On some new low storage implementations of time advancing Runge–Kutta methods

Abstract

In this paper, explicit Runge–Kutta (RK) schemes with minimum storage requirements for systems with very large dimension that arise in the spatial discretization of some partial differential equations are considered. A complete study of all four stage fourth-order schemes of the minimum storage families of Williamson (1980) [2], van der Houwen (1977) [8] and Ketcheson (2010) [12] that require only two storage locations per variable is carried out. It is found that, whereas there exist no schemes of this type in the Williamson and van der Houwen families, there are two isolated schemes and a one parameter family of fourth-order schemes in four stage Ketcheson’s family. This available parameter is used to obtain the optimal scheme taking into account the ‖⋅‖2 norm of the coefficients of the leading error term. In addition a new alternative minimum storage family to the s-stage Ketcheson that depends also on 3s−3 free parameters is proposed. This family contains both the Williamson and van der Houwen schemes but it is not included in Ketcheson’s family. Finally, the results of some numerical experiments are presented to show the behavior of fourth-order optimal schemes for some nonlinear problems.