Jacobian-free High Order Local Linearization methods for large systems of initial value problems.

In this paper, the class of Jacobian-free High Order Local Linearization (HOLL) methods is introduced for integrating large systems of initial value problems. Unlike other high order exponential integrators, this new class of methods involves the approximation of just a single phi-function times vector and does not require the evaluation and storing of Jacobian matrices, which result more efficient in terms of flops operations and computer memory. Specifically, the new Jacobian-free integrators are constructed by approximating the products of Jacobian matrix times vector that appear in the ordinary HOLL methods. A general result on the convergence rate of the new methods is derived in terms of the convergence rate of the mentioned approximations, resulting in a simple order condition that preserves the order of the ordinary HOLL methods. To design effective integrators of the new class, a novel Matrix-free Krylov-Padé approximation to the product of phi-function times vector is also proposed as well as an adaptive strategy for the automatic selection of the Krylov dimension and Padé order. As a particular instance, a class of Jacobian-free Locally Linearized Runge-Kutta schemes is developed, and schemes of third to fifth order explicitly constructed. Numerical simulations are provided illustrating the theoretical findings concerning the convergence rate of the introduced methods as well as their performance in comparison with other Jacobian-free integrators. • Jacobian-free high order exponential integrator with a single phi-function times vector per integration step. • Only one Krylov subspace approximation and matrix exponential per integration step. • Convergence rate with a single order condition. • Computation of phi-functions times vectors with high order Matrix-free Krylov-Padé approximation. • Adaptive estimation of Krylov subspaces dimensions with low fluctuations. [ABSTRACT FROM AUTHOR]