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The main goal of this article is to establish a new 4th-order numerical differential formula to approximate Riesz derivatives which is obtained by means of a newly established generating function. Then the derived formula is used to solve the Riesz space fractional advection-dispersion equation. Meanwhile, by the energy method, it is proved that the difference scheme is unconditionally stable and convergent with order 𝓞(τ2 + h4). Finally, several numerical examples are given to show that the numerical convergence orders of the numerical differential formulas and the finite difference scheme are in line with the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, high‐order numerical methods for time‐Caputo and space‐Riesz fractional Bloch‐Torrey equations in one‐ and two‐dimensional space are constructed, where the second‐order backward fractional difference operator and the sixth‐order fractional‐compact difference operator are applied to approximate the time and space fractional derivatives, respectively. The stability and convergence of the methods are analyzed and it is shown that the convergence orders are higher than the earlier work. Finally, some numerical experiments are presented to demonstrate the effectiveness of the methods and confirm our theoretical results. [ABSTRACT FROM AUTHOR]

The main goal of this article is to establish a new 4th-order numerical differential formula to approximate Riesz derivatives which is obtained by means of a newly established generating function. Then the derived formula is used to solve the Riesz space fractional advection-dispersion equation. Meanwhile, by the energy method, it is proved that the difference scheme is unconditionally stable and convergent with order 𝓞(τ2 + h4). Finally, several numerical examples are given to show that the numerical convergence orders of the numerical differential formulas and the finite difference scheme are in line with the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, applying a novel second-order numerical approximation formula for the Riesz derivative and Crank–Nicolson technique for the temporal derivative, a numerical algorithm is constructed for the two-dimensional spatial fractional diffusion equation with convergence order, whereτ,andare the temporal and spatial step sizes, respectively. It is proved that the proposed algorithm is unconditionally stable and convergent by using the energy method. Meanwhile, by adding the high-order perturbation items for the above numerical scheme, an alternating direction implicit difference scheme is also constructed. Finally, some numerical results are presented to demonstrate the validity of theoretical analysis and show the accuracy and effectiveness of the method described herein. [ABSTRACT FROM AUTHOR]