An implicit method using contravariant velocity components and its application to the calculations in a harbour-channel area

The key problem in the computation of fluid dynamics using fine boundary-fitted grids is how to improve the numerical stability and decrease the calculating quantity.To solve this problem,implicit schemes should be adopted since explicit schemes may bring about a great increase in computation quantity according to the Courant-Friedrichs-Lewy condition.Whereas the adoption of implicit schemes is difficult to be realized because of the existence of two partial derivatives of surface elevations with respect to variables of alternative direction coordinates in each momentum equation in non-rectangular coordinates.With an aim to design an implicit scheme in non-rectangular coordinate,new momentum equations with the contravariant components of velocity vector are derived based on the shallow water dynamic equations in generalized curvilinear coordinates.In each equation the coefficients before the two derivatives of surface elevations have different orders ofmagnitude,i.e.,the derivative with the larger coefficient may play a more important role than that with the smaller one.With this advantage,the ADI scheme can then be easily employed to improve the numerical stability and decrease the calculating quantity.The calculation in a harbour and a channel in Macao nearshore area shows that the implicit model is effective in calculating current fields in small size areas.