Implicit Multiderivative Collocation Solvers for Linear Partial Differential Equations with Discontinuous Galerkin Spatial Discretizations

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 unkowns 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.