Implicit multiderivative collocation solvers for linear partial differential equations with discontinuous Galerkin spatial discretizations
This work provides a novel discretization that reduces computational cost for solving linear PDEs with discontinuous Galerkin methods, but the results are limited to linear problems and the improvement over existing methods is not quantified.
The authors develop new multiderivative time integration methods for the unsteady convection-diffusion equation, achieving sixth-order accuracy with only three stages and allowing arbitrarily large time steps.
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.