Numerical Simulation of Coastal Flows in Open Multiply-Connected Irregular Domains

We develop a numerical method for the simulation of coastal flows in multiply-connected domains with irregular boundaries that may contain both closed and open segments. The governing equations are the shallow-water model. Our method involves splitting of the original nonlinear operator by physical processes and by coordinates. Specially constructed finite-difference approximations provide second-order unconditionally stable schemes that conserve the mass and the total energy of the discrete inviscid unforced shallow-water system, while the potential enstrophy results to be bounded, oscillating in time within a narrow range. This allows numerical simulation of coastal flows adequate both from the mathematical and physical standpoints. Several numerical experiments, including those with complex boundaries, demonstrate the skilfulness of the method.

[1]  J. Oliger,et al.  Theoretical and practical aspects of some initial-boundary value problems in fluid dynamics , 1976 .

[2]  Y. Skiba,et al.  Conservative arbitrary order finite difference schemes for shallow-water flows , 2008 .

[3]  Y. Skiba,et al.  Simulation of soliton‐like waves generated by topography with conservative fully discrete shallow‐water arbitrary‐order schemes , 2009 .

[4]  Vladimir Zeitlin,et al.  Frontal geostrophic adjustment and nonlinear wave phenomena in one-dimensional rotating shallow water. Part 2. High-resolution numerical simulations , 2004, Journal of Fluid Mechanics.

[5]  R. Heikes,et al.  Numerical Integration of the Shallow-Water Equations on a Twisted Icosahedral Grid , 1995 .

[6]  Harry Yeh,et al.  Advanced numerical models for simulating tsunami waves and runup , 2008 .

[7]  R. LeVeque,et al.  HIGH-RESOLUTION FINITE VOLUME METHODS FOR THE SHALLOW WATER EQUATIONS WITH BATHYMETRY AND DRY STATES , 2008 .

[8]  Rick Salmon.  A shallow water model conserving energy and potential enstrophy in the presence of boundaries , 2009 .

[9]  A. Arakawa,et al.  A Potential Enstrophy and Energy Conserving Scheme for the Shallow Water Equations , 1981 .

[10]  Roger Temam,et al.  Low-Frequency Variability in Shallow-Water Models of the Wind-Driven Ocean Circulation. Part I: Steady-State Solution* , 2003 .

[11]  C. Vreugdenhil.  Numerical methods for shallow-water flow , 1994 .

[12]  Vladimir Zeitlin,et al.  Frontal geostrophic adjustment, slow manifold and nonlinear wave phenomena in one-dimensional rotating shallow water. Part 1. Theory , 2003, Journal of Fluid Mechanics.

[13]  D. Randall,et al.  A Potential Enstrophy and Energy Conserving Numerical Scheme for Solution of the Shallow-Water Equations on a Geodesic Grid , 2002 .

[14]  A. A. Samarskii,et al.  Completely conservative difference schemes , 1969 .

[15]  B. Cisewski,et al.  Mixing in cyclonic eddies in the Antarctic Circumpolar Current , 2009 .

[16]  R. Sadourny.  The Dynamics of Finite-Difference Models of the Shallow-Water Equations , 1975 .