We study space--time isogeometric discretizations of the linear acoustic wave equation that use B-splines of arbitrary degree $p$, both in space and time. We propose a space--time variational formulation that is obtained by adding a non-consistent penalty term of order $2p+2$ to the bilinear form coming from integration by parts. This formulation, when discretized with tensor-product spline spaces with maximal regularity in time, is unconditionally stable: the mesh size in time is not constrained by the mesh size in space. We give extensive numerical evidence for the good stability, approximation, dissipation and dispersion properties of the stabilized isogeometric formulation, comparing against stabilized finite element schemes, for a range of wave propagation problems with constant and variable wave speed.

翻译：我们研究线性声波方程的时空等几何离散化，该方法在空间和时间上均使用任意次数$p$的B样条。我们提出一种时空变分公式，通过在分部积分得到的双线性形式中添加一个$2p+2$阶的非协调惩罚项而获得。当使用时间上具有最大正则性的张量积样条空间进行离散化时，该公式是无条件稳定的：时间网格尺寸不受空间网格尺寸的约束。我们通过大量数值实验证明了该稳定等几何公式在稳定性、逼近性、耗散和色散方面的良好性能，并将其与稳定化有限元方案进行比较，涵盖了多种恒定和变化波速的波动传播问题。