On an Adaptive Time Stepping Strategy for Solving Nonlinear Diffusion Equations

A new time step selection procedure is proposed for solving nonlinear diffusion equations. It has been implemented in the ASWR finite element code of Lorenz and Svoboda [10] for 2D semiconductor process modelling diffusion equations. The strategy is based on equidistributing the local truncation errors of the numerical scheme. The use of B-splines for interpolation (as well as for the trial space) results in a banded and diagonally dominant matrix. The approximate inverse of such a matrix can be provided to a high degree of accuracy by another handed matrix, which in turn can be used to work out the approximate finite difference scheme corresponding to the ASWR finite element method, and further to calculate estimates of the local truncation errors of the numerical scheme. Numerical experiments on six full simulation problems arising in semiconductor process modelling have been carried out. Results show that our proposed strategy is more efficient and better conserves the total mass.