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# Communications in Mathematical Sciences

## Volume 15 (2017)

### Number 5

### High-order quasi-compact difference schemes for fractional diffusion equations

Pages: 1183 – 1209

DOI: https://dx.doi.org/10.4310/CMS.2017.v15.n5.a1

#### Authors

#### Abstract

The continuous time random walk (CTRW) underlies many fundamental processes in non-equilibrium statistical physics. When the jump length of CTRW obeys a power-law distribution, its corresponding Fokker–Planck equation has a space fractional derivative, which characterizes Lévy flights. Sometimes the infinite variance of Lévy flight discourages it as a physical approach; exponentially tempering the power-law jump length of CTRW makes it more ‘physical’ and the tempered space fractional diffusion equation appears. This paper provides the basic strategy of deriving the high-order quasi-compact discretizations for the space fractional derivative and the tempered space fractional derivative. The fourth-order quasi-compact discretization for the space fractional derivative is applied to solve a space fractional diffusion equation, and the unconditional stability and convergence of the scheme are theoretically proved and numerically verified. Furthermore, the tempered space fractional diffusion equation is effectively solved by its counterpart, the fourth-order quasi-compact scheme, and the convergence orders are verified numerically.

#### Keywords

space fractional derivative, tempered space fractional derivative, WSGD discretization, quasi-compact difference scheme, numerical stability and convergence

#### 2010 Mathematics Subject Classification

26A33, 65M06, 65M12

This work was supported by the National Natural Science Foundation of China under Grant Nos. 11271173, 11471150, and 11671182.

Received 27 August 2014

Published 26 June 2017