Fully-Discrete Lyapunov Consistent Discretizations for Parabolic Reaction-Diffusion Equations with r Species

doi:10.1007/s42967-024-00425-7

Abstract

Abstract: Reaction-diffusion equations model various biological, physical, sociological, and environmental phenomena. Often, numerical simulations are used to understand and discover the dynamics of such systems. Following the extension of the nonlinear Lyapunov theory applied to some class of reaction-diffusion partial differential equations (PDEs), we develop the first fully discrete Lyapunov discretizations that are consistent with the stability properties of the continuous parabolic reaction-diffusion models. The proposed framework provides a systematic procedure to develop fully discrete schemes of arbitrary order in space and time for solving a broad class of equations equipped with a Lyapunov functional. The new schemes are applied to solve systems of PDEs, which arise in epidemiology and oncolytic M1 virotherapy. The new computational framework provides physically consistent and accurate results without exhibiting scheme-dependent instabilities and converging to unphysical solutions. The proposed approach represents a capstone for developing efficient, robust, and predictive technologies for simulating complex phenomena.

Key words: Reaction-diffusion, Lyapunov functional, Fully discrete Lyapunov consistent discretizations, Epidemiology, Tumor growth, Oncolytic M1 virotherapy

CLC Number:

Rasha Al Jahdali, David C. Del Rey Fernández, Lisandro Dalcin, Mohammed Sayyari, Peter Markowich, Matteo Parsani. Fully-Discrete Lyapunov Consistent Discretizations for Parabolic Reaction-Diffusion Equations with r Species[J]. Communications on Applied Mathematics and Computation, 2026, 8(1): 195-231.

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