A fast difference scheme for the multi-term time fractional advection-diffusion equation with a non-linear source term

In field experiments involving the movement of solutes through heterogeneous porous and fractured media, it has been observed that the contaminant plumes can transition between various diffusive states. This current research is dedicated to the development of a fast difference scheme designed for a multi -term time -fractional order advection-diffusion model that incorporates a nonlinear source term. This model is an instrumental tool in explaining the underlying dynamics of transport. To effectively address the substantial computational and storage challenges inherent in this problem, we present an efficient algorithm specifically designed for the fast computation of the Caputo derivative. This algorithm is based on the summation of exponentials technique. We thoroughly explore the theoretical aspects of our scheme including its unconditional stability and convergence with rigorous proofs. The proposed numerical scheme minimizes the CPU time and storage requirements when compared to the conventional direct difference scheme.