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-Pade 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 Pade 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.(c) 2023 IMACS. Published by Elsevier B.V. All rights reserved.

Automated summary

This paper introduces a new class of Jacobian-free High Order Local Linearization (HOLL) methods for integrating large systems of initial value problems. These methods approximate a single phi-function times vector, eliminating the need for evaluating and storing Jacobian matrices, leading to more efficient computations and memory usage. The convergence rate and order preserving condition of the new methods are derived, and a novel Matrix-free Krylov-Pade approximation and an adaptive strategy for selecting the Krylov dimension and Pade order are proposed. Numerical simulations validate the theoretical findings and demonstrate the performance of the Jacobian-free integrators compared to others.