A vertex-centred finite volume method for the 3D multi-term time and space fractional Bloch–Torrey equation with fractional Laplacian

Abstract

We investigate the suitability of a fractional Laplacian in the construction of a three-dimensional (3D) multi-term time–space fractional Bloch–Torrey equation. Instead of using the Riesz fractional derivative as in most previous research, we consider the fractional Laplacian in space and multi-term Caputo fractional derivative in time. A vertex-centred finite volume method is firstly derived to discretise this 3D fractional dynamic system in combination with the matrix transfer technique and the weighted and shifted Grünwald–Letnikov formula. Since computing the fractional power of the matrix representation m(−Δ) of the Laplacian operator results in a highly dense matrix, we use matrix function approximations developed in a Krylov subspace framework and exploit the sparsity of m(−Δ) to reduce the computational overhead. Then the Lanczos method is employed to approximate vector products of matrix functions. A preconditioner is applied to accelerate the convergence process, which shows excellent performance and reduces the memory considerably. In particular, we introduce block matrix functions to deal with the coupled system. Furthermore, the analytical solution of the two-term time–space fractional diffusion equation is derived and expressed by the multinomial Mittag-Leffler function. The feasibility and effectiveness of the proposed numerical techniques are verified by some numerical examples. Compared with two-dimensional fractional models or single-term time fractional dynamic systems, this multi-term time–space fractional model provides more flexibility for the characterisation of anomalous diffusion in biological tissues.