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 the plane. Existence, uniqueness and convergence of semidiscrete solutions are proved by using Schaefer’s fixed point theorem and H estimates of the discretized hyperbolic–parabolic system. For practical computation, the semidiscrete method is further discretized in space, resulting in a fully discrete energy-decaying finite element scheme. A fixed-point iterative method is proposed for solving the nonlinear algebraic system. The numerical results show that the proposed method requires only a few iterations to achieve the desired accuracy, with second-order convergence in time, and preserves energy decay well.

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