本文選題：平均向量場方法　切入點(diǎn)：保能量方法　出處：《海南大學(xué)》2017年碩士論文　論文類型：學(xué)位論文

【摘要】：在數(shù)學(xué)物理的研究中,許多偏微分方程可以表示成哈密爾頓系統(tǒng)的辛結(jié)構(gòu)形式或多辛結(jié)構(gòu)形式,如復(fù)修正KdV方程,KGS方程,耦合薛定諤Boussinesq方程,BBM方程等.這些偏微分方程具有能量守恒特性,在數(shù)值計(jì)算中,盡量構(gòu)造其保持能量守恒特性的數(shù)值算法在數(shù)值模擬微分方程的行為中具有重要的意義.哈密爾頓系統(tǒng)的辛幾何算法在1984年首次由馮康院士及其研究小組提出,在辛幾何算法的基礎(chǔ)之上Bridges和Reich等人在1997年構(gòu)造出了多辛算法.以上兩種算法具有長時(shí)間精確計(jì)算的優(yōu)點(diǎn),但不足之處在于這兩種算法只能近似保持方程的能量守恒.近年來,許多學(xué)者構(gòu)造了保持這些偏微分方程的保能量算法.在1999年保持哈密爾頓系統(tǒng)能量守恒的平均向量場方法被Quispel和McLachlan等人提出了,王雨順利用平均向量場方法構(gòu)造了多辛結(jié)構(gòu)偏微分方程的保能量方法.本文利用平均向量場方法和擬譜方法構(gòu)造耦合和復(fù)偏微分方程的高階保能量格式和多辛整體保能量格式,利用這些新格式對(duì)這些偏微分方程進(jìn)行數(shù)值模擬,并對(duì)數(shù)值結(jié)果進(jìn)行分析.在第一章,時(shí)間上利用四階平均向量場方法,空間上利用傅里葉擬譜方法對(duì)復(fù)修正KdV方程進(jìn)行離散,構(gòu)造了復(fù)修正KdV方程的高階保能量格式,利用構(gòu)造的高階保能量格式數(shù)值模擬孤立波的演化行為.數(shù)值結(jié)果表明復(fù)修正KdV方程的高階保能量格式可以很好地模擬孤立波的演化行為,并且可以精確地保持了方程的離散能量.在第二章,對(duì)于求解耦合偏微分方程,我們?cè)跁r(shí)間上利用四階平均向量場方法,空間上利用傅里葉擬譜方法對(duì)KGS方程和CSBE方程構(gòu)造高階保能量格式,并模擬孤立波的演化行為.數(shù)值結(jié)果表明KGS方程和CSBE方程的高階保能量格式可以很好地達(dá)到預(yù)期效果.在第三章,利用二階平均向量場方法,擬譜方法對(duì)具有多辛結(jié)構(gòu)的一維偏微分方程:BBM方程和復(fù)修正KdV方程構(gòu)造多辛整體保能量格式,利用構(gòu)造的多辛整體保能量格式數(shù)值模擬孤立波的演化行為,并證明了新格式能保方程離散的整體能量守恒特性.數(shù)值結(jié)果表明BBM方程和復(fù)修正KdV的多辛整體保能量格式可以很好地模擬孤立波的演化行為,并且可以精確地保持BBM方程和復(fù)修正KdV的離散整體能量守恒特性.在整體保能量守恒特性方面,BBM方程和復(fù)修正KdV方程新構(gòu)造的格式比已有的經(jīng)典的多辛格式更加精確,計(jì)算時(shí)間也比高階平均向量場方法大大縮短.在第四章,對(duì)具有多辛結(jié)構(gòu)的二維偏微分方程:ZK方程構(gòu)造多辛整體保能量格式,利用構(gòu)造的多辛整體保能量格式數(shù)值模擬孤立波的演化行為.數(shù)值結(jié)果表明整體保能量方法不僅可以長時(shí)間的模擬孤立波的演化行為,而且也可以精確地保持ZK方程的離散能量守恒.
[Abstract]:In the study of mathematical physics, many partial differential equations can be expressed as symplectic structure of Hamiltonian system or multi-symplectic structure, such as complex modified KdV equation and KGS equation. Coupled Schrodinger Boussinesq equation and so on. These partial differential equations have the characteristic of energy conservation. It is of great significance to construct a numerical algorithm for preserving the conservation of energy in numerical simulation of differential equations. In 1984, the symplectic geometry algorithm of Hamilton system was first proposed by academician Feng Kang and his research group. On the basis of symplectic geometry algorithm, Bridges and Reich constructed multi-symplectic algorithm in 1997. These two algorithms have the advantage of long time accurate calculation, but the disadvantage of these two algorithms is that they can only keep the energy conservation of equation approximately in recent years. In 1999, the mean vector field method for preserving the energy conservation of Hamiltonian systems was proposed by Quispel and McLachlan et al. Wang Yu-shun constructed the energy-preserving method for multi-symplectic partial differential equations using the mean vector field method. In this paper, the high-order energy-preserving schemes and the multi-symplectic global energy-preserving schemes for coupled and complex partial differential equations are constructed by means of the mean vector field method and the pseudo-spectral method. These partial differential equations are numerically simulated by these new schemes, and the numerical results are analyzed. In chapter 1, the fourth order average vector field method is used to discretize the complex modified KdV equation in space by Fourier pseudo-spectral method. A high order energy preserving scheme for complex modified KdV equations is constructed. The evolution behavior of solitary waves is numerically simulated by using the high-order energy-conserving scheme. The numerical results show that the high-order energy-conserving scheme of complex modified KdV equation can well simulate the evolution behavior of solitary waves. In chapter 2, we use the fourth-order average vector field method to solve coupled partial differential equations. In this paper, Fourier pseudospectral method is used to construct high-order energy-conserving schemes for KGS equation and CSBE equation. The numerical results show that the high order energy preserving schemes of KGS equation and CSBE equation can achieve the desired results. In chapter 3, the second order mean vector field method is used. The pseudospectral method is used to construct multi-symplectic global energy-conserving schemes for one-dimensional partial differential equations with multi-symplectic structure: BBM equation and complex modified KdV equation. The evolution behavior of solitary waves is numerically simulated by using the constructed multi-symplectic global energy-conserving schemes. It is proved that the new scheme can preserve the global energy conservation property of the discrete equation, and the numerical results show that the BBM equation and the multi-symplectic global energy preserving scheme of complex modified KdV can well simulate the evolution behavior of solitary waves. Moreover, the discrete global energy conservation properties of BBM equation and complex modified KdV can be accurately preserved. The new schemes of BBM equation and complex modified KdV equation are more accurate than the classical multi-symplectic schemes. The computational time is also much shorter than that of the high-order average vector field method. In Chapter 4th, the multi-symplectic global energy-preserving scheme is constructed for the 2-D partial differential equation with multi-symplectic structure: ZK equation. The evolution behavior of solitary waves is numerically simulated by using the multi-symplectic global energy preserving scheme. The numerical results show that the global energy conservation method can not only simulate the evolution behavior of solitary waves for a long time. Moreover, the discrete energy conservation of ZK equation can be kept accurately.
【學(xué)位授予單位】：海南大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O241.82

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1 蔣朝龍;高階保能量平均向量場方法的理論分析和應(yīng)用[D];海南大學(xué);2015年

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本文編號(hào)：1586249