SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 265-288, January 2021.
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.

中文翻译：

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粘性旋转浅水方程的二阶能量衰减方法的收敛性

SIAM数值分析学报，第59卷，第1期，第265-288页，2021年1月。
针对平面上粘性旋转浅水方程，构造了隐式能量衰减的改进的Crank-Nicolson时间步长方法。利用Schaefer的不动点定理和离散双曲抛物系统的$ H ^ 2 $估计，证明了半离散解的存在性，唯一性和收敛性。对于实际计算，半离散方法在空间中进一步离散化，从而导致完全离散的能量衰减有限元方案。提出了求解非线性代数系统的定点迭代方法。数值结果表明，所提出的方法只需要进行几次迭代就可以达到所需的精度，并具有二阶时间收敛性，并很好地保留了能量衰减。