一些非线性方程的粒子方法

发布时间：2017-12-28 07:01

本文关键词：一些非线性方程的粒子方法　出处：《电子科技大学》2015年硕士论文　论文类型：学位论文

【摘要】：近些年来,粒子法对于求取一些非线性偏微分方程的近似解是一种很有效的方法,并且在理论与实际的应用中都得到了比较好的发展。所谓粒子法,就是将方程的解表示为一些粒子点的位置函数()ix t与权值函数()ip t的乘积的和的表示方法。那么,方程就可以描述带有位置与权值函数的粒子点随着时间变化的动力学原理。由于粒子法的这种拉格朗日表示的本质,我们可以使用相对很少的粒子点来表示方程的小范围的解。在这篇文章里,我们主要针对一些非线性发展方程的粒子法建立最优误差分析,其中包括Camassa-Holm和Degasperis-Procesi方程,以及二维Euler-Poincare方程等。具体的做法就是利用给定方程在Largrangre坐标下的表示X(ξ,t),p(ξ,t),通过适当的变换来分别代表粒子点的位置函数与权值函数。再通过涉及核函数1||()2 2()expxG xαα-=的数值积分计算的方法,对这类方程近似求取粒子解;由于求取的粒子解可能出现不光滑,不稳定或是粒子点跳跃等现象,所以,采取引入具体柔化算子ρ(x)??的方法对核函数进行磨光,进而提高粒子解的精确性。伴随着理论的分析得出,我们的粒子法对区间步长h是可以达到二阶收敛的,对柔化指标??是可以达到一阶收敛的,即表示为2O(+h??)。最后,我们通过将粒子法应用到具体的C-H方程,D-P方程以及E-P方程的求解中,对方程磨光前后的粒子解与准确解进行比较,求得误差的1l-范数,再通过对计算结果进行系统地分析来验证我们的方法。进而依次说明,我们的粒子法的收敛阶。
[Abstract]:In recent years, particle method is a very effective method for finding approximate solutions of some nonlinear partial differential equations, and has been well developed in theoretical and practical applications. The so-called particle method is a representation of the sum of the solution of the equation as the sum of the product of the position function of some particle points () IX T and the weight function () IP t. Then, the equation can describe the dynamic principle of the particle point with position and weight function with time change. Because of the essence of this Lagrange representation of particle method, we can use relatively few particle points to represent the small range solution of the equation. In this article, we mainly establish the optimal error analysis for some nonlinear evolution equations by particle method, including the Camassa-Holm and Degasperis-Procesi equations, and the two-dimensional Euler-Poincare equation. X said the specific approach is to use the given equation in Largrangre coordinates (E, t), P (E, t), the position function and weight function through the appropriate transform to represent the particle. The kernel function of 1|| (22) to () method for numerical calculation of integral expxG x alpha alpha = the approximation of this equation is obtained to calculate the particle solution; because particle solutions may be not smooth, unstable or particle jumps and other phenomena, so take the introduction of specific soft count Zi (x)?? method of smoothing kernel function, and improve the accuracy of the particle solution. With theoretical analysis, particle method of interval length is h we can achieve two order convergence, to soften the index?? is first order convergence, which is expressed as 2O (+h??). Finally, we will through the particle method is applied to the C-H equation the D-P equation and the solution of E-P equation, comparing before and after polishing particle equation solution and the exact solution, 1l- norm error, then the calculation results were systematically analyzed to validate our method. Then the convergence order of our particle method is explained in turn.
【学位授予单位】：电子科技大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O241.82

【共引文献】