Parameter-uniformly convergent exponential spline difference scheme for singularly perturbed semilinear reaction–diffusion problems

We consider a Dirichlet boundary value problem for singularly perturbed semilinear reaction–diffusion equation. The problem is discretized using an exponential spline difference scheme derived on the basis of splines in tension on piecewise-uniform Shishkin type mesh. The convergence analysis is given and the method is shown to be almost second order accurate in the discrete maximum norm, uniformly in the perturbation parameter ε . Numerical experiments are conducted to demonstrate the theoretical results.