A solution to the time-dependent two-dimensional Navier-Stokes equation in a rectangular domain using the Adomian decomposition method and theory of Gröbner Basis

In the present work we propose a modified decomposition method to derive approximate solutions for non-linear problems. Depending on the type of non-linearity, the source terms of the differential equations to be solved in each recursion step may result in extensive expressions, impractical for computational implementations and applications. This shortcomings are circumvented by the present methodology, which contemplates as a solution procedure in each recursion step a combined variable separation method together with Duhamel's principle, where the non-linearity appears as inhomogeneity. The source terms of the equation in each step of recursion are interpolated by polynomials and, using the Gröbner basis of the set points, the polynomial of reduced degree is obtained so that the integration may be carried out easily. As an application we considered a simplified version of the Navier-Stokes equation, which was used to simulate the wind field making use of the micrometeorological data from the Copenhagen experiment. The derived solution was evaluated against these experimental data from the field experiments showed that the computed results are acceptable and thus the solution may be considered an acceptable one and may be used as a simulation device for these type of field experiments. For almost all experiments twenty eigenvalues and ten recursion steps were sufficient. As results the wind speed at certain positions was simulated and compared to the measured values. The results obtained allow us to affirm that the presented methodology works satisfactorily and, therefore, can be considered a promising tool for solving non-linear problems, which are not tractable with the conventional decomposition method.