A parallel, implicit reconstructed discontinuous Galerkin method for the compressible flows on 3D arbitrary grids

A Navier-Stokes flow solver RDGFLO based on a third-order accurate reconstructed discontinuous Galerkin (RDG) method is presented for modeling the compressible flows on 3D arbitrary grids. A remarkable feature of this solver is that the 1D, 2D and 3D flow problems can be simulated with the very same code, thanks to the adopted Taylor basis for the various types of elements, i.e., tetrahedron, pyramid, prism, and hexahedron. A quadratic polynomial (P₂) solution is obtained from the underlying linear polynomial (P₁) DG solution via a hierarchical WENO reconstruction scheme, termed as HWENO(P₁P₂), which is designed in order to not only increase the accuracy of the underlying DG method, but also ensure the non-linear stability of the RDG method. Both multi-stage explicit Runge-Kutta and implicit backward Euler methods are implemented for time advancement in the RDG method. In the implicit method, the flux Jacobian matrix is well approximated via an automatic differentiation engine TAPENADE, which is the key to a robust and effcient linear solver. The approximate system of linear equations arising from Newton linearization is solved by an LU-SGS (lower-upper symmetric Gauss-Seidel) preconditioned GMRES (general minimum residual) algorithm, termed as GMRES+LU-SGS. The parallelization in the RDG method is based on a message passing interface (MPI) programming paradigm, where the METIS library is used for the partitioning of a grid into subdomain grids of approximately the same size. The developed solver is used to compute a variety of compressible flow problems to demonstrate its accuracy, robustness and scalability. The numerical experiments indicate that RDGFLO is able to offer a cost-effective solution to complicated flows of practical importance. Overall, the developed RDG method shows a great potential to become a viable, attractive, competitive and ultimately superior DG method over the current state-of-the-art second-order finite volume methods.