Convergence of a Second-order Energy-decaying Method for the Viscous Rotating Shallow Water Equation

An implicit energy-decaying modified Crank--Nicolson time-stepping method is constructed for the viscous rotating shallow water equation on plane. Existence, uniqueness, and convergence of semidiscrete solutions are proved by using Schaefer's fixed point theorem $H^2$ estimates discretized hyperbolic--parabolic system. For practical computation, further in space, resulting a fully discrete finite element scheme. A fixed-point iterative proposed solving nonlinear algebraic The numerical results show that requires only few iterations to achieve desired accuracy, with second-order time, preserves energy decay well.