The compact gradient recovery discontinuous Galerkin method for diffusion problems

Abstract

We introduce a new family of schemes, labeled Compact Gradient Recovery (CGR), for handling parabolic partial differential equations within the discontinuous Galerkin (DG) framework, with as the ultimate goal the discretization of diffusive flux terms in advection-diffusion systems such as the compressible Navier-Stokes equations. Like other DG approaches for diffusion, this family of schemes is based on the mixed formulation, where an auxiliary variable is introduced (but not stored between timesteps) to approximate the solution gradient. To maximize accuracy, wherever interface approximations are necessary within the mixed formulation, our schemes apply the Recovery concept originally introduced by Van Leer and Nomura [1]. However, unlike Recovery DG, our new family of schemes is based on a nearest-neighbor stencil and does not require differentiation of the recovered solution, which can introduce large errors for shear diffusion problems. By design, the new schemes fill the void between the highly accurate Recovery DG schemes and existing mixed-form and penalty-based formulations. Fourier analysis is performed on Cartesian and simplex meshes to determine the order of accuracy and timestep-size stability limits for different solution polynomial orders p. Similar to other DG approaches for diffusion, our proposed schemes make use of a jump penalization factor when solving for the gradient along interfaces; the effect of this parameter is explored through Fourier analysis, allowing an informed recommendation regarding its value. In addition to exploration of the new scheme, the analysis includes previously unknown properties of the widely used BR2 scheme under a broad set of configurations. Our new approach is verified using a comprehensive suite of test problems (scalar Laplacian diffusion, shear diffusion, and compressible Navier-Stokes) on different mesh types and compared to the gold-standard, nearest-neighbor approach for diffusion, the BR2 scheme; our new scheme performs favorably with regard to accuracy of the solution, accuracy of the solution gradient, and timestep size without a substantial increase in computational cost.