Hp-Version Continuous Petrov–Galerkin Time-Stepping for the Diffusive-Viscous Wave Equation: Unconditional Stability and Optimal a Priori Error Estimates

In this paper, we develop and analyze an unconditionally stable fully discrete method for the diffusive-viscous wave equation, combining an hp-version continuous Petrov–Galerkin time discretization with an hp-conforming finite element method in space. We first reformulate the problem as a first-order system to facilitate the time-stepping construction; the combination of the hp-version temporal and spatial discretizations then yields simultaneous high-order accuracy in both time and space. Unconditional stability is established via an energy argument that employs a special linear weight function, and the discrete energy is proved to be monotonically non-increasing at the time nodes. A rigorous hp-version a priori error analysis yields optimal convergence rates in the L2(L2) and L2(H1) norms, with constants explicitly independent of the spatio-temporal discretization parameters. Extensive numerical experiments confirm the theoretical results, demonstrating optimal algebraic convergence under h-refinement and exponential convergence under p-refinement. Further numerical examples validate the method’s long-time stability, energy dissipation properties, and its ability to handle both homogeneous and heterogeneous media with discontinuous coefficients.