Jacobian-free Locally Linearized Runge–Kutta method of Dormand and Prince for large systems of differential equations

Abstract

This paper introduces Jacobian-free Locally Linearized Runge–Kutta formulas of Dormand and Prince for integrating large systems of initial value problems. With these new Jacobian-free formulas, an adaptive time-stepping variable-order scheme with regulated Krylov dimension is constructed with new developments in the computation of the phi-function action over vectors through Jacobian-free Krylov–Padé approximations, and in the procedures for controlling the approximation errors and for estimating the dimension of the Krylov subspaces. At each integration step, the implemented scheme performs only one Krylov subspace decomposition and a few computations of low dimensional exponential matrices, which contrasts with the implementation of other exponential-type integrators of high order. Regarding to existing schemes, the performed numerical study demonstrates a higher effectiveness of the constructed scheme in the integration of a variety of test equations.