Numerical schemes for modelling time-fractional dynamics of non-isothermal diffusion in soils

Abstract

In this paper we consider the issues for increasing computation schemes performance while modelling non-isothermal convective diffusion using Caputo fractional derivative model. We propose to solve the considered problem using locally one-dimensional finite difference scheme. As the numerical computation of fractional derivative has the biggest influence on computational complexity, an additional approximation procedure of logarithmic complexity is proposed. The procedure represents finite difference form of the fractional derivative as two sets of partial Taylor series which are modified in the process of sequential computations during modelling. As the considered model is non-isothermal, two coupled processes of different speeds are considered. In this case the use of non-uniform time steps can be efficient to improve performance. The proposed approximation procedure is modified for the case of non-uniform time steps and the trial and error method is used to dynamically change time step length during modelling. Time interval in which such schemes are efficient is experimentally determined.