A Second Order Approximation for Quasilinear Non-Fickian Diffusion Models

In this paper initial boundary value problems, defined using quasilinear diffusion equations of Volterra type, are considered. These equations arise for instance to describe diffusion processes in viscoelastic media whose behaviour is represented by a Voigt-Kelvin model or a Maxwell model. We propose a finite difference discretization defined on a general nonuniform grid and we show second convergence order. The analysis does not follows the usual splitting of the global error using the solution of an elliptic equation induced by the integro-differential equation. The new approach enables us to reduce the smoothness required to the theoretical solution when the usual split technique is used. Numerical simulations which shows the effectiveness of the method are included.