Two-derivative Runge–Kutta methods for PDEs using a novel discretization approach

Abstract

We develop a novel and general approach to the discretization of partial differential equations. This approach overcomes the rigid restriction of the traditional method of lines (MOL) and provides flexibility in the treatment of spatial discretization. This method is essential for developing efficient numerical schemes for PDEs based on two-derivative Runge–Kutta (TDRK) methods, where the first and second derivatives must be discretized in an efficient way. This is unlikely to be achieved by using MOL. We then apply the explicit TDRK methods to the advection equations and analyze the numerical stability in the linear advection equation case. We conduct numerical experiments on the Burgers' equation using the TDRK methods developed. We also apply a two-stage semi-implicit TDRK method of order-4 and stage-order-4 to the heat equation. A very significant improvement in the efficiency of this TDRK method is observed when compared to the popular Crank-Nicolson method. This paper is partially based on the work in Tsai's PhD thesis (2011) [10].