Efficient second-order accurate exponential time differencing for time-fractional advection–diffusion–reaction equations with variable coefficients

Time-fractional advection–diffusion–reaction type equations are useful for characterizing anomalous transport processes. In this paper, linearly implicit as well as explicit generalized exponential time differencing (GETD) schemes are proposed for solving a class of such equations having time–space dependent coefficients. The implicit scheme, being unconditionally stable, is robust in handling the numerical instabilities in problems where the advection term is dominant. Regarding the error analysis, uniformly optimal second-order convergence rates are derived using time-graded meshes to counter the effect of the inherent singularity of the continuous solution. Implementation of generalized exponential integrators requires computing the action of Mittag-Leffler function of matrices on a vector, or on a matrix in the case of the implicit scheme. For cost-effective implementation, using global Padé approximants these computation tasks get reduced to solving linear systems. A new approach based on Sylvester equation formulation of the resulting linear systems is developed in this paper. This technique leads to significantly faster algorithms for implementing the GETD schemes. Numerical experiments are provided to illustrate the theoretical findings and to assert the efficiency of the Sylvester equation based approach. Application of this approach to an existing GETD scheme for solving a nonlinear subdiffusion problem is also discussed.