Numerical methods and analysis for a multi-term time–space variable-order fractional advection–diffusion equations and applications

Abstract

Field experiments of solute transport through heterogeneous porous and fractured media show that the growth of contaminant plumes may convert between diffusive states. In this paper, we propose a multi-term time–space variable-order fractional advection–diffusion model (MTT-SVO-FADM) to describe the underlying transport dynamics. We consider a numerical approach based on the implicit numerical method for numerical solution of this model. A fully-discrete numerical scheme is developed by using the classical finite difference method. The unconditional stability and convergence of the scheme are discussed and theoretically proved. We use a modified grid approximation method (MGAM) to estimate the model’s parameters. The MTT-SVO-FADM is then applied to describe transient dispersion observed at a field tracer test and four numerical experiments. The results show that this model can simulate the experimental data more accurately and can efficiently quantify these transitions. • A multi-term time–space variable-order fractional advection–diffusion model is proposed. • A fully-discrete numerical scheme is developed. • The unconditional stability and convergence of the scheme are discussed and proved. • A modified grid approximation method is used to estimate the model’s parameters. • This model is applied to describe transient dispersion observed at a field tracer test and four numerical experiments.