American Institute of Mathematical Sciences

2004, 4(4): 1143-1172. doi: 10.3934/dcdsb.2004.4.1143

Convergence analysis of the numerical method for the primitive equations formulated in mean vorticity on a Cartesian grid

 1 Department of Mathematics, University of Tennessee, Knoxville, TN 37996, United States

Received  September 2002 Revised  February 2004 Published  August 2004

A second order numerical method for the primitive equations (PEs) of large-scale oceanic flow formulated in mean vorticity is proposed and analyzed, and the full convergence in $L^2$ is established. In the reformulation of the PEs, the prognostic equation for the horizontal velocity is replaced by evolutionary equations for the mean vorticity field and the vertical derivative of the horizontal velocity. The total velocity field (both horizontal and vertical) is statically determined by differential equations at each fixed horizontal point. The standard centered difference approximation is applied to the prognostic equations and the determination of numerical values for the total velocity field is implemented by FFT-based solvers. Stability of such solvers are established and the convergence analysis for the whole scheme is provided in detail.
Citation: Cheng Wang. Convergence analysis of the numerical method for the primitive equations formulated in mean vorticity on a Cartesian grid. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1143-1172. doi: 10.3934/dcdsb.2004.4.1143
 [1] Cheng Wang. The primitive equations formulated in mean vorticity. Conference Publications, 2003, 2003 (Special) : 880-887. doi: 10.3934/proc.2003.2003.880 [2] Juan Li, Wenqiang Li. Controlled reflected mean-field backward stochastic differential equations coupled with value function and related PDEs. Mathematical Control & Related Fields, 2015, 5 (3) : 501-516. doi: 10.3934/mcrf.2015.5.501 [3] Chuchu Chen, Jialin Hong. Mean-square convergence of numerical approximations for a class of backward stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2051-2067. doi: 10.3934/dcdsb.2013.18.2051 [4] Evelyn Buckwar, Girolama Notarangelo. A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1521-1531. doi: 10.3934/dcdsb.2013.18.1521 [5] Silvia Sastre-Gomez. Equivalent formulations for steady periodic water waves of fixed mean-depth with discontinuous vorticity. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2669-2680. doi: 10.3934/dcds.2017114 [6] Chang-Shou Lin. An expository survey on the recent development of mean field equations. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 387-410. doi: 10.3934/dcds.2007.19.387 [7] Pierre-Emmanuel Jabin. A review of the mean field limits for Vlasov equations. Kinetic & Related Models, 2014, 7 (4) : 661-711. doi: 10.3934/krm.2014.7.661 [8] Michael Herty, Lorenzo Pareschi, Sonja Steffensen. Mean--field control and Riccati equations. Networks & Heterogeneous Media, 2015, 10 (3) : 699-715. doi: 10.3934/nhm.2015.10.699 [9] Makram Hamouda, Chang-Yeol Jung, Roger Temam. Asymptotic analysis for the 3D primitive equations in a channel. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 401-422. doi: 10.3934/dcdss.2013.6.401 [10] Y. Goto, K. Ishii, T. Ogawa. Method of the distance function to the Bence-Merriman-Osher algorithm for motion by mean curvature. Communications on Pure & Applied Analysis, 2005, 4 (2) : 311-339. doi: 10.3934/cpaa.2005.4.311 [11] Jinju Xu. A new proof of gradient estimates for mean curvature equations with oblique boundary conditions. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1719-1742. doi: 10.3934/cpaa.2016010 [12] Gabriella Tarantello. Analytical, geometrical and topological aspects of a class of mean field equations on surfaces. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 931-973. doi: 10.3934/dcds.2010.28.931 [13] Chiun-Chuan Chen, Chang-Shou Lin. Mean field equations of Liouville type with singular data: Sharper estimates. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1237-1272. doi: 10.3934/dcds.2010.28.1237 [14] Yves Achdou, Mathieu Laurière. On the system of partial differential equations arising in mean field type control. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 3879-3900. doi: 10.3934/dcds.2015.35.3879 [15] Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Mean-field backward stochastic Volterra integral equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1929-1967. doi: 10.3934/dcdsb.2013.18.1929 [16] Hailong Zhu, Jifeng Chu, Weinian Zhang. Mean-square almost automorphic solutions for stochastic differential equations with hyperbolicity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1935-1953. doi: 10.3934/dcds.2018078 [17] Oleksandr Misiats, Nung Kwan Yip. Convergence of space-time discrete threshold dynamics to anisotropic motion by mean curvature. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6379-6411. doi: 10.3934/dcds.2016076 [18] Michael Hutchings. Mean action and the Calabi invariant. Journal of Modern Dynamics, 2016, 10: 511-539. doi: 10.3934/jmd.2016.10.511 [19] Franco Flandoli, Matti Leimbach. Mean field limit with proliferation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3029-3052. doi: 10.3934/dcdsb.2016086 [20] Hongyong Deng, Wei Wei. Existence and stability analysis for nonlinear optimal control problems with $1$-mean equicontinuous controls. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1409-1422. doi: 10.3934/jimo.2015.11.1409

2016 Impact Factor: 0.994