Efficient Numerical Algorithms for Three-Dimensional Fractional Partial Differential Equations (Journal of Computational Mathematics, 2018)
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.
Keywords: Fractional partial differential equations; Numerical stability; Locally one dimensional method; Crank-Nicolson procedure; Alternating direction implicit method.