具Robin型阻尼边界波动方程的有限差分格式

发布时间：2018-04-20 21:34

本文选题：Robin型阻尼边界 + 波动方程　；参考：《山西大学》2017年硕士论文

【摘要】：波动方程的稳定化控制是分布参数控制理论的重要研究内容,其控制方程往往是带有反馈边界条件的波动方程初边值(IBV)问题.带有Robin型阻尼边界的波动方程IBV问题就是其中一类,对其数值算法的研究具有重要的理论意义与应用价值.首先,本文对如下Robin型阻尼边界条件一维波动方程初边值问题(?)构造了一个全离散的三层隐式有限差分格式,所构造的格式在每个时间层需要求解一个三对角线性方程组.通过离散能量方法证明所构造的差分格式在无穷范数意义下关于时间和空间方向都是二阶收敛的,并且关于初始条件和右端源项都是无条件稳定的.数值实验验证了理论结果.其次,通过引入中间变量把IBV问题(1)变成如下等价的弱耦合方程组(?)通过对(2)构造全离散隐式有限差分格式,得到IBV问题(1)的一个新的有限差分格式,这样避免了IBV问题(1)中的复杂的边界条件带来的困难.通过能量方法证明所构造的差分格式关于初始条件和右端项是无条件稳定的,且在L2范数意义下二阶收敛.数值实验验证了理论结果.最后,对带有Robin型阻尼边界IBV问题(1)构造了一个全离散的高阶紧致有限差分格式,通过数值实验验证了所构造的差分格式在无穷范数意义下关于空间方向是四阶收敛的,关于时间方向是二阶收敛的.
[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.
【学位授予单位】：山西大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8

【参考文献】