Asymptotic, numerical and approximate techniques for a free boundary problem arising in the diffusion of glassy polymers

This paper considers approximate solution methods for a one dimensional Stefan problem describing solvent diffusion in glassy polymers. Similar to the classic Stefan problem, the region initially has zero thickness and so must be analysed carefully before performing a numerical computation. A small-time analysis gives the correct starting solution which is then incorporated into the second order accurate Keller box finite difference scheme. We also consider a detailed analysis of small and large time expansions, as well as the large control parameter limit, and show that our generalised approach enables us to obtain higher order terms than given in previous studies. Finally, we apply the combined integral method (CIM) to this problem, which is a refinement of the popular heat balance integral method (HBIM), and compare both the CIM and asymptotic solutions to the numerical results.