局部化MAPS法求解时空偏微分方程

Local Method of Approximate Particular Solution for Time-Space Partial Differential Equation

用基于径向基函数的局部近似特别解法求解时空偏微分方程,并与局部Kansa方法进行比较,通过在局部区域内构造低阶矩阵,并推广到全局形式,构建一个全局稀疏矩阵,成功摆脱了求解病态线性方程组的困境,大大提高了计算的效率。采用Matern与MQ径向基函数求解偏微分方程,Matern径向基函数避免了对形状参数c的选择。在时间层划分方面采用四阶龙格—库塔(Runge-Kutta)方法。最后对数值例子的误差进行了比较分析,验证了方法的有效性。

Several time-space partial differential equations were solved by the localized method of approximation particular solution （LMAPS）. The numerical results that we obtained were compared with local Kansa method by structuring low-level matrix within the local area to promote the global form and to build a global sparse matrix, through which we can avoid the ill troubles of solving linear equations successfully and improve the efficiency of computing greatly. We choose Matern and MQ radial basis functions for solving partial differential equations, and the Matern radial basis function can avoid the option of parameter c. The fourth-order Runge-Kutta method was adopted in terms of time discretization level. At last, we compared the error of numerical examples, which verified the validity of the method.