Exponential techniques and implicit Runge-Kutta methods for singularly-perturbed volterra integro-differential equations

Numerical experiments performed with an exponential finite difference method in equally-spaced and piecewise-uniform meshes for both the inner and the outer layers and with an implicit Runge-Kutta-Radau IIA method for the outer layer of singularly-perturbed Volterra integro-differential equations are reported. The exponential finite difference technique is based on piecewise linear approximations and its linear stability has been analyzed. It is shown that the exponential method presented in this paper provides first-order accurate solutions for small values of the perturbation parameter, whereas the same technique in a piecewise-uniform mesh is almost second-order uniformly convergent because it does resolve the inner layer and, most importantly, because the finite difference equations are independent of the perturbation parameter in the inner layer. The implicit Runge-Kutta method for the outer layer yields errors that only depend on the step size if the number of stages is small or the step size is large, but depend on both the small perturbation parameter and the step size, otherwise.