Energy Diagnostics and Long-Time Behavior of Crank–Nicolson Schemes for Shallow Water Flows with Bottom Friction

We investigate the discrete energy behavior and long-time stability of a second-order Crank–Nicolson mixed finite element discretization for the shallow water equations with nonlinear bottom friction. The method combines a compatible BDM1–DG0 spatial approximation with a skew-symmetric formulation of the advective terms and a midpoint treatment of dissipative source terms. At the fully discrete level, we derive a precise mechanical energy identity showing that the scheme is energy-consistent;the discrete energy satisfies a balance law consisting of a nonnegative frictional dissipation term and a higher-order midpoint defect of the order O(Δt3). Although the method is not unconditionally energy-dissipative, we prove that strict Lyapunov decay holds under a mild CFL-type restriction on the time step. Furthermore, we establish uniform long-time boundedness of the discrete energy and asymptotic recovery of the continuous dissipation law as Δt→0. We also analyze the interaction between nonlinear solver tolerances and energy diagnostics, showing that the observed positive energy increments are controlled, non-accumulating, and intrinsic to the midpoint quadrature structure rather than solver artifacts. The scheme is proven to be precisely well balanced for lake-at-rest equilibria, including nonlinear bottom friction. Comprehensive numerical experiments confirm second-order temporal accuracy, robustness under friction, asymptotic monotonicity under time step refinement, and strict equilibrium preservation. The results provide a rigorous energy-diagnostic framework clarifying when Crank–Nicolson schemes are physically reliable despite the absence of unconditional discrete dissipation.