  This paper considers a two-step fourth-order modified explicit
Euler/Crank-Nicolson numerical method for solving the time-variable fractional
mobile-immobile advection-dispersion model subjects to suitable initial and
boundary conditions. Both stability and error estimates of the new approach are
deeply analyzed in the $L^{\infty}(0,T;L^{2})$-norm. The theoretical studies
show that the proposed technique is unconditionally stable with convergence of
order $O(k+h^{4})$, where $h$ and $k$ are space step and time step,
respectively. This result indicate that the two-step fourth-order formulation
is more efficient than a broad range of numerical schemes widely studied in the
literature for the considered problem. Numerical experiments are performed to
verify the unconditional stability and convergence rate of the developed
algorithm.
