数值求解分数阶偏微分方程的高精度快速算法研究

In the recent few years, effective numerical solutions for solving the fractional partial differential equation have been heavily favored. Main numerical algorithms cover the finite difference method, the finite element method, the spectral method, etc. Among which, the finite difference method is studied extensively due to its computational simplicity and easy-to-hand property. Classical difference methods for solving the fractional partial differential equation are usually with low-order accuracy, poor theoretical analysis, large memory and computational quantity. How to further improve the computational accuracy, speed the computation and perfect the theoretical analysis is one of the subject need to continue thorough investigation in the field. This project mainly aims to develop the fast numerical algorithms with high-order accuracy for time-fractional partial differential equations. The techniques of exploring superconvergence, high-order interpolation, weighed average for approximating the time-fractional derivative and high-order interpolation with derivatives for approximating the spatial derivatives in the compact stencil will be considered. The stability and convergence of the resultant algorithms will also be studied. The new developed algorithms will be expected to simply handle the non self-adjoint advection-diffusion problems with variable coefficients and derivative boundary conditions. Moreover, some fast algorithms will be investigated by using the FFT, pre-processing for the obtained numerical algorithms with high-order accuracy.

近年来，分数阶偏微分方程的有效数值求解倍受青睐．已有数值算法主要涵盖了有限差分法、有限元法和谱方法等，其中差分算法以其计算简单和易操作的优势被广泛研究．求解分数阶偏微分方程的经典差分算法数值精度偏低、理论分析匮乏、存储量大、计算耗时，如何进一步提高计算精度、加快计算速度和完善理论分析是该领域需要继续深入研究的课题之一．本项目拟主要针对时间分数阶偏微分方程，通过超收敛、高次插值和加权平均等技巧来提高时间分数阶导数离散精度，通过带导数的高次插值对空间导数紧致离散，从而建立高精度数值求解算法，并探讨算法的稳定性和收敛性．期待该算法可简单处理导数边界条件，并适用于求解变系数的非自共轭对流－扩散问题．同时，通过FFT、预处理等技巧对所得到的高精度数值算法进行加速计算，从而得到高精度快速算法．

近年来，分数阶微分方程的数值求解因其广泛的应用背景而备受关注。本项目主要研究了求解几类时间分数阶偏微分方程的高精度数值方法。通过分片高次插值、加权平均、超收敛和外推等技术提高了时间导数的离散精度，通过加权平均和组合紧致逼近等技巧实现了空间导数的高阶逼近，基本完成了预定研究目标。发表学术论文19篇，其中SCI论文16篇，2篇是高被引论文；在科学出版社合作出版专著一部。指导硕士研究生8名。取得的研究成果主要包括：建立了求解一类时间分数阶对流扩散方程的空间组合紧致差分格式、求解一类时间分数阶扩散波方程和时空分数阶Bloch-Torrey方程的高精度差分格式、求解二维分数阶Cattaneo方程的有效差分格式、求解几类时间分布阶扩散波方程的高阶差分格式和求解一类时间多项空间四阶分数阶扩散方程的差分方法等。运用离散能量方法或Fourier方法分析了所得算法的唯一可解性、稳定性和收敛性。所有算法均通过数值实验进行了验证。项目研究成果在数值求解分数阶偏微分方程领域有着重要意义，深受国内外同行关注和肯定。