We consider the classical wave equation in a Friedrichs-type formulation involving a skew-symmetric spatial differential operator. The focus is on the analysis of a fully discrete setting, employing third- and fourth-order explicit Runge–Kutta schemes (ERK3 and ERK4) for time discretization, combined with Hybrid High-Order (HHO) and Weak Galerkin (WG) methods for spatial discretization. A first objective is to establish key properties that address the static coupling between cell and face unknowns, which is intrinsic to hybrid methods.

We investigate two distinct strategies for error analysis: one based on bounding the operator norm of Taylor polynomials applied to the discrete differential operator, and another that involves testing the error equations with appropriately chosen increments. The effectiveness of these strategies depends on the specific interpolation operators involved and the properties that make such analyses viable. Finally, we outline how the same framework, using various HHO variants, extends to the acoustic-elastic interface problem, where the coupled terms contribute no additional error.

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