The conditioning and spectral properties of the mass and stiffness matrices for acoustic wave problems are here investigated when isogeometric analysis (IGA) collocation methods in space and Newmark methods in time are employed. Theoretical estimates and extensive numerical results are reported for the eigenvalues and condition numbers of the acoustic mass and stiffness matrices in the reference square domain with Dirichlet, Neumann, and absorbing boundary conditions. This study focuses in particular on the spectral dependence on the polynomial degree p, mesh size h, regularity k, of the IGA discretization and on the time step size \(\Delta t\) and parameter \(\beta \) of the Newmark method. Results on the sparsity of the matrices and the eigenvalue distribution with respect to the number of degrees of freedom d.o.f. and the number of nonzero entries nz are also reported. The results show that the spectral properties of the IGA collocation matrices are comparable with the available spectral estimates for IGA Galerkin matrices associated with the Poisson problem with Dirichlet boundary conditions, and in some cases, the IGA collocation results are better than the corresponding IGA Galerkin estimates, in particular for increasing p and maximal regularity \(k=p-1\).

这里研究了当采用空间中的等几何分析（IGA）配置方法和时间上的纽马克方法时，声波问题的质量和刚度矩阵的调节和谱特性。报告了具有狄利克雷、诺依曼和吸收边界条件的参考平方域中声质量和刚度矩阵的特征值和条件数的理论估计和广泛的数值结果。这项研究特别关注 IGA 离散化的多项式次数p、网格大小h、正则性k的谱依赖性以及Newmark 的时间步长\(\Delta t\)和参数\(\beta \)方法。还报告了矩阵稀疏性的结果以及关于自由度dof数量和非零条目nz数量的特征值分布。结果表明，IGA 配置矩阵的谱特性与与 Dirichlet 边界条件下的泊松问题相关的 IGA Galerkin 矩阵的可用谱估计相当，并且在某些情况下，IGA 配置结果优于相应的 IGA Galerkin 估计，特别是增加p和最大正则性\(k=p-1\)。