A Fast Temporal Second-Order Compact ADI Scheme for Time Fractional Mixed Diffusion-Wave Equations

Abstract

A fast temporal second-order compact alternating direction implicit (ADI) difference scheme is proposed and analysed for 2D time fractional mixed diffusion-wave equations. The time fractional operators are approximated by mixed fast $L2$-$1_σ$ and fast $L1$-type formulas derived by using the sum-of-exponentials technique. The spatial derivatives are approximated by the fourth-order compact difference operator, which can be implemented by an ADI approach with relatively low computational cost. The resulting fast algorithm is computationally efficient in long-time simulations since the computational cost is significantly reduced. Numerical experiments confirm the effectiveness of the algorithm and theoretical analysis.