Numerical Solution of Reaction–Diffusion Equations with Convergence Analysis

Heidari, Mohammad; Ghovatmand, M.; Skandari, Mohammad Hadi Noori; Băleanu, Dumitru

doi:10.1007/s44198-022-00086-1

Cited by 3 publications

(1 citation statement)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Heidari et al [22] utilized a Legendre-Gauss-Lobatto spectral collocation method to solve a general diffusion-reaction equation. Kolev et al [23] proposed a numerical method that preserves the nonnegative property of the variable of the solutions of a diffusion-reaction equation-system.…”

Section: Introductionmentioning

confidence: 99%

Analytical and Numerical Results for the Diffusion-Reaction Equation When the Reaction Coefficient Depends on Simultaneously the Space and Time Coordinates

et al. 2023

View full text Add to dashboard Cite

We utilize the travelling-wave Ansatz to obtain novel analytical solutions to the linear diffusion–reaction equation. The reaction term is a function of time and space simultaneously, firstly in a Lorentzian form and secondly in a cosine travelling-wave form. The new solutions contain the Heun functions in the first case and the Mathieu functions for the second case, and therefore are highly nontrivial. We use these solutions to test some non-conventional explicit and stable numerical methods against the standard explicit and implicit methods, where in the latter case the algebraic equation system is solved by the preconditioned conjugate gradient and the GMRES solvers. After this verification, we also calculate the transient temperature of a 2D surface subjected to the cooling effect of the wind, which is a function of space and time again. We obtain that the explicit stable methods can reach the accuracy of the implicit solvers in orders of magnitude shorter time.

show abstract

Section: Introductionmentioning

confidence: 99%

Analytical and Numerical Results for the Diffusion-Reaction Equation When the Reaction Coefficient Depends on Simultaneously the Space and Time Coordinates

et al. 2023

View full text Add to dashboard Cite

show abstract

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

Jahdali,

Fernández,

Dalcin

et al. 2024

Commun. Appl. Math. Comput.

View full text Add to dashboard Cite

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.

show abstract