p-Multigrid high-order discontinuous Galerkin solver for three-dimensional compressible turbulent flows

The study of turbulent flows through steady-state simulations based on the Reynolds-averaged Navier-Stokes equations and turbulence models can be considered the workhorse in different scientific and industrial applications. Among the different numerical approaches, discontinuous Galerkin methods demonstrated to be perfectly suited for high-order accurate numerical solutions on structured or arbitrary unstructured and non-conforming meshes, and high-performance computing with massively parallel processing. However, their computational cost increases rapidly when the solution is discretized with higher-order polynomial approximations. For this reason, many research efforts have been devoted to overcome this drawback. Literature shows many applications of p-multigrid algorithms for the solution of Euler and Navier-Stokes equations, while few works report the solution of the Reynolds-averaged Navier-Stokes equations with p-multigrid algorithms. In fact, different authors highlighted a lack of performance for the stiffness associated with the discretized RANS equations, and for highly stretched meshes, typically used for an accurate resolution of turbulent boundary layers. This work presents the implementation of an improved p-multigrid algorithm based on the nonlinear full approximation scheme in a discontinuous Galerkin solver for the solution of the three-dimensional and compressible Reynolds-Average Navier-Stokes equations. The performance of the algorithm with different smoothers is compared with the implicit (single-order) time integration on many test cases with different flow conditions, domains, and meshes, showing an average reduction of the computing time around 75% with respect to single-order implicit solvers.