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：本文研究线性声波波动方程在空间和时间维度上均采用任意p次样条的时空等几何离散化。我们提出一种时空变分公式，通过在分部积分所得双线性形式中添加2p+2阶非一致罚项得到。当采用时间方向最大正则张量积样条空间进行离散化时，该公式具有无条件稳定性：时间网格尺寸不受空间网格尺寸约束。通过大量数值实验，我们证明了该稳定化等几何公式在常波速和变波速波动传播问题中良好的稳定性、逼近性、耗散性和色散特性，并与稳定化有限元方案进行了对比。