Error Analysis of a Finite Difference Method on Graded Meshes for a Time-Fractional Diffusion Equation

By Martin Stynes, Eugene O'Riordan, J. Gracia. SIAM Journal on Numerical Analysis, 2017

784 citations recorded by the source corpus.

A reaction-diffusion problem with a Caputo time derivative of order $\alpha\in (0,1)$ is considered. The solution of such a problem is shown in general to have a weak singularity near the initial time $t=0$, and sharp pointwise bounds on certain derivatives of this solution are derived. A new analysis of a standard finite difference method for the problem is given, taking into account this initial singularity. This analysis encompasses both uniform meshes and meshes that are graded in time, and includes new stability and consistency bounds. The final convergence result shows clearly how the regularity of the solution and the grading of the mesh affect the order of convergence of the difference scheme, so one can choose an optimal mesh grading. Numerical results are presented that confirm the sharpness of the error analysis.

Subfields: Mechanics of Materials, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/903c31f6fd7282cbae1ad363beff667828f7582f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Mechanics of Materials

1. Minhao Cai
2. Zhibo Wang
3. William McLean
4. Hong-lin Liao
5. Jiwei Zhang
6. Pin Lyu
7. Yuxiang Liang
8. Martin Stynes

In Modeling and Simulation

1. Minhao Cai
2. Martin Stynes
3. Jiwei Zhang
4. Hong-lin Liao
5. William McLean
6. Zhibo Wang
7. Pin Lyu
8. Yuxiang Liang

In Numerical Analysis

1. Minhao Cai
2. Zhibo Wang
3. William McLean
4. Jiwei Zhang
5. Hong-lin Liao
6. Pin Lyu
7. Yuxiang Liang
8. Martin Stynes

Citations in the corpus

• Too much regularity may force too much uniqueness (2016, background): “In [15] convergence of a finite difference scheme for a class of problems that includes Example 1.”
• A Second-Order Scheme with Nonuniform Time Steps for a Linear Reaction-Subdiffusion Problem (2018, positive): “Actually, such meshes have long been used in the numerical solution of Fredholm [4] and Volterra [2] integral equations, and their use for time-fractional PDEs is now well established [8, 10, 15, 19].”
• Numerical Approaches to Fractional Integrals and Derivatives: A Review (2020, background): “One can refer to [72,76,77] for more details in this respect.”
• A fast linearized finite difference method for the nonlinear multi-term time-fractional wave equation (2019, methodology): “It is worth to mention that the solution of many time-fractional differential equations typically displays a weak singularity near the initial time [8, 9, 10, 19, 20, 21, 26, 29, 30], which leads to the loss of time accuracy for many related high-order numerical methods.”