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.

翻译：本文件考虑了一种经过两步四级修改的明确的Euler/Crank-Nicolson数字方法,用于根据适当的初始条件和边界条件解决时间可变的移动-移动消化-分散模型主题,新办法的稳定性和误差估计均在$L ⁇ infty}(0,T;L ⁇ 2})$-norm中进行深入分析。理论研究表明,拟议的技术无条件稳定,与O(k+h ⁇ 4})o(k+h ⁇ 4})ocolson(美元)ocolent)o(美元)col-ocolson)cold(美元)colental(美元)color)o(美元)colum(美元)为空间步骤和时间步骤。这一结果表明,两步四阶四级配制比文献中广泛研究的关于所考虑的问题的广泛数字方案更有效。进行了数字实验,以核查发达算法的无条件稳定和趋同率。