A High-Order Method of Lines with SSPRK54 Time Integration for the One-Dimensional Nonlinear Fisher Equation: Accuracy, Stability, and Benchmarking
DOI:
https://doi.org/10.61231/nkww1e18Keywords:
Fisher Equation, Reaction–Diffusion, Method of lines, SSP Runge–Kutta, SSP RK54, Traveling Wave, Numerical Stability, Convergence AnalysisAbstract
We present a rigorously documented, reproducible, and benchmarked high‑order numerical framework for the one‑dimensional nonlinear Fisher (Fisher–KPP) equation. The method combines second‑order central finite differences for spatial discretization within the method‑of‑lines (MOL) framework and a five‑stage, fourth‑order strong stability‑preserving Runge–Kutta integrator (SSP‑RK54) for time integration. The resulting semi‑discrete system is advanced with an explicit SSP integrator whose coefficients are given in full; we provide the Butcher tableau, algorithmic pseudocode, and practical guidance for CFL selection. We verify the method on two canonical traveling‑wave test problems with known closed‑form solutions, perform systematic spatial and temporal convergence studies, and compare cost‑normalized accuracy against representative published methods. A detailed truncation‑error derivation, Von Neumann linear stability analysis, and nonlinear positivity discussion are included. Numerical experiments demonstrate fourth‑order temporal convergence and second‑order spatial convergence, with error levels consistent with the theoretical O(Δt4+h2) behavior. We provide reproducibility materials, implementation notes for MATLAB, and recommendations for extensions to stiff regimes, IMEX variants, and multi‑dimensional problems
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