CONVERGENCE OF A SECOND-ORDER ENERGY-DECAYING METHOD FOR THE VISCOUS ROTATING SHALLOW WATER EQUATION

摘要

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-2 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.