A non-uniform time-stepping method for time-fractional diffusion-wave equations with variable coefficients arising in fluid flow problems

Purpose The time-fractional diffusion-wave equation (TFDWE) models fluid flow in inhomogeneous media where transport phenomena exhibit anomalous diffusion processes, such as super-diffusive behavior seen in fusion plasmas, molecular movement in living cells and more. This paper aims to present two efficient schemes to solve TFDWE with spatially variable coefficients for non-smooth solutions having singularity at initial time t = 0. Design/methodology/approach To quantify the solution’s initial layer, the authors apply a half-point discretization process to the time domain on a non-uniform temporal mesh using the L1 finite difference scheme. Central and compact difference schemes are proposed subject to Dirichlet boundary conditions. The energy method is used to prove the stability and convergence of the proposed schemes. Furthermore, a central difference scheme is also presented to solve the two-dimensional (2D) TFDWE with variable coefficients for non-smooth solutions. Finally, numerical examples are presented to verify the convergence order and consistency of the proposed schemes. Findings The central and compact difference schemes achieve convergence orders O(N−(3−α)+h2) and O(N−(3−α)+h4), respectively, where N,α and h represent the number of temporal grids, order of the fractional derivative and spatial grid size, respectively. The results show that the compact difference scheme reduces the computational cost and storage requirement. Originality/value A high-order finite difference scheme is used to solve TFDWE in both 1D and 2D cases with variable coefficients. The spatial accuracy has improved using the compact difference scheme. In addition, the authors tested one example with an initial discontinuity for 1D and 2D cases.