Modeling and Computing of Fractional Convection Equation

doi:10.1007/s42967-019-00019-8

Communications on Applied Mathematics and Computation ›› 2019, Vol. 1 ›› Issue (4): 565-595.doi: 10.1007/s42967-019-00019-8

• ORIGINAL PAPER • 上一篇下一篇

Modeling and Computing of Fractional Convection Equation

Changpin Li, Qian Yi

Department of Mathematics, Shanghai University, Shanghai 200444, China

收稿日期:2018-10-07 修回日期:2019-01-09 出版日期:2019-12-30 发布日期:2019-10-16
通讯作者: Changpin Li, Qian Yi E-mail:lcp@shu.edu.cn;yiqianqianyi@163.com
基金资助:
The work was partially supported by the National Natural Science Foundation of China under Grant nos. 11671251 and 11632008.

Modeling and Computing of Fractional Convection Equation

Changpin Li, Qian Yi

Department of Mathematics, Shanghai University, Shanghai 200444, China

Received:2018-10-07 Revised:2019-01-09 Online:2019-12-30 Published:2019-10-16
Contact: Changpin Li, Qian Yi E-mail:lcp@shu.edu.cn;yiqianqianyi@163.com
Supported by:
The work was partially supported by the National Natural Science Foundation of China under Grant nos. 11671251 and 11632008.

摘要/Abstract

摘要： In this paper, we derive the fractional convection (or advection) equations (FCEs) (or FAEs) to model anomalous convection processes. Through using a continuous time random walk (CTRW) with power-law jump length distributions, we formulate the FCEs depicted by Riesz derivatives with order in (0, 1). The numerical methods for fractional convection operators characterized by Riesz derivatives with order lying in (0, 1) are constructed too. Then the numerical approximations to FCEs are studied in detail. By adopting the implicit Crank-Nicolson method and the explicit Lax-Wendrof method in time, and the secondorder numerical method to the Riesz derivative in space, we, respectively, obtain the unconditionally stable numerical scheme and the conditionally stable numerical one for the FCE with second-order convergence both in time and in space. The accuracy and efciency of the derived methods are verifed by numerical tests. The transport performance characterized by the derived fractional convection equation is also displayed through numerical simulations.

关键词: Continuous time random walk, Fractional convection equation, Power-law distribution, Riesz derivative

Abstract: In this paper, we derive the fractional convection (or advection) equations (FCEs) (or FAEs) to model anomalous convection processes. Through using a continuous time random walk (CTRW) with power-law jump length distributions, we formulate the FCEs depicted by Riesz derivatives with order in (0, 1). The numerical methods for fractional convection operators characterized by Riesz derivatives with order lying in (0, 1) are constructed too. Then the numerical approximations to FCEs are studied in detail. By adopting the implicit Crank-Nicolson method and the explicit Lax-Wendrof method in time, and the secondorder numerical method to the Riesz derivative in space, we, respectively, obtain the unconditionally stable numerical scheme and the conditionally stable numerical one for the FCE with second-order convergence both in time and in space. The accuracy and efciency of the derived methods are verifed by numerical tests. The transport performance characterized by the derived fractional convection equation is also displayed through numerical simulations.

Key words: Continuous time random walk, Fractional convection equation, Power-law distribution, Riesz derivative

Changpin Li, Qian Yi. Modeling and Computing of Fractional Convection Equation[J]. Communications on Applied Mathematics and Computation, 2019, 1(4): 565-595.

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Modeling and Computing of Fractional Convection Equation

Modeling and Computing of Fractional Convection Equation

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