Fast Difference Scheme Research Articles

A fast difference scheme on a graded mesh for time-fractional and space distributed-order diffusion equation with nonsmooth data

The time-fractional and space distributed-order diffusion equation with a singularity at the initial time is investigated in this paper. For the solution of this equation, we propose a new difference scheme on a graded mesh. The theoretical underpinning for the suggested procedure is presented, which includes the stability and convergence of the difference scheme. Furthermore, the difference strategy proves to be unconditionally stable in this research. Additionally, we demonstrate that the difference technique is convergent and that the temporal convergence rate is faster than when a uniform mesh uses. Finally, a test example is provided to ensure that the theoretical analysis is accurate.

An H2N2 Interpolation for Caputo Derivative with Order in (1, 2) and Its Application to Time-Fractional Wave Equations in More Than One Space Dimension

In this paper, a new derived method is developed for a known numerical differential formula of the Caputo fractional derivative of order $$\gamma \in (1,2)$$ (Li and Zeng in Numerical methods for fractional calculus. Chapman & Hall/CRC numerical analysis and scientific computing, CRC Press, Boca Raton, 2015) by means of the quadratic interpolation polynomials, and a concise expression of the truncation error is given. This new method will be called as the H2N2 method because of the application of the quadratic Hermite and Newton interpolation polynomials. A finite difference scheme with a second order accuracy in space and a $$(3-\gamma )$$-th order accuracy in time based on the H2N2 method is constructed for the initial boundary value problem of time-fractional wave equations. The stability and convergence of the difference scheme are proved. Furthermore, in order to increase computational efficiency, using the sum-of-exponentials to approximate the kernel $$t^{1-\gamma }$$, a fast difference scheme is presented. The problem with weak regularity at the initial time is also discussed with the help of the graded meshes. At each time level, the difference scheme is solved with a fast Poisson solver. Numerical results show the effectiveness of the two difference schemes and confirm our theoretical analysis.

The Temporal Second Order Difference Schemes Based on the Interpolation Approximation for the Time Multi-term Fractional Wave Equation

In this paper, two temporal second-order schemes are derived and analyzed for the time multi-term fractional diffusion-wave equation based on the order reduction technique. The weighted average at two time levels is applied to the discretization of the spatial derivative, in which the weight coefficient corresponds to the optimal point for the time discretization. The two difference schemes are proved to be uniquely solvable. The stability and convergence are rigorously investigated utilizing the energy method. In addition, a fast difference scheme is also presented. The applicability and the accuracy of the schemes are demonstrated by several numerical experiments.