New efficient linearly implicit numerical methods for stiff huge differential problems
Giovanni Pagano (Department of Mathematics, University of Salerno), Lidia Aceto, Dajana Conte, Beatrice Paternoster
Partial Differential Equations (PDEs) are used for modeling various phenomena. We are interested in the efficient numerical solution of PDEs from applications, such as: a DIB (Dual-Ion Batteries) model for the formation of spatio-temporal patterns in electrodeposition in batteries [3]; models for the evolution of vegetation in environments characterized by specific conditions such as soil aridity, rainfall periodicity, and so on [5]. The spatial discretization of these models leads to huge stiff initial value problems. In this talk, we derive a new class of linearly implicit numerical methods capable of solving the mentioned problems accurately, employing reasonable computing times, and reproducing the expected Turing patterns.
The proposed methods constitute a generalization of the TASE-RK (Time-Accurate and highly-Stable Explicit Runge-Kutta) numerical schemes introduced by Bassenne et al. and Calvo et al. in 2021 [2, 4]. The latter make use of appropriate preconditioners, called TASE operators, to improve the stability of explicit RK methods. The new methods, called GTRK (Generalized TASE-RK) [1], are derived using different TASE operators for each stage of the underlying explicit RK scheme. By exploiting connections between GTRK methods and W-methods for the study of consistency, we show that it is possible to drastically reduce the number of linear systems required by classical TASE-RK schemes to reach order \(p \leq 4\). Furthermore, we construct A-stable GTRK methods of order \(p=2,3,4\). The numerical experiments show the better efficiency of the proposed methods over the classical TASE-RK schemes, and over other linearly implicit numerical methods from the scientific literature.
References
L. Aceto, D. Conte, and G. Pagano. Generalized TASE Runge-Kutta methods for integrating stiff differential problems. Submitted.
M. Bassenne, L. Fu, and A. Mani. Time-accurate and highly-stable explicit operators for stiff differential equations. J. Comput. Phys., 424:Paper No. 109847, 24, 2021.
B. Bozzini, D. Lacitignola, and I. Sgura. Spatio-temporal organization in alloy electrodeposition: a morphochemical mathematical model and its experimental validation. J. Solid State Electr., 17, 467–479, 2013.
M. Calvo, J. I. Montijano, and L. Rández. A note on the stability of time-accurate and highly-stable explicit operators for stiff differential equations. J. Comput. Phys., 436:Paper No. 110316, 13, 2021.
D. Conte, G. Pagano, and B. Paternoster. Nonstandard finite differences numerical methods for a vegetation reaction-diffusion model. J. Comput. Appl. Math., 419:Paper No. 114790, 17, 2023.
This talk is part of a research activity within the project PRIN PNRR 2022 P20228C2PP (CUP: F53D23010020001) “BAT-MEN (BATtery Modeling, Experiments & Numerics)”.