We suggest a new multigrid preconditioning strategy for use in Jacobian-free Newton-Krylov (JFNK) methods for the solution of algebraic equation systems arising from implicit Discontinuous Galerkin (DG) discretisations. To define the new preconditioner, use is made of an auxiliary first-order finite volume discretisation that refines the original DG mesh, but can still be implemented algebraically. As smoother, we consider the pseudo-time iteration W3 with a symmetric Gauss-Seidel-type approximation of the Jacobian. As a proof of concept numerical tests are presented for the one-dimensional Euler equations, demonstrating the potential of the new approach.
Our aim is to construct efficient preconditioners for high order discontinuous Galerkin (DG) methods. We consider the DG spectral element method with Gauss-Lobatto-Legendre nodes (DGSEM-GL) for the 1D linear advection equation. It has been shown in [4] that DGSEM-GL has the summation-by-parts (SBP) property and an equivalent finite volume (FV) discretization is presented in [3]. Thus we present a multigrid (MG) preconditioner based on a simplified FV discretization.An efficient implicit DG variant is DGSEM, where the interpolation of the flux is collocated with the numerical quadrature used for the inner products, [4]. Key to an efficient algorithm is a fast solver with low memory footprint. Our aim is to constuct a matrix-free preconditioner using approximations to the FV discretization. We solve the 1D advection equation with DGSEM using a right preconditioner based on an agglomeration multigrid method. We show first results for 2-, 3-and 4-stage Runge-Kutta (RK) smoothers with optimized parameters from [1].
In this paper we present a Local Fourier Analysis of a space-time multigrid solver for a hyperbolic test problem. The space-time discretization is based on arbitrarily high order discontinuous Galerkin spectral element methods in time and a first order finite volume method in space. We apply a block Jacobi smoother and consider coarsening in space-time, as well as temporal coarsening only. Asymptotic convergence factors for the smoother and the two-grid method for both coarsening strategies are presented. For high CFL numbers, the convergence factors for both strategies are 0.5 for first order, and 0.375 for second order accurate temporal approximations. Numerical experiments in one and two spatial dimensions for space-time DG-SEM discretizations of varying order gives even better convergence rates of around 0.3 and 0.25 for sufficiently high CFL numbers.
We discuss two approaches for the formulation and implementation of space-time discontinuous Galerkin spectral element methods (DG-SEM). In one, time is treated as an additional coordinate direction and a Galerkin procedure is applied to the entire problem. In the other, the method of lines is used with DG-SEM in space and the fully implicit Runge-Kutta method Lobatto IIIC in time. The two approaches are mathematically equivalent in the sense that they lead to the same discrete solution. However, in practice they differ in several important respects, including the terminology used to describe them, the structure of the resulting software, and the interaction with nonlinear solvers. Challenges and merits of the two approaches are discussed with the goal of providing the practitioner with sufficient consideration to choose which path to follow. Additionally, implementations of the two methods are provided as a starting point for further development. Numerical experiments validate the theoretical accuracy of these codes and demonstrate their utility, even for 4D problems.
The goal of our research is the construction of efficient Jacobian-free preconditioners for high order Discontinuous Galerkin (DG) discretizations with implicit time integration. We are motivated by three-dimensional unsteady compressible flow applications, which often result in large stiff systems. Implicit time integrators overcome the impact upon restrictive CFL conditions on explicit ones but leave the problem to solve huge nonlinear systems. In this paper we consider a multigrid preconditioning strategy for Jacobian-free Newton-Krylov (JFNK) methods for the solution of algebraic equation systems arising from implicit Discontinuous Galerkin (DG) discretizations. The preconditioner is defined by an auxiliary first order Finite Volume (FV) discretization that refines the original DG mesh, but can still be implemented algebraically. Different options exist to define the grid transfer between DG and FV. We suggest an ad hoc assignment of the unknowns as well as L 2 projections. We present new numerical results for the two-dimensional convection-diffusion equation in combination with the different transfer options, which demonstrate the quality and efficiency of the suggested preconditioner with regards to convergence speed up and CPU time. The suggested L 2 projection from this paper result in the best convergence speed up.
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