Convergence Analysis and Approximate Optimal Temporal Step Sizes for Some Finite Difference Methods Discretising Fisher's Equation - Agbavon, Koffi Messan; Appadu, Appanah Rao; Inan, Bilge; Tenkam, Herve Michel (Frontiers in Applied Mathematics and Statistics, 2022)
In this work, we obtain 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, stability and obtain convergence estimates. We then obtain values of L1 and L1 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. Lastly, we compute the rate of convergence in time using L1 and L1 errors for all the three methods for the three cases.