Implicit multiderivative collocation solvers for linear partial differential equations with discontinuous Galerkin spatial discretizations

TL;DR

This paper introduces novel implicit multiderivative collocation methods for linear PDEs with discontinuous Galerkin discretizations, enabling larger time steps and reduced unknowns while maintaining accuracy.

Contribution

The work develops new multiderivative collocation schemes that reduce computational complexity and allow for larger time steps in solving linear PDEs with discontinuous Galerkin methods.

Findings

Methods achieve expected order of accuracy

Large time steps are feasible without loss of stability

Reduced number of unknowns improves computational efficiency

Abstract

In this work, we construct novel discretizations for the unsteady convection-diffusion equation. Our discretization relies on multiderivative time integrators together with a novel discretization that reduces the total number of unknowns for the solver. These type of temporal discretizations come from an umbrella class of methods that include Lax-Wendroff (Taylor) as well as Runge-Kutta methods as special cases. We include two-point collocation methods with multiple time derivatives as well as a sixth-order fully implicit collocation method that only requires a total of three stages. Numerical results for a number of sample linear problems indicate the expected order of accuracy and indicate we can take arbitrarily large time steps.

Figures

Full text

See pages 1-last of preprint.pdf