Finite-Difference Method of Solution of the Shallow Water Equations on an Unstructured Mesh
In the chapter we consider a linearized system of shallow water equations. Since this problem should be solved in domains being seas and oceans (or their parts), then solving this problem should use unstructured meshes to approximate domains under conside
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Finite-Difference Method of Solution of the Shallow Water Equations on an Unstructured Mesh G. M. Kobelkov and A. V. Drutsa
Abstract In the chapter we consider a linearized system of shallow water equations. Since this problem should be solved in domains being seas and oceans (or their parts), then solving this problem should use unstructured meshes to approximate domains under consideration properly. This problem was studied in the papers [1–4]. Here we consider finite-difference approximation of these equations, prove convergence of approximate solution to the differential one, and provide a number of numerical experiments confirming theoretical results. We also carried out some numerical experiments for real geographic objects.
8.1 Introduction In the chapter we consider a linearized system of shallow water equations. Since this problem should be solved in domains being seas and oceans (or their parts), then solving this problem should use unstructured meshes to approximate domains under consideration properly. This problem was studed in the papers [1–4]. Here we consider finite-difference approximation of these equations, prove convergence of approximate solution to the differential one, and provide a number of numerical experiments confirming theoretical results. We also carried out some numerical experiments for real geographic objects.
8.2 Formulation of the Problem Let us consider the system of shallow water equations in 2D Cartesian coordinates (see, e.g., [4–6]): (8.1) ut = g∇ζ − Ru − λk¯ × u + f, G. M. Kobelkov (B) · A. V. Drutsa Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, GSP-1, Leninskie Gory, Moscow, Russian Federation 119991 e-mail: [email protected] M. Z. Zgurovsky and V. A. Sadovnichiy (eds.), Continuous and Distributed Systems, Solid Mechanics and Its Applications 211, DOI: 10.1007/978-3-319-03146-0_8, © Springer International Publishing Switzerland 2014
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G. M. Kobelkov and A. V. Drutsa
ζt = divH u;
(8.2)
¯ here u = (u, v) is a velocity vector, ζ —height of a tidal wave, k—a unite vector in Oz direction, R, λ, g—some constants, H (x, y) is a depth—a function of coordinates. These equations are considered in a bounded domain Ω with the boundary Γ1 Γ2 . Boundary conditions are of the form of impermeability conditions or fixed wave height: (8.3) u · n ≡ un 1 + vn 2 = 0, on Γ1 ;
Initial conditions are
ζ = 0, on Γ1 ;
(8.4)
u(x, y, 0) = u0 (x, y), ζ (x, y, 0) = ζ0 (x, y),
(8.5)
where ζ0 (x, y) is some initial distribution of flow level while u0 (x, y) is some velocity vector field. Our aim is to approximate problem (8.1)–(8.5).
8.3 Mesh and Mesh Operators Triangulate Ω in such a way that Ω h (triangulation of the original domain) contains acute triangles only. A boundary of Ω h is denoted as Γ h = Γ1h Γ1h . A mesh is constructed in the following way: a cell center is a center of circumference described round a triangle, intersection point of a segment connecting centers of two neighboring cells and their common side is called flow node. Since
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