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

This repo holds the original source code used for the Local Linearization papers

This Matlab toolbox provides the two implementations of the Jacobian-free Locally Linearized Runge-Kutta method of Dormand and Prince (LLDP) described in [1] for the integration of large systems of initial value problems.

Furthermore, the toolbox provides the implementation of the Concurrent Jacobian-free Locally Linearized Runge-Kutta method of Dormand and Prince (LLDP) described in [2] for the integration of the same type of equations.

[1] Naranjo-Noda, F. S., & Jimenez, J. C. (2024). Jacobian-free Locally Linearized Runge-Kutta method of Dormand and Prince for large systems of differential equations. Journal of Computational and Applied Mathematics, 449, 115974.

[2] New class of Jacobian-free high order local Linearization methods for differential equations by F.S. Naranjo-Noda and J.C. Jimenez

Jacobian-free HOLL

LLDP1 variable step-size Jacobian-free Locally Linearized Runge-Kutta method of Dormand and Prince with variable order finite difference in the Arnoldi algorithm

LLDP2 variable step-size Jacobian-free Locally Linearized Runge-Kutta method of Dormand and Prince with fixed order 1 finite difference in the Arnoldi algorithm

LLDP variable step-size Concurrent Jacobian-free Locally Linearized Runge-Kutta method of Dormand and Prince

Demos

run_examples_lldp_fj.m generates the Tables 3-5 of [1] illustrating the performance of the Jacobian-free Locally Linearized Runge-Kutta method of Dormand and Prince schemes LLDP1 and LLDP2 in the integration of test examples.

run_large_examples_lldp_fj.m generates the Table 7 of [1] illustrating the performance of the Jacobian-free Locally Linearized Runge-Kutta method of Dormand and Prince schemes LLDP1 and LLDP2 in the integration of test examples with larger dimensions.

run_examples_lldp_fj_i.m generates the Tables 1-2 of [2] illustrating the performance of the Concurrent Jacobian-free Locally Linearized Runge-Kutta method of Dormand and Prince scheme LLDP in the integration of test examples.