A New Family of Discontinuous Galerkin Schemes for Diffusion Problems

Abstract

Novel Discontinuous Galerkin formulations for parabolic partial differential equations, such as the unsteady diffusion equation, are introduced. Fourier analysis of the various schemes is presented to determine order of accuracy and spectral radius. Additionally, computational examples for the 1D scalar diffusion equation are presented. The method's performance for 2D problems is also demonstrated. Some members of the family exhibit favorable performance in comparison to commonly used methods with regard to accuracy and computational efficiency. Additionally, the method can be adapted to fit within the Flux Reconstruction description of Huynh for diffusion problems; we explore the corresponding 1D formulation. Figure 1: The Recovery process in 2D for a p2 discretization.in Figure 1 for a p2 discretization, taken from our presentation at AIAA Aviation 2016. 8 The recovered solution corresponding to the interface I AB between two neighboring elements Ω A and Ω B , denoted f AB , recovers a very good approximation for a smooth exact solution by applying the recovery procedure to the discontinuous solution representation of U h A and U h B over Ω A ∪ Ω B . The Recovery DG method, extended to multidimensional problems by Lo & Van Leer, 9 shows great promise for purely parabolic PDEs in multiple dimensions, including initial boundary value problems (IBVPs) with time-dependent Dirichlet conditions. 8 However, for combined advection-diffusion problems, where the Recovery DG scheme is employed to handle the diffusive fluxes, it is difficult to design a complete advectiondiffusion DG scheme that captures the advective fluxes with the same accuracy that Recovery DG achieves for the diffusive terms. While the accuracy of the combined method is constrained by the standard DG method's order of accuracy in capturing the advective fluxes, the cost and complication of the method is dominated by the Recovery DG scheme for the diffusive fluxes. This pair of performance bottlenecks is undesirable.With these issues in mind, the aim of our recent studies has been to design new DG schemes for diffusion problems that benefit from the high-order reconstruction provided by the recovery concept while avoiding the complications of Lo & Van Leer's 9 Recovery DG formulation. Our umbrella term for this new family of schemes is "Interface Gradient Recovery," abbreviated IGR. Most of the schemes in this family maintain smaller spectral radii than presently popular DG schemes for diffusion. Simplicity is maintained by starting from the "mixed form" approach for parabolic PDEs, which is familiar to many DG practitioners. Within the mixed form, the IGR family is distinguished by the use of recovery for handling interface terms.The focus of this work is the analysis and performance of the scheme for time-dependent diffusion problems. In section II, the scheme will be described for the multidimensional and 1D cases. Then, we show results from Fourier analysis, providing comparisons in terms of spectral radius and order of accuracy. Next, computationa...