Locally Linearized Runge-Kutta method of Dormand and Prince for large systems of initial value problems

Abstract

In this paper, new Locally Linearized Runge-Kutta formulas of Dormand and Prince are introduced with the purpose of integrating large systems of initial value problems. The new formulas are obtained by replacing the Padé approximation for matrix exponential by a Krylov-Padé approximation in the embedded formulas proposed in a previous work. Unlike other high order exponential integrators, these new formulas involve the approximation of just a single phi-function times vector, which makes them particularly attractive from a computational viewpoint. With these new formulas, adaptive schemes are constructed with novelties in the approximation of phi-functions times vectors by Krylov-Padé approximations, and in the strategies for controlling the approximation errors, for estimating the dimension of the Krylov subspaces, and for the reuse of the Jacobians. In addition, numerical experiments are performed to show the potential of the new embedded formulas and adaptive codes in the integration of known physical, biophysical, and physical-chemistry models.