双曲型问题基于自然边界归化的人工边界条件

【作者】 艾焰；

【导师】 杜其奎；

【作者基本信息】 南京师范大学 ， 计算数学， 2006， 硕士

【摘要】 本文研究双曲型初边值问题基于自然边界归化的人工边界条件及其数值方法。 第一部分，研究无界外区域双曲型初边值问题基于自然边界归化的人工边界条件及其数值方法。利用Newmark方法对问题进行时间的离散化(半离散化)，在每个时间步求解一个椭圆外问题。引入一条人工边界B_R，通过自然边界归化获得人工边界B_R上精确的人工边界条件。借助人工边界条件给出半离散化问题的变分形式，讨论了所得的变分问题解的存在性与唯一性。详细分析了半离散化格式的稳定性，给出了格式的稳定性条件。用有限元方法求解有界区域上的问题，给出了全离散化问题的误差估计。对于不同内边界的无界区域问题，给出了一些数值例子。数值结果表明文中所提出的方法是十分有效的。 第二部分，研究一类具有凹角的无界外区域的双曲型初边值问题基于自然边界归化的人工边界条件及其数值方法。同样用Newmark方法对时间进行离散化，在每个时间步求解一个椭圆外问题；然后引入一条人工边界B_R，并通过自然边界归化获得B_R上精确的人工边界条件；给出半离散化问题的变分形式，证明了变分问题的适定性，并给出了误差估计；最后通过数值例子，说明该方法的可行性与有效性。更多还原

【Abstract】 In this paper, we investigate the artificial boundary conditions and its numerical methods based on natural boundary reduction for hyperbolic initial boundary value problems.Part I, the artificial boundary condition and its numerical methods based on natural boundary reduction for hyperbolic initial boundary value problems on unbounded domains is studied. The problem is first discretized in time by the Newmark time-marching scheme, leading to a time-stepping scheme, where an exterior elliptic problem has to be solved in each time step. An artificial boundary B_r is introduced, and an exact boundary condition by the natural boundary reduction is obtained. At each time step, the variational formulation of the semidiscrete problem is derived. Existence and uniqueness of the variational solution are established. Error estimates of the fully discrete problem is performed. Stability criteria of the semi-discrete scheme is also obtained. Numerical results are presented to demonstrate the performance of our method.Part II, the artificial boundary condition and its numerical methods based on natural boundary reduction for a hyperbolic initial boundary value problem in an infinite domain with a concave angle is investigated. The governing equation is first discreted in time by Newmark method, leading to a time-stepping scheme, where an exterior elliptic problem has to be solved in each time step. Then an artificial boundary Br is introduced, an exact artificial boundary condition is obtained. The variational problem of semi-discrete problem is given, the well-posedness of the variational problem is proved, and some error estimates are presented. Finally, some numerical examples are presented to demonstrate the performance of the method.更多还原