First-order decoupled method of the three-dimensional primitive equations of the ocean
Date
2016
Authors
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Journal ISSN
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Publisher
SIAM Scientific Computing
Abstract
This paper is concerned with a first-order fully discrete decoupled method for solving the three-dimensional (3D) primitive equations of the ocean with the Dirichlet boundary conditions on the side, where a decoupled semi-implicit scheme is used for the time discretization, and the $P_1(P_1)-P_1-P_1(P_1)$ finite element for velocity, pressure, and density is used for the spatial discretization of these equations. The $H^1-L^2-H^1$ optimal error estimates for the numerical solution $(u_h^n,p_h^n,\theta_h^n)$ and the $L^2$ optimal error estimate for $(u^n_h,\theta_h^n)$ are established under the restriction of $0<h\le \beta_1$ and $0<\tau\le \beta_2$ for some positive constants $\beta_1$ and $\beta_2$. Moreover, numerical investigations are provided to show that the first-order decoupled method is of almost unconditional convergence with accuracy $\mathcal{O}(h+\tau)$ in the $H^1$-norm and $\mathcal{O}(h^2+\tau)$ in the $L^2$-norm for solving the 3D primitive equations of the ocean. Numerical results are given to verify the theoretical analysis.