Numerical Solution of the Shallow Water Equations Using Compact Delta Formulation

摘要

The objective of the present work is to develop a highly accurate code for numerical solution of the conservation-law form of the shallow water equations in the beta plane. The second-order delta formulation of the trapezoidal time differencing scheme is used. The Hirsch fourth-order compact finite-difference method is also applied to discretize the spatial factored form of the equations obtained using the ADI method. Because of the large aliasing error introduced by the fourth-order scheme a very selective low-pass filter is used to overcome the error generated by the interaction of the nonlinear terms of the equations. It was found that the best results can be obtained by a periodical application of the filter. The integral invariants of the shallow water equations, i.e. the total energy and enstrophy are well conserved during the numerical integration. This fact shows that the nonlinear structure of the equations is correctly modeled. The validation of the fourth-order compact results are investigated by comparing them with an accurate nonlinear ADI method performed by Gustafs-son. In addition, a comparison is made between the results of the fourth-order compact scheme and a second-order finite-difference method (developed by authors) for different grid resolutions to confirm the high accuracy of the compact scheme.