Numerical Computation and Stability Analysis for the Fractional Subdiffusions with Spatial Variable Coefficients

In this paper, we propose an efficient compact finite difference method for a class of time-fractional subdiffusion equations with spatially variable coefficients. Based on the L2-1σ approximation formula of the time-fractional derivative and a fourth-order compact finite difference approximation to the spatial derivative, an efficient compact finite difference method is developed. The local truncation error and the solvability of the developed method are discussed in detail. The unconditional stability of the resulting scheme and also its convergence of second-order in time and fourth-order in space are rigorously proved using a discrete energy analysis method. Numerical examples are provided to demonstrate the accuracy and the theoretical results.

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An Efficient Compact Difference Method for Temporal Fractional Subdiffusion Equations

Advances in Mathematical Physics ◽

10.1155/2019/3263589 ◽

2019 ◽

Vol 2019 ◽

pp. 1-9 ◽

Cited By ~ 1

Author(s):

Lei Ren ◽

Lei Liu

Keyword(s):

Finite Difference ◽

Fourth Order ◽

Difference Method ◽

Second Order ◽

Finite Difference Approximation ◽

Difference Approximation ◽

Approximation Formula ◽

Discrete Energy ◽

Compact Finite Difference ◽

Second Order In Time

In this paper, a high-order compact finite difference method is proposed for a class of temporal fractional subdiffusion equation. A numerical scheme for the equation has been derived to obtain 2-α in time and fourth-order in space. We improve the results by constructing a compact scheme of second-order in time while keeping fourth-order in space. Based on the L2-1σ approximation formula and a fourth-order compact finite difference approximation, the stability of the constructed scheme and its convergence of second-order in time and fourth-order in space are rigorously proved using a discrete energy analysis method. Applications using two model problems demonstrate the theoretical results.

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Optimal Error Estimates of Compact Finite Difference Discretizations for the Schrödinger-Poisson System

Communications in Computational Physics ◽

10.4208/cicp.251011.270412a ◽

2013 ◽

Vol 13 (5) ◽

pp. 1357-1388 ◽

Cited By ~ 9

Author(s):

Yong Zhang

Keyword(s):

Finite Difference ◽

Finite Difference Method ◽

Error Estimates ◽

Difference Method ◽

Finite Difference Approximation ◽

Difference Approximation ◽

Optimal Error Estimates ◽

Optimal Error ◽

Compact Finite Difference ◽

Compact Finite Difference Method

AbstractWe study compact finite difference methods for the Schrodinger-Poisson equation in a bounded domain and establish their optimal error estimates under proper regularity assumptions on wave function and external potential V(x). The Crank-Nicolson compact finite difference method and the semi-implicit compact finite difference method are both of order Ҩ(h4 + τ2) in discrete l2,H1 and l∞ norms with mesh size h and time step t. For the errors ofcompact finite difference approximation to the second derivative andPoisson potential are nonlocal, thus besides the standard energy method and mathematical induction method, the key technique in analysisis to estimate the nonlocal approximation errors in discrete l∞ and H1 norm by discrete maximum principle of elliptic equation and properties of some related matrix. Also some useful inequalities are established in this paper. Finally, extensive numerical re-sults are reported to support our error estimates of the numerical methods.

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