  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.

Numerical Analysis;Numerical Analysis