本文關(guān)鍵詞：一些非線性方程的粒子方法　出處：《電子科技大學》2015年碩士論文　論文類型：學位論文

  更多相關(guān)文章： 粒子法 Camassa-Holm方程 Euler-Poincare方程 雙哈密頓性

【摘要】：近些年來,粒子法對于求取一些非線性偏微分方程的近似解是一種很有效的方法,并且在理論與實際的應用中都得到了比較好的發(fā)展。所謂粒子法,就是將方程的解表示為一些粒子點的位置函數(shù)()ix t與權(quán)值函數(shù)()ip t的乘積的和的表示方法。那么,方程就可以描述帶有位置與權(quán)值函數(shù)的粒子點隨著時間變化的動力學原理。由于粒子法的這種拉格朗日表示的本質(zhì),我們可以使用相對很少的粒子點來表示方程的小范圍的解。在這篇文章里,我們主要針對一些非線性發(fā)展方程的粒子法建立最優(yōu)誤差分析,其中包括Camassa-Holm和Degasperis-Procesi方程,以及二維Euler-Poincare方程等。具體的做法就是利用給定方程在Largrangre坐標下的表示X(ξ,t),p(ξ,t),通過適當?shù)淖儞Q來分別代表粒子點的位置函數(shù)與權(quán)值函數(shù)。再通過涉及核函數(shù)1||()2 2()expxG xαα-=的數(shù)值積分計算的方法,對這類方程近似求取粒子解;由于求取的粒子解可能出現(xiàn)不光滑,不穩(wěn)定或是粒子點跳躍等現(xiàn)象,所以,采取引入具體柔化算子ρ(x)??的方法對核函數(shù)進行磨光,進而提高粒子解的精確性。伴隨著理論的分析得出,我們的粒子法對區(qū)間步長h是可以達到二階收斂的,對柔化指標??是可以達到一階收斂的,即表示為2O(+h??)。最后,我們通過將粒子法應用到具體的C-H方程,D-P方程以及E-P方程的求解中,對方程磨光前后的粒子解與準確解進行比較,求得誤差的1l-范數(shù),再通過對計算結(jié)果進行系統(tǒng)地分析來驗證我們的方法。進而依次說明,我們的粒子法的收斂階。
[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.
【學位授予單位】：電子科技大學
【學位級別】：碩士
【學位授予年份】：2015
【分類號】：O241.82

【共引文獻】

相關(guān)碩士學位論文 前1條

1 王麗娜;基于隨機游動模型的北部灣水體動力特征解析[D];廈門大學;2014年

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本文編號：1345020