A Higher-Order Uniformly Convergent Numerical Method for a Singularly Perturbed Nonlinear Reaction–Diffusion Equation

This study presents a higher-order uniformly convergent numerical method for a singularly perturbed nonlinear reaction–diffusion equation. The quasilinearization technique is used to transform a nonlinear term into a linear boundary value problem. The equivalent singularly perturbed reaction–diffusion differential equation is discretized using a central finite-difference method on the Shishkin and Bakhvalov meshes. The stability and parameter-uniform convergence are rigorously analyzed and established. Theoretically, the proposed method is of second order. The application of Richardson extrapolation demonstrates that the Bakhvalov mesh achieves a pure CM−4 rate, whereas the Shishkin mesh attains an ε-uniform convergence rate of CM−4ln4M. Three numerical examples have been solved to corroborate the theoretical findings.