A class of second order difference approximations for solving space fractional diffusion equations

By Wenyi Tian, Weihua Deng. Mathematics of Computation, 2012

661 citations recorded by the source corpus.

A class of second order approximations, called the weighted and shifted Grunwald difference (WSGD) operators, are proposed for Riemann-Liouville fractional derivatives, with their effective applications to numerically solving space fractional diffusion equations in one and two dimensions. The stability and convergence of our difference schemes for space fractional diffusion equations with constant coefficients in one and two dimensions are theoretically established. Several numerical examples are implemented to test the efficiency of the numerical schemes and confirm the convergence order, and the numerical results for variable coefficients problem are also presented.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/ab0eb1436747eb13a9f093d4a67ab33e3583c25d

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In Applied Mathematics

1. Changpin Li
2. Lijing Zhao
3. Minhao Cai
4. G. Em Karniadakis
5. Xiaozhe Hu
6. Wei Cai
7. Xuan Zhao
8. Weihua Deng

In Modeling and Simulation

1. Minhao Cai
2. Changpin Li
3. Weihua Deng
4. Lijing Zhao
5. G. Em Karniadakis
6. Wei Cai
7. Xiaozhe Hu
8. Xuan Zhao

In Numerical Analysis

1. Minhao Cai
2. G. Em Karniadakis
3. Changpin Li
4. Lijing Zhao
5. Wei Cai
6. Xiaozhe Hu
7. Xuan Zhao
8. Weihua Deng

Citations in the corpus

• Adaptive finite element method for fractional differential equations using hierarchical matrices (2016, background): “However, as the size of the problem increases, the time required to solve the final system of equations increases considerably due to the nonlocality of the fractional differential operators [4, 9–11, 22, 25, 29, 35, 36, 39, 42].”
• A series of high‐order quasi‐compact schemes for space fractional diffusion equations based on the superconvergent approximations for fractional derivatives (2013, methodology): “By assembling the Grünwald difference operators with different weights and shifts, a class of stable second order discretizations for Riemann-Liouville space fractional derivative is developed in [33,17], and an abstract way of discussing the stability and convergence of the discretizations can be s”
• Numerical Approaches to Fractional Integrals and Derivatives: A Review (2020, background): “[103], in the spirit of eliminating the first order terms in the truncated errors of the shifted Grünwald–Letnikov formulae.”