Modelling chemically reacting flow systems—II: An Adaptive spatial mesh techniques for problems with discontinuities and steep fronts

Abstract

An algorithm is developed for the numerical solution of nonlinear time-varying partial differential equations in one space dimension, whose solution exhibits moving spatial derivative discontinuities and steep fronts. A finite-element technique dynamically maps a static uniform mesh onto a moving mesh. This gives the capability of tracking discontinuities and steep fronts and assigning variable subintervals. The technique used is a combination of spatial discretization by Galerkins method, using B-splines, and solution in time by a variable order, variable time-step extrapolation procedure. The two spatial meshes are coupled through a set of ordinary differential equations defining either conditions in the solution function (discontinuities, steep fronts), or direct values of the adaptive mesh. This set of ordinary differential equations is coupled to the partial differential equations defining the problem and is solved simultaneously with it. The technique was applied to the problem of migration of copper salts through polyethylene, and enabled us to handle successfully the phenomena of diffusion, reaction, and precipitation, which mathematically translate into a steep front, boundary layer, and a derivative discontinuity respectively. The simulation reproduced the main features found experimentally, lending credence to the correctness of the model and the validity of the calculated results.