In this paper, a novel compact operator is derived for the approximation of Riesz derivative with order $\alpha\in(1,2].$ The proved fourth-order accuracy. Combining in space discretization, linearized difference scheme proposed two-dimensional nonlinear fractional Schrödinger equation. It that uniquely solvable, stable, and convergent $O(\tau^2+h^4)$, where $\tau$ time step size, $h=\max\{h_1,h_2\}$, $h_1,\,h_2$ are grid sizes $x$ direction $y$ direction, respectively. Based on scheme,...

Abstract Alternating direction implicit (ADI) schemes are computationally efficient and widely utilized for numerical approximation of the multidimensional parabolic equations. By using discrete energy method, it is shown that ADI solution unconditionally convergent with convergence order two in maximum norm. Considering an asymptotic expansion difference solution, we obtain a fourth‐order, both time space, by one Richardson extrapolation. Extension our technique to higher‐order compact also...

In this paper, we consider the numerical method for solving two-dimensional fractional diffusion-wave equation with a time derivative of order $\alpha$ ($1<\alpha<2$). A difference scheme combining compact approach spatial discretization and alternating direction implicit (ADI) in stepping is proposed analyzed. The unconditional stability $H^1$ norm convergence are proved rigorously. $\mathcal {O}(\tau^{3-\alpha}+h_1^4+h^4_2),$ where $\tau$ temporal grid size $h_1,h_2$ sizes $x$ $y$...

A Crank–Nicolson-type difference scheme is proposed for solving the subdiffusion equation with fractional derivative, and truncation error analyzed in detail. At each temporal level, only a tridiagonal linear system needs to be solved Thomas algorithm may used. The solvability, unconditional stability, $H^1$ norm convergence are proved. order ${\rm min}\{2-{\gamma}/{2},\;1+\gamma\}$ direction two spatial direction. By Sobolev embedding inequality, we obtain maximum estimate. compact based on...

Abstract The fractional derivatives include nonlocal information and thus their calculation requires huge storage computational cost for long time simulations. We present an efficient high-order accurate numerical formula to speed up the evaluation of Caputo derivative based on L 2-1 σ proposed in [A. Alikhanov, J. Comput. Phys. , 280 (2015), pp. 424-438], employing sum-of-exponentials approximation kernel function appeared derivative. Both theoretically numerically, we prove that while...

A fourth-order compact difference scheme is proposed for two-dimensional linear Schrödinger equations with periodic boundary conditions. By using the discrete energy method, it proven that uniquely solvable, unconditionally stable, and convergent. maximum norm error estimate thus an asymptotic expansion of solution are also obtained. Using solution, high-order approximations could be achieved by Richardson extrapolations. Extension to three-dimensional problems discussed. Numerical...

In this article, motivated by Alikhanov's new work (Alikhanov, J Comput Phys 280 (2015), 424–438), some difference schemes are proposed for both one-dimensional and two-dimensional time-fractional wave equations. The obtained can achieve second-order numerical accuracy in time space. unconditional convergence stability of these the discrete H1-norm proved energy method. spatial compact with results on also presented. addition, three-dimensional problem is briefly mentioned. Numerical...