On a uniformly accurate finite difference approximation of a singularly perturbed reaction–diffusion problem using grid equidistribution

摘要

We examine the convergence properties of a finite difference approximation of a singularly perturbed reaction–diffusion boundary value problem using a nonuniform grid. The grid is based on the equidistribution of a positive monitor function that is a linear combination of a constant floor and a power of the second derivative of the solution. Analysis shows how the monitor function can be chosen to ensure that the accuracy of the numerical approximation is insensitive to the size of the singular perturbation parameter. The use of equidistribution principles appears in many practical grid adaption schemes and our analysis provides insight into the convergence behaviour on such grids. Numerical results are given that confirm the uniform convergence rates.