在修改的时空稀疏网格上的一个高效的数值方法,用于使用非光滑数据的时间微分扩散方程.
Bi-Yun Zhu1, Ai-Guo Xiao1, Xue-Yang Li1
1School of Mathematics and Computational Science & National Center for Applied Mathematics in Hunan & Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, Hunan 411105 China.
Numerical algorithms
|June 26, 2023
概括
本研究介绍了一种修改的时空稀疏网格 (STSG) 方法,以高效地解决d维时间分数扩散方程 (TFDE). 改进的算法提高了合率,特别是在初始时间附近的规律性较低的解决方案中.
科学领域:
- 数字分析 数字分析
- 部分微分方程部分微分方程.
- 计算数学是指计算数学.
背景情况:
- 时间分数扩散方程 (TFDE) 在建模异常扩散过程中至关重要.
- 由于TFDE解决方案的规律性较低,通常是由于初始条件不平滑,这显著阻碍了数值方法的融合.
- 现有的数值方法在处理TFDE解决方案固有的低规律性时,难以保持准确性和效率.
研究的目的:
- 开发一种高效的算法来解决d维时间分数扩散方程 (TFDE).
- 提高TFDE的数值方法的融合率,特别是解决低规律性解决方案所带来的挑战.
- 引入和分析为TFDE量身定制的修改时空稀疏网格 (STSG) 方法.
主要方法:
- 开发一个时空稀疏网格 (STSG) 方法,使用正弦基础进行空间离谱化和线性元素基础进行时间离谱化.
- 通过空间多层次和时间层次基础的张量积来构建STSG.
- 将全网格方法集成到STSG中,以创建修改后的STSG,以快速变化的初始解决方案解决准确性问题.
主要成果:
- 在某些条件下,标准的STSG方法可以在减少的自由度 (DOF) 中实现特定的精度顺序.
- 修改后的STSG方法显示了更高的准确性和稳定性,特别是对于TFDE,其解决方案在初始时刻快速变化.
- 比较的数值实验验证了修改后的STSG方法在标准方法上的显著优势.
结论:
- 修改后的STSG方法为解决d维TFDE提供了可靠和高效的数值方案.
- 这种方法有效地克服了标准STSG方法在处理低规律性的解决方案时的局限性.
- 开发的算法为准确模拟异常扩散现象提供了一个有前途的工具.
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