Parameter-uniform numerical method for global solution and global normalized flux of singularly perturbed boundary value problems using grid equidistribution

Abstract

In this paper, we present the analysis of an upwind scheme for obtaining the global solution and the normalized flux for a convection–diffusion two-point boundary value problem. The solution of the upwind scheme is obtained on a suitable nonuniform mesh which is formed by equidistributing the arc-length monitor function. It is shown that the discrete solution obtained by the upwind scheme and the global solution obtained via interpolation converges uniformly with respect to the perturbation parameter. In addition, we prove the uniform first-order convergence of the weighted derivative of the numerical solution on this nonuniform mesh and the uniform convergence of the global normalized flux on the whole domain. Numerical results are presented that demonstrate the sharpness of our results.