Stable discretisations of high-order discontinuous Galerkin methods on equidistant and scattered points

TL;DR

This paper introduces stable, high-order collocation discretisations for the discontinuous Galerkin method using discrete least squares, enabling accurate, stable computations on arbitrary points and extending to complex nonlinear problems.

Contribution

It develops a novel framework combining discrete least squares with discontinuous Galerkin methods for stability and high-order accuracy on scattered points.

Findings

Proves conservation and L^2 stability of the discretisations.

Demonstrates high accuracy and stability in nonlinear and long-time simulations.

Extends the method to variable coefficient problems in two dimensions.

Abstract

In this work, we propose and investigate stable high-order collocation-type discretisations of the discontinuous Galerkin method on equidistant and scattered collocation points. We do so by incorporating the concept of discrete least squares into the discontinuous Galerkin framework. Discrete least squares approximations allow us to construct stable and high-order accurate approximations on arbitrary collocation points, while discrete least squares quadrature rules allow us their stable and exact numerical integration. Both methods are computed efficiently by using bases of discrete orthogonal polynomials. Thus, the proposed discretisation generalises known classes of discretisations of the discontinuous Galerkin method, such as the discontinuous Galerkin collocation spectral element method. We are able to prove conservation and linear $L^2$-stability of the proposed discretisations. Finally, numerical tests investigate their accuracy and demonstrate their extension to nonlinear conservation laws, systems, longtime simulations, and a variable coefficient problem in two space dimensions.