We study space--time isogeometric discretizations of the linear acoustic wave equation that use 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：本文研究了线性声波方程的时空等几何离散化方法，该方法在空间和时间上均采用任意p阶样条函数。我们提出了一种时空变分格式，该格式通过在分部积分所得双线性形式中添加阶数为2p+2的非协调罚项而获得。当时域采用具有最大正则性的张量积样条空间进行离散时，该格式具有无条件稳定性：时间步长不受空间网格尺寸的限制。通过对比稳定化有限元格式，针对常波速与变波速情形下的多种波传播问题，我们提供了大量数值实验证据，证明了稳定化等几何格式在稳定性、逼近性、耗散性和色散性方面的优越特性。