A second-order decoupled implicit/explicit method of the 3D primitive equations of ocean II: finite element spatial discretization

Abstract

Summary A fully discrete second-order decoupled implicit/explicit method is proposed for solving 3D primitive equations of ocean in the case of Dirichlet boundary conditions on the side, where a second-order decoupled implicit/explicit scheme is used for time discretization, and a finite element method based on the P1(P1) − P1−P1(P1) elements for velocity, pressure and density is used for spatial discretization of these primitive equations. Optimal H1−L2−H1 error estimates for numerical solution uhn,phn,θhn and an optimal L2 error estimate for uhn,θhn are established under the convergence condition of 0 < h≤β1,0 < τ≤β2, and τ≤β3h for some positive constants β1,β2, and β3. Furthermore, numerical computations show that the H1−L2−H1 convergence rate for numerical solution uhn,phn,θhn is of O(h + τ2) and an L2 convergence rate for uhn,θhn is O(h2+τ2) with the assumed convergence condition, where h is a mesh size and τ is a time step size. More practical calculations are performed as a further validation of the numerical method. Copyright © 2016 John Wiley & Sons, Ltd.