Rothe time discretization and weak solutions for a cutoff Westervelt system

Abstract

We study a fully implicit Rothe time discretization for a cutoff first-order formulation of the Westervelt equation. The key ingredients are the enthalpy variable and the primitive mobility variable, which turn each nonlinear time step into a uniformly monotone elliptic problem and avoid higher-order energy estimates and inverse inequalities. For every time step, the discrete problem reduces to a monotone elliptic equation, which yields well-posedness of the Rothe scheme by the Browder-Minty theorem. We derive a discrete energy inequality, establish compactness for the transformed variable by an Alt-Luckhaus type argument, and pass to the limit to obtain existence of weak solutions for the cutoff first-order system. We prove a weak-strong uniqueness principle for the cutoff problem and formulate a conditional a posteriori criterion under which the cutoff is inactive. For sufficiently regular solutions of the forced cutoff problem, we establish consistency estimates which identify the temporal residuals produced by the Rothe approximation in the natural discrete-in-time weak norms. These estimates provide a rigorous consistency basis for the observed temporal behavior. Finally, numerical experiments illustrate the stability of the scheme, its observed near first-order temporal behavior, and inactivity of the cutoff along the computed discrete trajectories in the tested regimes.