Comparison of the finite difference and Galerkin methods as applied to the solution of the hydrodynamic equations

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Abstract

The three-dimensional nonlinear hydrodynamic equations which describe wind induced flow in a homogeneous sea are transformed from Cartesian coordinates into sigma coordinates. The solution of these equations in the horizontal is accomplished using a standard finite difference grid and established finite difference methods. The accuracy and computational efficiency, in terms of both computer time and main memory requirements, of using either the Galerkin method or a finite difference grid through the vertical is considered. Calculations, using the same number of functions in the Galerkin method as grid bases through the vertical shows that the Galerkin method has superior accuracy over the grid box method. Hence, for a given accuracy a smaller number of functions than grid boxes may be used, with associated saving in computational resources. For the case in which the vertical variation of eddy viscosity is fixed, an eigenvalue problem can be solved to yield a set of eigenfunctions. Using these eigenfunctions as a basis set with the Galerkin approach, a Galerkin-eigenfunction method is developed. Calculations show that the Galerkin-eigenfunction technique is accurate and in a linear model is clearly computationally more economic than the use of grid boxes through the vertical. © 1983.

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Davies, A. M., & Stephens, C. V. (1983). Comparison of the finite difference and Galerkin methods as applied to the solution of the hydrodynamic equations. Applied Mathematical Modelling, 7(4), 226–240. https://doi.org/10.1016/0307-904X(83)90076-8

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