Two Derivative Runge Kutta Methods and the Multidimensional Generalized Riemann Problem

Abstract: Traditional timestepping methods for hyperbolic systems rely on Runge-Kutta methods, which reduce the timestepping problem to a sequence of first order stages. Alternatively, they rely on ADER schemes which provide a single step update, even at higher order. Higher order Runge-Kutta schemes run into the problem that they have Butcher barriers. As a result, progressively higher order Runge-Kutta schemes become progressively inefficient. ADER schemes do not have that problem, but the higher order ADER schemes can require several Picard iterations to converge.

This suggests a halfway house approach. The approach is based on the realization that a generalized Riemann problem (GRP) solver that is second order accurate can indeed be constructed for any hyperbolic PDE. As a result, this GRP yields not just the numerical flux but also its first derivative. This opens the door to two derivative Runge-Kutta (TDRK) schemes. Such schemes use not just the numerical flux but also the time derivative of the flux. We describe such schemes, and the efficient GRP solvers that make this possible. Furthermore, a multidimensional GRP is designed and presented, which opens the door to treating involution-constrained PDEs.