Compact finite difference scheme for the solution of time fractional advection‐dispersion equation

In this paper, a compact finite difference method is proposed for the solution of time fractional advection‐dispersion equation which appears extensively in fluid dynamics. In this approach the time fractional derivative of mentioned equation is approximated by a scheme of order O(τ ^{2 − α} ), 0 < α < 1, and spatial derivatives are replaced with a fourth order compact finite difference scheme. We will prove the unconditional stability and solvability of proposed scheme. Also we show that the method is convergence with convergence order O(τ ^{2 − α} + h ⁴). Numerical examples confirm the theoretical results and high accuracy of proposed scheme.