两类耦合的非线性偏微分方程组的微分求积法

发布时间：2018-04-25 01:27

本文选题：耦合非线性方程组 + 改进的三次B-样条函数　；参考：《中国矿业大学》2017年硕士论文

【摘要】：本篇论文主要研究两类耦合的非线性偏微分方程组:广义Zakharov方程组和Klein-Gordon-Zakharov方程组的Dirichlet初边值问题的数值解法。在这里,我们采用高精度的微分求积法求解上述两类偏微分方程组,在微分求积法中应用改进的三次B-样条函数确定加权系数。于是偏微分方程组转化为常微分方程组系统,最终我们使用最优四阶,保时间步长三阶的强稳定性的龙格库塔法求解这个系统。绪论部分简单的介绍了广义Zakharov方程组和Klein-Gordon-Zakharov方程组的研究背景以及研究内容和一些预备知识。本文中我们将微分求积法和改进的三次B-样条函数结合起来应用在二维、三维的广义Zakharov方程组中求解该方程组的数值解。同时我们继续使用改进的三次B-样条函数微分求积法求解Klein-Gordon-Zakharov方程组。紧接着我们对这两个方程组进行了数值模拟,将本文提出的数值方法和已有的研究这两类方程组的有限差分法得到的数值解与精确解进行比较。从结果可以看出,与有限差分法相比,我们用改进的三次B-样条函数微分求积法得到的数值解更加地接近于精确解,即误差也相对的较小。同时我们也绘出了两个方程组的数值解与精确解的图形,从图形可直观的看出由我们的方法得到的数值解图形与精确解的图形吻合的很好。尤其对于Klein-Gordon-Zakharov方程组,我们也模拟出了单个波的传播过程以及在三维的情况下的数值解与精确解的图形。从数值实验的结果可以看出我们的方法的有效性,以及与差分法相比,可得出我们的方法的准确性。最后我们对本篇论文进行了总结。
[Abstract]:In this paper, we study two kinds of coupled nonlinear partial differential equations: generalized Zakharov equations and Dirichlet initial-boundary value problems for Klein-Gordon-Zakharov equations. Here, we use the high-precision differential quadrature method to solve the above two kinds of partial differential equations, and apply the improved cubic B-spline function to determine the weighting coefficient in the differential quadrature method. So the system of partial differential equations is transformed into a system of ordinary differential equations. Finally, we use the Runge-Kutta method, which is an optimal fourth-order and three-order preserving time step, to solve the system. The introduction briefly introduces the research background of generalized Zakharov equations and Klein-Gordon-Zakharov equations, as well as the research contents and some preliminary knowledge. In this paper, the differential quadrature method and the improved cubic B-spline function are combined to solve the numerical solutions of the equations in two-dimensional and three-dimensional generalized Zakharov equations. At the same time, we continue to use the improved cubic B-spline function differential quadrature method to solve the Klein-Gordon-Zakharov equations. Then we carry on the numerical simulation to these two equations, and compare the numerical solution with the exact solution obtained by the numerical method proposed in this paper and the finite difference method which has been used to study these two kinds of equations. It can be seen from the results that compared with the finite difference method, the numerical solution obtained by using the improved cubic B-spline differential quadrature method is closer to the exact solution, that is, the error is also relatively small. At the same time, we also draw the figure of the numerical solution and the exact solution of the two equations. From the figure, we can directly see that the figure of the numerical solution obtained by our method is in good agreement with the figure of the exact solution. Especially for Klein-Gordon-Zakharov equations, we also simulate the propagation process of single wave and the figure of numerical solution and exact solution in three dimensional case. The results of numerical experiments show the validity of our method and the accuracy of our method compared with the difference method. Finally, we summarize this paper.
【学位授予单位】：中国矿业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82

【参考文献】