Intrinsic unconditional stability in space-time isogeometric approximation of the acoustic wave equation in second-order formulation

Abstract

We present a novel space-time isogeometric discretization of the acoustic wave equation in second-order formulation that is intrinsically unconditionally stable. The method relies on a variational framework inspired by [Walkington 2014], with an exponential weight introduced in the time integrals. Conformity requires at least $C^1$ regularity in time and $C^0$ in space. The approximation in time is carried out using spline functions. The unconditional stability of the space-time method for conforming discrete spaces arises naturally from the variational structure itself, rather than from any artificial stabilization mechanisms. The analysis of an associated ordinary differential equation problem in time yields error estimates with respect to the mesh size that are suboptimal by one order in standard Sobolev norms. However, for certain choices of approximation spaces, it achieves quasi-optimal estimates. In particular, we prove this for $C^1$-regular splines of even polynomial degree, and provide numerical evidence suggesting that the same behavior holds for splines with maximal regularity, irrespective of the degree. The error analysis is extended to the full space-time problem with tensor-product approximation spaces. Numerical results are provided to support the theoretical findings and demonstrate the sharpness of the estimates.