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By John Ya-Otto, Ole Gjerstad, Michael Mercer

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See Fig. 8. On a rectangular grid, there are multiple possibilities for the spatial arrangement of the dependent variables, as reviewed, for example, in [4, 6]. In the following Physical and Computational Issues in Ocean Modeling 21 Fig. 8 A tripolar rectangular grid. In the northern hemisphere, the grid has two singular points, and these are located outside the fluid domain. (Figure provided by Philip W. Jones, Los Alamos National Laboratory) descriptions, regard grid rectangles as mass cells.

17) it can be seen that at any point x p ∈ [x0 , x1 ], ∂x q(x ˜ p , ·, ·) = ˜ q(x Q˜ (x p ) is an eigenvector r0 (q(x ˜ p )) of A( ˜ p )) associated with a zero eigenvalue, λ0 (q(x ˜ p )) = 0. 18) where α (x) is a scalar proportionality constant. The solution varies smoothly along the curve in state space as x varies. The relevant discontinuous steady-state solution with a single jump in topography is therefore a jump discontinuity between two states lying on the same integral curve of r0 (q), ˜ which connects q(x ˜ 0 ) and q(x ˜ 1 ).

A. U. W. L. Schnoor, Environ. Sci. Technol. 37, 5314 (2003) 2. J. Stoker, Water Waves: The Mathematical Theory with Applications (Interscience Publishers, New York, 1957) 3. G. Whitham, Linear and Nonlinear Waves (John Wiley and Sons, New York, 1974) 4. T. J. Westerink, Physics Today pp. 33–38 (2008) 5. T. Mandli, Finite volume methods for the multilayer shallow water equations with applications to storm surges. D. thesis, University of Washington, Seattle, WA (2011) 6. R. Abgrall, S. Karni, SIAM Journal on Scientific Computing 31(3), 1603 (2009) 7.

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