DG P-multigrid. Efficient solvers and complex applications

These lecture notes will focus on some efficiency-related topics relevant to an effective implementation of the p-multigrid technique, presented in the lecture notes “p-Multigrid For Discontinuous Galerkin Methods”, in a high-order DG method. Actually, the underlying building blocks of the p-multigrid solver are those of an implicit DG code, named MIGALE, developed over the past years and more recently within the ADIGMA project. The code solves the Euler, Navier-Stokes and the coupled RANS and k-! turbulence model equations. The implicit implementation of the DG method is based on the exact linearization of residuals and on linearly implicit Runge-Kutta schemes for time integration. All the boundary conditions are also implemented implicitly. Based on the framework of the implicit code, the matrix blocks of the semi-implicit p-multigrid solver are nothing but the matrix blocks of the implicit scheme local to the elements, i.e., the diagonal blocks. On the other hand, the backward Euler smoother employed at the lowest degree of the p-multigrid algorithm is exactly the same used by the implicit solver. These notes summarize recent results of numerical investigation on how to improve the efficiency of our DG implementation, with special attention to benefits for the p-multigrid approach. For example, the polynomial approximations based on nodal collocation presented in the following should be particularly effective if coupled with the p-multigrid technique. The notes present also new advancements and results of closer investigations on efficiency of the implicit, high-order DG method, which are expected to be important for the p-multigrid and that will serve for assessing its efficiency. The current capability of the implicit DG approach will be demonstrated by presenting some recent results of high-order turbulent flow computations of fairly complex test cases proposed within the ADIGMA project.