A high-order numerical method for solving the 2D fourth-order reaction-diffusion equation

作者：Zhang, Haixiang; Yang, Xuehua^*; Xu, Da^*

来源：Numerical Algorithms, 2019, 80(3): 849-877.

DOI：10.1007/s11075-018-0509-z

• Fourth-order fractional equation
• Orthogonal spline collocation
• Finite difference method
• Error estimate

In the present work, orthogonal spline collocation (OSC) method with convergence order O((3-) + h(r+1)) is proposed for the two-dimensional (2D) fourth-order fractional reaction-diffusion equation, where , h, r, and are the time-step size, space size, polynomial degree of space, and the order of the time-fractional derivative (0 < <1), respectively. The method is based on applying a high-order finite difference method (FDM) to approximate the time Caputo fractional derivative and employing OSC method to approximate the spatial fourth-order derivative. Using the argument developed recently by Lv and Xu (SIAM J. Sci. Comput. 38, A2699-A2724, 2016) and mathematical induction method, the optimal error estimates of proposed fully discrete OSC method are proved in detail. Then, the theoretical analysis is validated by a number of numerical experiments. To the best of our knowledge, this is the first proof on the error estimates of high-order numerical method with convergence order O((3-) + h(r+1)) for the 2D fourth-order fractional equation.

• 出版日期2019-3
• 单位湖南师范大学; 北京应用物理与计算数学研究所; 湖南工业大学