Convergence Analysis and Approximate Optimal Temporal Step Sizes for Some Finite Difference Methods Discretising Fisher's Equation

• Published in: Frontiers in applied mathematics and statistics Vol. 8
• Main Authors: Agbavon, Koffi Messan, Appadu, Appanah Rao, Inan, Bilge, Tenkam, Herve Michel
• Format: Journal Article
• Language: English
• Published: Frontiers Media S.A 13.07.2022
• Subjects: convergence estimate; EEFD; Fisher's equation; FTCS; NSFD; optimal
• ISSN: 2297-4687; 2297-4687
• DOI: 10.3389/fams.2022.921170

In this study, we obtain a numerical solution for Fisher's equation using a numerical experiment with three different cases. The three cases correspond to different coefficients for the reaction term. We use three numerical methods namely; Forward-Time Central Space (FTCS) scheme, a Nonstandard Finite Difference (NSFD) scheme, and the Explicit Exponential Finite Difference (EEFD) scheme. We first study the properties of the schemes such as positivity, boundedness, and stability and obtain convergence estimates. We then obtain values of L 1 and L ∞ errors in order to obtain an estimate of the optimal time step size at a given value of spatial step size. We determine if the optimal time step size is influenced by the choice of the numerical methods or the coefficient of reaction term used. Finally, we compute the rate of convergence in time using L 1 and L ∞ errors for all three methods for the three cases.