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

In this paper, 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 N. J. Walkington, Combined DG-CG time stepping for wave equations, SIAM J. Numer. Anal. 52 (2014) 1398–1417, with an exponential weight introduced in the time integrals. Consistency requires Formula: see text regularity in time and Formula: see text in space. 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 error analysis is developed in the case of tensor-product approximation spaces with approximation in time carried out using spline functions. In particular, we prove optimal convergence rates for Formula: see text-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. Numerical results are provided to support the theoretical findings and demonstrate the sharpness of the estimates.