Stability, convergence, and geometric properties of second-order-in-time space-time discretizations for linear and semilinear wave equations

Abstract

We revisit second-order-in-time space-time discretizations of the linear and semilinear wave equations by establishing precise equivalences with first-order-in-time formulations. Focusing on schemes using continuous piecewise-polynomial trial functions in time, we analyze their stability, convergence, and geometric properties. We consider first a weak space-time formulation with test functions projected onto discontinuous polynomials of one degree lower in time, showing that it is equivalent to the scheme proposed in [French, Peterson 1996] in the linear case, and extended in [Karakashian, Makridakis 2005] to the semilinear case. In particular, this equivalence shows that this method conserves energy at mesh nodes but is not symplectic. We then introduce two symplectic variants, obtained through Gauss-Legendre and Gauss-Lobatto quadratures in time, and show that they correspond to specific Runge-Kutta time integrators. These connections clarify the geometric structure of the space-time methods considered.