This paper is concerned with the numerical approximation of initial-boundary-value problems of a three-parameter family of Bona-Smith systems, derived as a model for the propagation of surface waves under a physical Boussinesq regime. The work proposed here is focused on the corresponding problem with Dirichlet boundary conditions and its approximation in space with spectral methods based on Jacobi polynomials, which are defined from the orthogonality with respect to some weighted $L^{2}$ inner product. Well-posedness of the problem on the corresponding weighted Sobolev spaces is first analyzed and existence and uniqueness of solution, locally in time, are proved. Then the spectral Galerkin semidiscrete scheme and some detailed comments on its implementation are introduced. The existence of numerical solution and error estimates on those weighted Sobolev spaces are established. Finally, the choice of the time integrator to complete the full discretization takes care of different stability issues that may be relevant when approximating the semidiscrete system. Some numerical experiments illustrate the results.

翻译：本文研究三参数Bona-Smith族初边值问题的数值逼近，该模型源于物理Boussinesq机制下表面波传播的建模。本文工作聚焦于具有Dirichlet边界条件的对应问题，并采用基于Jacobi多项式的谱方法进行空间逼近——这些多项式通过关于某加权$L^{2}$内积的正交性定义。首先分析问题在相应加权Sobolev空间中的适定性，并证明解的局部时间存在唯一性。随后引入谱Galerkin半离散格式及其实现细节，建立数值解的存在性及加权Sobolev空间中的误差估计。最后，为完成全离散化所选择的时间积分器需处理逼近半离散系统时可能出现的不同稳定性问题。数值实验验证了所得结果。