【摘要】 提出了一种用广义函数δ序列求解偏微分方程的数值方法.首先对一阶B样条函数N1(x)进行卷积得到四阶B样条函数N4(x),用N4(x)的线性组合构造出三次样条插值基函数;然后用样条插值基序列逼近δ函数,利用δ函数的性质构造插值样条δ序列,该δ序列具有对称、Riesz基和插值性质.以非线性对流扩散方程(伯格方程)为例,用插值样条δ序列离散该方程的空间形式,用四阶龙格库塔方法描述发展过程,取得了较好的精度.为减少计算量,加快插值函数的收敛速度,进一步提高求解精度,对δ序列进行了改进,对同一算例进行数值实验,结果表明,改进后的算法求解过程稳定发展,能够有效描述局部快速变化的情况.更多还原

【Abstract】 A method for solving PDE numerically was studied by using delta-sequence. The fourth-order cardinal B-splines N₄（x） was obtained by convoluting the first order cardinal B-splines N₁（x）, and cubic spline interpolation cardinal function obtained by using linear combination of N₄（x）. This delta-sequence based on cubic spline interpolation cardinal function showed the properties of symmetry, Riesz basis and interpolation. Non linear convection diffusion equation （Burgers′ equation） was used as an example. Interpolation spline delta-sequance was used to discrete the spatial derivatives, while the fourth order rugne Kutta convergence rate of the interpolation function and improve computational accuracy, we modified this delta sequence, the same example was used in numerical application. The numerical results varied that the process of the numerically solution was stable. This method could describe the situation of fast variety in local effectively.更多还原