Year: 2011
International Journal of Numerical Analysis and Modeling, Vol. 8 (2011), Iss. 2 : pp. 226–251
Abstract
The problem of non-local nonlinear non-Fickian polymer diffusion as modelled by a diffusion equation with a nonlinearly coupled boundary value problem for a viscoelastic 'pseudostress' is considered (see, for example, DA Edwards in Z. angew. Math. Phys., 52, 2001, pp. 254-288). We present two numerical schemes using the implicit Euler method and also the Crank-Nicolson method. Each scheme uses a Galerkin finite element method for the spatial discretisation. Special attention is paid to linearising the discrete equations by extrapolating the value of the nonlinear terms from previous time steps. A priori error estimates are given, based on the usual assumptions that the exact solution possesses certain regularity properties, and numerical experiments are given to support these error estimates. We demonstrate by example that although both schemes converge at their optimal rates the Euler method may be more robust than the Crank-Nicolson method for problems of practical relevance.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2011-IJNAM-684
International Journal of Numerical Analysis and Modeling, Vol. 8 (2011), Iss. 2 : pp. 226–251
Published online: 2011-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 26
Keywords: a priori error estimates nonlinear diffusion non-Fickian diffusion finite element method linearisation extrapolation implicit Euler Crank-Nicolson.