A two-stage fourth-order modified explicit Euler/Crank-Nicolson approach for a time-variable fractional transport model

This paper considered a two-stage fourth-order modified explicit Euler/Crank-Nicolson numerical method for solving a time-variable fractional transport equation subject to suitable initial and boundary conditions. Both stability and error estimates of the new approach were deeply analyzed in the $ L^{\infty}(0, T; L^{2}) $-norm. The analysis suggests that the proposed technique is unconditionally stable and converges with order $ O(\max\{k^{2-\overline{\beta}}, k^{\frac{3}{2}}\}+h^{4}) $, where $ \overline{\beta} $ is a positive constant satisfying $ 0 < \overline{\beta} < 1 $, $ h $ and $ k $ are the space step and time step, respectively. This result indicates that the two-stage fourth-order formulation is fast and efficient. Numerical experiments were performed to verify the unconditional stability and convergence rate of the developed algorithm.