Volume 9, Issue 2
A New Family of Difference Schemes for Space Fractional Advection Diffusion Equation

Can Li & Weihua Deng

Adv. Appl. Math. Mech., 9 (2017), pp. 282-306.

Published online: 2018-05

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  • Abstract

The second order weighted and shifted Gr ¨unwald difference (WSGD) operators are developed in [Tian, Zhou and Deng, Math. Comput., 84 (2015), pp. 1703– 1727] to solve space fractional partial differential equations. Along this direction, we further design a new family of second order WSGD operators; by properly choosing the weighted parameters, they can be effectively used to discretize space (RiemannLiouville) fractional derivatives. Based on the new second order WSGD operators, we derive a family of difference schemes for the space fractional advection diffusion equation. By von Neumann stability analysis, it is proved that the obtained schemes are unconditionally stable. Finally, extensive numerical experiments are performed to demonstrate the performance of the schemes and confirm the convergence orders.

  • Keywords

Riemann-Liouville fractional derivative, WSGD operator, fractional advection diffusion equation, finite difference approximation, stability.

  • AMS Subject Headings

26A33, 65M06, 65M12

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-9-282, author = {}, title = {A New Family of Difference Schemes for Space Fractional Advection Diffusion Equation}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {9}, number = {2}, pages = {282--306}, abstract = {

The second order weighted and shifted Gr ¨unwald difference (WSGD) operators are developed in [Tian, Zhou and Deng, Math. Comput., 84 (2015), pp. 1703– 1727] to solve space fractional partial differential equations. Along this direction, we further design a new family of second order WSGD operators; by properly choosing the weighted parameters, they can be effectively used to discretize space (RiemannLiouville) fractional derivatives. Based on the new second order WSGD operators, we derive a family of difference schemes for the space fractional advection diffusion equation. By von Neumann stability analysis, it is proved that the obtained schemes are unconditionally stable. Finally, extensive numerical experiments are performed to demonstrate the performance of the schemes and confirm the convergence orders.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.2015.m1069}, url = {http://global-sci.org/intro/article_detail/aamm/12149.html} }
TY - JOUR T1 - A New Family of Difference Schemes for Space Fractional Advection Diffusion Equation JO - Advances in Applied Mathematics and Mechanics VL - 2 SP - 282 EP - 306 PY - 2018 DA - 2018/05 SN - 9 DO - http://doi.org/10.4208/aamm.2015.m1069 UR - https://global-sci.org/intro/article_detail/aamm/12149.html KW - Riemann-Liouville fractional derivative, WSGD operator, fractional advection diffusion equation, finite difference approximation, stability. AB -

The second order weighted and shifted Gr ¨unwald difference (WSGD) operators are developed in [Tian, Zhou and Deng, Math. Comput., 84 (2015), pp. 1703– 1727] to solve space fractional partial differential equations. Along this direction, we further design a new family of second order WSGD operators; by properly choosing the weighted parameters, they can be effectively used to discretize space (RiemannLiouville) fractional derivatives. Based on the new second order WSGD operators, we derive a family of difference schemes for the space fractional advection diffusion equation. By von Neumann stability analysis, it is proved that the obtained schemes are unconditionally stable. Finally, extensive numerical experiments are performed to demonstrate the performance of the schemes and confirm the convergence orders.

Can Li & Weihua Deng. (2020). A New Family of Difference Schemes for Space Fractional Advection Diffusion Equation. Advances in Applied Mathematics and Mechanics. 9 (2). 282-306. doi:10.4208/aamm.2015.m1069
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