具有滑移壁边界条件的规则浅水波系统：理论与数值分析
A regularized shallow-water waves system with slip-wall boundary conditions in a basin: Theory and numerical analysis
摘要：
在有界区域中模拟长的非线性色散波通常需要使用滑移壁边界条件。文献中出现的Boussinesq系统在施加这样的边界条件时通常不是适定的，或者如果它们是适定的，则在数值逼近中实现边界条件是非常麻烦的。本文提出了一种新的Boussinesq系统，用于研究需要滑动壁边界条件时盆地中的小振幅长波。新的系统推导出使用渐近技术的假设下，小的水深变化，和一个数学证明的适定性的新系统的开发。新的系统也解决了数值使用伽辽金有限元法，其中的边界条件施加的帮助下，Nitsche的方法。分析了该数值方法的收敛性，并给出了精确的误差估计。然后实现该方法，并通过数值实验验证了收敛性。本文还对平面斜坡上的孤波浅滩化进行了数值模拟。将计算结果与实验数据进行了比较，结果吻合良好。
英文摘要：
The simulation of long, nonlinear dispersive waves in bounded domains usually requires the use of slip-wall boundary conditions. Boussinesq systems appearing in the literature are generally not well-posed when such boundary conditions are imposed, or if they are well-posed it is very cumbersome to implement the boundary conditions in numerical approximations. In the present paper a new Boussinesq system is proposed for the study of long waves of small amplitude in a basin when slip-wall boundary conditions are required. The new system is derived using asymptotic techniques under the assumption of small bathymetric variations, and a mathematical proof of well-posedness for the new system is developed. The new system is also solved numerically using a Galerkin finite-element method, where the boundary conditions are imposed with the help of Nitsche's method. Convergence of the numerical method is analyzed, and precise error estimates are provided. The method is then implemented, and the convergence is verified using numerical experiments. Numerical simulations for solitary waves shoaling on a plane slope are also presented. The results are compared to experimental data, and excellent agreement is found.
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