The new godunov-type high order method via flux-reconstruction, are capable of unifiying several popular methods including the discontinuous Galerkin, staggered-grid multi-domain method, or the spectral difference/spectral volume methods into a single family. Compared to discontinuous galerkin method, this new method retain the arbitrary order and compact properties but written in differential form and without mass matrix. The flux-reconstrcution method using general unstructured 2D/3D mesh including triangles, quadrilaterals, tetrahedrons, pyramids, prisms and hexahedrons are implemented. Compared to discontinuous galerkin method, FR is differential form without any quandature, so it runs faster and is easier to implement on GPU to achieve higher speed up ratio. The present paper investigates three different solving methods for this new method in order to cut down the hugh computational cost and memory requirement for fully implicit methods. Explicit multi-stage Runge-Kutta, nonlinear Lower-Upper Symmetry Gauss-Seidel (LU-SGS) and generalized minimal residual (GMRES) with matrix-free preconditioning methods are implemented and all these three methods use p-multigrid to smooth low-frequency errors on lower order. The results of 2D steady-state external/internal viscous flows are presented and the convergence properties for different solving method are compared. Efficiency and robustness are improved through symmetric Gauss-Seidel (SGS) iterations as preconditioning, a remarkable feature of the present GMRES+SGS method is that the storage of the full Jacobian matrix can be eliminated and only its diagonal stored. A laplacian artificial viscosity is tested to capture shock in one element and performs well for this new high order method, an L2 projection method is used to improve the perfomance of aliasing error in conservative form. The one equation Spalart-Allmaras turbulence model is unsed to solve the Raynolds average navier-stokes equations. MPI is used for parallel simulations for ralatively complex 3D internal flow when using high order method. Compared to other high order methods, results in this paper show that Flux-Reconstruction methods with efficient solving method performs well for 2D/3D general unstructured mesh in wide range of reynorlds number, and is also potential for “real geometry” simulations.
On the Implementation of a Multiscale Hybrid High-Order Method
