本文選題：Robin型阻尼邊界 + 波動方程��； 參考：《山西大學(xué)》2017年碩士論文

【摘要】：波動方程的穩(wěn)定化控制是分布參數(shù)控制理論的重要研究內(nèi)容,其控制方程往往是帶有反饋邊界條件的波動方程初邊值(IBV)問題.帶有Robin型阻尼邊界的波動方程IBV問題就是其中一類,對其數(shù)值算法的研究具有重要的理論意義與應(yīng)用價值.首先,本文對如下Robin型阻尼邊界條件一維波動方程初邊值問題(?)構(gòu)造了一個全離散的三層隱式有限差分格式,所構(gòu)造的格式在每個時間層需要求解一個三對角線性方程組.通過離散能量方法證明所構(gòu)造的差分格式在無窮范數(shù)意義下關(guān)于時間和空間方向都是二階收斂的,并且關(guān)于初始條件和右端源項(xiàng)都是無條件穩(wěn)定的.數(shù)值實(shí)驗(yàn)驗(yàn)證了理論結(jié)果.其次,通過引入中間變量把IBV問題(1)變成如下等價的弱耦合方程組(?)通過對(2)構(gòu)造全離散隱式有限差分格式,得到IBV問題(1)的一個新的有限差分格式,這樣避免了IBV問題(1)中的復(fù)雜的邊界條件帶來的困難.通過能量方法證明所構(gòu)造的差分格式關(guān)于初始條件和右端項(xiàng)是無條件穩(wěn)定的,且在L2范數(shù)意義下二階收斂.數(shù)值實(shí)驗(yàn)驗(yàn)證了理論結(jié)果.最后,對帶有Robin型阻尼邊界IBV問題(1)構(gòu)造了一個全離散的高階緊致有限差分格式,通過數(shù)值實(shí)驗(yàn)驗(yàn)證了所構(gòu)造的差分格式在無窮范數(shù)意義下關(guān)于空間方向是四階收斂的,關(guān)于時間方向是二階收斂的.
[Abstract]:Stabilization control of wave equation is an important research content of distributed parameter control theory. The governing equation of wave equation is usually the initial and boundary value of wave equation with feedback boundary condition (IBV) problem. The IBV problem of wave equation with Robin damping boundary is one of them. The study of its numerical algorithm has important theoretical significance and application value. First of all, this paper deals with the initial boundary value problem of one-dimensional wave equation with Robin damping boundary conditions. In this paper, a fully discrete three-layer implicit finite difference scheme is constructed. The scheme needs to solve a tridiagonal linear equation system at each time level. It is proved by the discrete energy method that the difference scheme is second-order convergent in the sense of infinite norm with respect to the direction of time and space, and that the initial conditions and the source term at the right end are unconditionally stable. The theoretical results are verified by numerical experiments. Secondly, by introducing intermediate variables, the IBV problem is transformed into the following equivalent weakly coupled equations. A new finite difference scheme for the IBV problem is obtained by constructing a fully discrete implicit finite difference scheme, which avoids the complex boundary conditions in the IBV problem. It is proved by the energy method that the difference scheme is unconditionally stable for the initial conditions and the right term, and the second order convergence is obtained in the sense of L 2 norm. The theoretical results are verified by numerical experiments. Finally, a fully discrete high order compact finite difference scheme is constructed for the IBV problem with Robin damping boundary. Numerical experiments show that the scheme is fourth-order convergent in the sense of infinite norm. The time direction is second order convergent.
【學(xué)位授予單位】：山西大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O241.8

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本文編號：1779563