Digital simulations on unequally spaced grids. Part 3. Attaining exponential convergence for the discretisation error of the flux as a new strategy in digital simulations of electrochemical experiments

It is a well-known phenomenon, called superconvergence, in the mathematical literature that the error level of an integral quantity can be much smaller than the magnitude of the local errors involved in the computation of this quantity. When discretising an integrated form of Fick's second law of diffusion the local errors reflect the accuracy of individual concentration points while the integral quantity has the physical meaning of the flux. The paper demonstrates how an extraordinarily fast exponential convergence towards zero can be achieved for the discretisation error of the flux by subjecting the partial differential equations to a suitable variable transformation. The variable transformation actually used in the present paper is not new. However, the paper presents a theoretical concept for explaining why the accuracy of the simulated flux can be preserved even on strongly expanding space grids when discretising the partial differential equations (PDEs) in the optimal way. (C) 2004 Elsevier B.V. All rights reserved.