TY - JOUR
T1 - Efficient Numerical Algorithms for Three-Dimensional Fractional Partial Differential Equations
JO - Journal of Computational Mathematics
VL - 4
SP - 371
EP - 391
PY - 2014
DA - 2014/08
SN - 32
DO - http://doi.org/10.4208/jcm.1401-m3893
UR - https://global-sci.org/intro/article_detail/jcm/9893.html
KW - Fractional partial differential equations, Numerical stability, Locally one dimensional method, Crank-Nicolson procedure, Alternating direction implicit method.
AB - <p style="text-align: justify;">This paper detailedly discusses the locally one-dimensional numerical methods for efficiently solving the three-dimensional fractional partial differential equations, including fractional advection diffusion equation and Riesz fractional diffusion equation. The second order finite difference scheme is used to discretize the space fractional derivative and the Crank-Nicolson procedure to the time derivative. We theoretically prove and numerically verify that the presented numerical methods are unconditionally stable and second order convergent in both space and time directions. In particular, for the Riesz fractional diffusion equation, the idea of reducing the splitting error is used to further improve the algorithm, and the unconditional stability and convergency are also strictly proved and numerically verified for the improved scheme.</p>