Sharp error estimates for spatial-temporal finite difference approximations to fractional sub-diffusion equation without regularity assumption on the exact solution

02/06/2023
by   Daxin Nie, et al.
0

Finite difference method as a popular numerical method has been widely used to solve fractional diffusion equations. In the general spatial error analyses, an assumption u∈ C^4(Ω̅) is needed to preserve 𝒪(h^2) convergence when using central finite difference scheme to solve fractional sub-diffusion equation with Laplace operator, but this assumption is somewhat strong, where u is the exact solution and h is the mesh size. In this paper, a novel analysis technique is proposed to show that the spatial convergence rate can reach 𝒪(h^min(σ+1/2-ϵ,2)) in both l^2-norm and l^∞-norm in one-dimensional domain when the initial value and source term are both in Ĥ^σ(Ω) but without any regularity assumption on the exact solution, where σ≥ 0 and ϵ>0 being arbitrarily small. After making slight modifications on the scheme, acting on the initial value and source term, the spatial convergence rate can be improved to 𝒪(h^2) in l^2-norm and 𝒪(h^min(σ+3/2-ϵ,2)) in l^∞-norm. It's worth mentioning that our spatial error analysis is applicable to high dimensional cube domain by using the properties of tensor product. Moreover, two kinds of averaged schemes are provided to approximate the Riemann–Liouville fractional derivative, and 𝒪(τ^2) convergence is obtained for all α∈(0,1). Finally, some numerical experiments verify the effectiveness of the built theory.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
12/15/2021

A Higher Order Resolvent-positive Finite Difference Approximation for Fractional Derivatives

We develop a finite difference approximation of order α for the α-fracti...
research
09/07/2021

Numerical approximations for the fractional Fokker-Planck equation with two-scale diffusion

Fractional Fokker-Planck equation plays an important role in describing ...
research
05/31/2022

A sharp α-robust L1 scheme on graded meshes for two-dimensional time tempered fractional Fokker-Planck equation

In this paper, we are concerned with the numerical solution for the two-...
research
08/24/2019

Analysis of a time-stepping discontinuous Galerkin method for fractional diffusion-wave equation with nonsmooth data

This paper analyzes a time-stepping discontinuous Galerkin method for fr...
research
08/30/2019

Numerical analysis of a semilinear fractional diffusion equation

This paper considers the numerical analysis of a semilinear fractional d...
research
04/12/2023

A quadrature scheme for steady-state diffusion equations involving fractional power of regularly accretive operator

In this paper, we construct a quadrature scheme to numerically solve the...
research
08/25/2022

Uniform error estimate of an asymptotic preserving scheme for the Lévy-Fokker-Planck equation

We establish a uniform-in-scaling error estimate for the asymptotic pres...

Please sign up or login with your details

Forgot password? Click here to reset