Uniform convergence analysis of a new adaptive upwind finite difference method for singularly perturbed convection-reaction-diffusion boundary value problems

In this paper, a new adaptive upwind finite difference method based on the arc-length equidistribution principle is studied for solving the general linear singularly perturbed convection-reaction-diffusion two-point boundary value problem. Under the discrete comparison principle and its derived properties, the a posteriori error estimate of the upwind finite difference scheme on an arbitrary mesh is obtained by using a linear interpolation, the existence of the solution of the adaptive upwind finite difference method is proved without the presupposition, and the boundedness of the arc-length of the numerical solution and then the uniform first-order convergence of the adaptive upwind finite difference method are achieved by the discrete Green’s function. Finally, the proposed method is verified to be uniformly first-order convergent and is compared with the existing method in numerical examples