非线性分数阶偏微分方程的高效数值算法研究

In recent years, developing effective numerical algorithms for solving the fractional partial differential equations is one of the hot issues in the field of computational mathematics. The finite difference method is studied extensively due to its easy implementation. However, it has some limitations; such as poor adaptability on complex geometric domain, low-order accuracy, large memory cost, time-consuming computation and so on. How to improve calculation accuracy and expedite calculation speed still needs further study. Therefore, this project mainly aims at developing the fast algorithms with high-order accuracy for the nonlinear fractional partial differential equations. Firstly, with the help of the techniques of using the weighted average for the L1 approximation formula of superconvergence points and high order interpolation approximation, we improve the time fractional order derivative of discrete accuracy. Secondly, using the integral identities techniques improve the accuracy of spatial direction, the high accuracy scheme for the nonlinear time fractional partial differential equations is derived. Then the global superconvergence results are obtained through the post-processing techniques. Thirdly, we establish time and space high accuracy numerical algorithms for the nonlinear space-time fractional partial differential equations. The stability and convergence of the derived algorithms will also be studied. Moreover, some fast algorithms will be investigated by using the FFT, pre-processing for the obtained numerical schemes in order to achieve high-order accuracy methods.

近年来，发展分数阶偏微分方程的高效数值算法是计算数学领域的一个热点问题．其中有限差分法以其计算简单和易操作的优势被广泛研究，但它存在数值精度偏低、存储量大、计算耗时等缺点．如何进一步提高计算精度和加快计算速度仍需深入研究，因此本项目主要探索非线性分数阶偏微分方程的高效数值求解算法.首先，通过利用加权平均的技巧来获得L1逼近公式的超收敛点以及高次插值逼近等途径，提高时间分数阶导数的离散精度.其次，利用积分恒等式技巧和平均值技巧提高非线性空间方向的逼近精度，得到非线性时间分数阶偏微分方程的高精度数值格式，然后利用后处理技巧得到整体的超收敛结果.再次，建立非线性时间—空间分数阶偏微分方程的时空高精度数值求解算法，并探讨算法的稳定性和收敛性．最后，通过FFT、预处理等技巧，对建立的高精度数值格式，进行加速计算，从而得到高精度快速算法．

近年来，发展分数阶偏微分方程的高效数值算法是计算数学领域的一个热点问题。其中有限差分法以其计算简单和易操作的优势被广泛研究，但它存在数值精度偏低、存储量大、计算耗时等缺点。如何进一步提高计算精度和加快计算速度仍需深入研究，因此本项目主要探索非线性分数阶偏微分方程的高效数值求解算法。首先对时间分数阶慢扩散方程和扩散波方程构造高精度的有限元快速算法。利用积分恒等式技巧，证明了格式的无条件稳定性和超收敛性。为了降低算法的存储量和计算时间，我们采用SOE快算法逼近L1公式，构造快速的有限元数值格式，并给出误差分析。最后通过数值算例验证了算法的精度，并且数值例子说明SOE快速算法在减少计算时间和降低存储量上是有效的。对非线性时间分数阶次扩散方程构造了几个线性化的有限元数值格式，利用L1型离散的Gronwall不等式，严格地证明了格式的无条件收敛性。采用SOE快速算法逼近L1公式，构造快速的有限元数值格式，并给出误差分析。通过数值算例验证了算法的精度和有效性。