The tesis was write in Spanish but all the papers are write English. You can get the las version from the Release assets
Partial Differential Equations (PDEs) are used to describe various physical, biophysical, and physical-chemical phenomena, but analytical solutions are only known for a few PDEs defined on simple geometries. For this reason, numerical methods such as finite elements, finite differences, and method of lines are used to approximate the solutions. However, most conventional numerical integrators involve solving large systems of ill-conditioned algebraic equations, which leads to the use of specific preconditioning matrices for each type of PDE. As an exception, exponential integrators for ordinary differential equations (ODEs) that result from the method of lines calculate matrix exponentials and/or functions φ by vectors instead of solving algebraic equations. Exponential integrators are available in two types: Globally Linearized (GL) and Locally Linearized (LL). GL integrators work well for ODEs with small or bounded nonlinear terms by the linear part, while LL integrators improve on GL integrators and, in general, on traditional integrators used with the method of lines to solve PDEs. However, these integrators involve the calculation of several products of functions φ by vector with a high computational cost. The absence of an adaptive strategy for the integration step size and the need to evaluate the exact Jacobian matrix of the ODE's vector field are other practical limitations of these methods. For small-dimensional ODE systems, higher-order LL methods (LLOS) improve the efficiency of the classical LL method of order 2 by adding an approximation to the integral representation or to the differential of its residual. LLOS methods derived from the differential form contain only one product of function φ by vector in each integration step, which distinguishes them positively from those derived from the integral representation and motivates the research of this thesis.
In this thesis, for large dimensional ODE systems, the adaptive implementation of the Runge-Kutta embedded formulas of Dormand and Prince Locally Linearized is proposed, using a higher-order Krylov approximation for the products of function φ by vector, a new error measure and a new way to estimate the optimal Krylov dimension. A new family of Jacobian-Free LLOS methods is also proposed for cases where it is not feasible to evaluate and store the Jacobian matrix of the ODE. For this purpose, a new type of Krylov approximation is introduced, the Jacobian-free, and a criterion for estimating the Krylov dimension. The class of Locally Linearized Jacobian-Free Runge-Kutta schemes is presented and third to fifth order schemes are explicitly constructed. In addition, an adaptive variable-order Jacobian-free scheme is implemented by modifying the Dormand and Prince Locally Linearized Runge-Kutta embedded formulas. Numerical experiments show the effectiveness of new schemes in the integration of known test equations and is compared with that of other exponential integrators.
- F.S. Naranjo-Noda, J.C. Jimenez, Locally Linearized Runge-Kutta method of Dormand and Prince for large systems of initial value problems, Journal of Computational Physics, Volume 426, 2021, 109946, ISSN 0021-9991, https://doi.org/10.1016/j.jcp.2020.109946.
- F.S. Naranjo-Noda, J.C. Jimenez, Computing high dimensional multiple integrals involving matrix exponentials, Journal of Computational and Applied Mathematics, Volume 421, 2023, 114844, ISSN 0377-0427, https://doi.org/10.1016/j.cam.2022.114844.
- F.S. Naranjo-Noda, J.C. Jimenez, Jacobian-free High Order Local Linearization methods for large systems of initial value problems, Applied Numerical Mathematics, Volume 187, 2023, Pages 158-175, ISSN 0168-9274, https://doi.org/10.1016/j.apnum.2023.02.009.
- F.S. Naranjo-Noda, J.C. Jimenez, Jacobian-free Locally Linearized Runge-Kutta method of Dormand and Prince for large systems of differential equations, Journal of Computational and Applied Mathematics, Volume 449, 2024, 115974, ISSN 0377-0427, https://doi.org/10.1016/j.cam.2024.115974.