Time Step Restrictions for Strong-Stability-Preserving Multistep Runge–Kutta Discontinuous Galerkin Methods

Discontinuous Galerkin finite element spatial discretizations are often used in a method-of-lines approach with an ordinary differential equation solver to step the solution forward in time. Explicit strong-stability-preserving time steppers are a popular choice because they provably preserve the nonlinear stability properties of the forward Euler method applied to discontinuous Galerkin semi-discretized equations, subject to a time step constraint. While nonlinear stability is guaranteed by strong-stability-preservation, a separate condition for linear stability of the combined scheme must also be satisfied. In this work, we assess the linear stability properties of discontinuous Galerkin spatial discretizations with a set of strong-stability-preserving multistep Runge–Kutta methods. We find that, in all cases, the constraint for linear stability is more strict than that for strong-stability-preservation. For each order, from the set of multistep Runge–Kutta methods, we select an optimal time stepper that requires the fewest evaluations of the discontinuous Galerkin operator. All methods are tested for convergence in application to both a linear and a nonlinear partial differential equation, and all methods are found to converge in both cases using the maximum stable time step as determined by the stability constraints found in this paper.