# 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.

## Key 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.