Unconditional Stability Of A Two-Step Fourth-Order Modified Explicit Euler/Crank-Nicolson Approach For Solving Time-Variable Fractional Mobile-Immobile Advection-Dispersion Equation

TL;DR

This paper introduces a new two-step fourth-order explicit Euler/Crank-Nicolson method for solving time-variable fractional advection-dispersion equations, demonstrating unconditional stability and high convergence rate.

Contribution

The paper presents a novel unconditionally stable, fourth-order accurate two-step numerical scheme for fractional advection-dispersion equations, with rigorous stability and error analysis.

Findings

The method is unconditionally stable in the $L^{ abla}(0,T;L^{2})$-norm.

Convergence order is $O(k+h^{4})$, confirming high accuracy.

Numerical experiments verify theoretical stability and convergence results.

Abstract

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.