| 钟霖,马淑芳,莱蒙.无网格法求解一类分段连续型延迟偏微分方程[J].计算力学学报,2021,38(2):160~165 |
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| 无网格法求解一类分段连续型延迟偏微分方程 |
| Meshless method for delay partial differential equations with piecewise continuous arguments |
| 投稿时间：2019-12-17 修订日期：2020-06-13 |
| DOI：10.7511/jslx20191217001 |
| 中文关键词: 延迟偏微分方程 无网格插值法 θ-加权有限差分法 MQ径向基函数 稳定性 |
| 英文关键词:delay partial differential equation meshless interpolation method θ-weighted finite difference method multiquadric (MQ) radial basis function stability |
| 基金项目:中央高校基本科研业务费专项资金（2572018BC19）资助项目. |
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| 中文摘要: |
| 无网格法是基于散点信息求解偏微分方程问题的数值方法，无网格法可减少或完全消除对网格的依赖，数值实施更加灵活。因此，考虑采用基于径向基函数的无网格插值法求解一类分段连续型延迟偏微分方程。首先，利用θ-加权有限差分法得到方程时间上的离散格式，利用基于径向基函数的无网格插值法近似空间导数，得到了全离散数值格式。采用的基函数是Multiquadric（MQ）径向基函数，MQ径向基函数在精度及稳定性等方面都优于其他径向基函数。其次，采用傅里叶分析方法对该方法进行稳定性分析，得到了该方法稳定的条件，且该条件只与时间步长有关。最后，通过数值算例验证了方法的收敛性和稳定性，从而说明了方法的有效性和适用性。 |
| 英文摘要: |
| A meshless method is a numerical method for solving partial differential equations based on scatter information.It can reduce or completely eliminate the dependence on the grid of the FEM,and the numerical implementation is more flexible.Therefore,the meshless interpolation method based on radial basis function is considered to solve a class of delay partial differential equations with piecewise continuous arguments.Firstly,the θ-weighted finite difference method is used to obtain the time-discrete scheme of the equation,then the spatial derivative is approximated by meshless interpolation method based on radial basis function,and the fully discrete numerical scheme is obtained.The basis function adopted is MQ radial basis function,which is superior to other radial basis functions in precision and stability.Secondly,the stability of the method is analyzed by using the Fourier analysis method,and the stability conditions of the method are obtained,which are only related to the time step.Finally,the convergence and stability of the method are verified by numerical examples,which show the effectiveness and applicability of the method. |
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