An affine reconstructed algorithm for diffusion on triangular grids using the nodal discontinuous Galerkin method

TL;DR

This paper introduces an affine reconstructed nodal DG method for diffusion problems on triangular grids, enhancing accuracy and efficiency by reconstructing smooth solutions on parallelograms within unstructured meshes.

Contribution

It presents the first practical guideline for applying reconstructed algorithms to nodal DG methods for diffusion, focusing on accuracy and computational efficiency.

Findings

Demonstrates improved accuracy on benchmark cases.

Effective handling of highly disparate diffusion parameters.

Reconstruction method promotes computational efficiency.

Abstract

This work discusses the application of an affine reconstructed nodal DG method for unstructured grids of triangles. Solving the diffusion terms in the DG method is non-trivial due to the solution representations being piecewise continuous. Hence, the diffusive flux is not defined on the interface of elements. The proposed numerical approach reconstructs a smooth solution in a parallelogram that is enclosed by the quadrilateral formed by two adjacent triangle elements. The interface between these two triangles is the diagonal of the enclosed parallelogram. Similar to triangles, the mapping of parallelograms from a physical domain to a reference domain is an affine mapping, which is necessary for an accurate and efficient implementation of the numerical algorithm. Thus, all computations can still be performed on the reference domain, which promotes efficiency in computation and storage. This reconstruction does not make assumptions on choice of polynomial basis. Reconstructed DG algorithms have previously been developed for modal implementations of the convection-diffusion equations. However, to the best of the authors' knowledge, this is the first practical guideline that has been proposed for applying the reconstructed algorithm on a nodal discontinuous Galerkin method with a focus on accuracy and efficiency. The algorithm is demonstrated on a number of benchmark cases as well as a challenging substantive problem in HED hydrodynamics with highly disparate diffusion parameters.