Numerical Solution of the Experimental Model: Traditional Approach Versus Optimization

Abstract

This paper proposes a near-Newton approach, using uni-dimensional Rossby-Obukhov and Korteweg-deVries (KdV) equations as models for the solution of the partial differential equations (PDE) -controlled problem. One extremely important aspect of the process is that the notation of differential equations does not restrict the stability of the step size. The traditional approach to solve regional models, the boundary conditions are obtained by interpolating the solution provided by the global model on the coarse mesh. Essentially, the global points within the regional domain are discarded. With the use of optimization tools we can reformulate the problem in order to take advantage of all available data. The intention of this work is to formulate an optimization problem that seeks to solve the model in such a way that the solution is as close as possible to the data contained in the domain, which can be interpreted as the interpolation of the data by solving the model equations. Currently, there is no theory to explain the stable behavior of the non-evolutionary problem. However, we can observe that the global data interpolated in the problem (weather forecast) act as a regularization, demanding that the solution does not deviate too far from the global data and does not present oscillatory behavior. In this paper it was present two comparative experiments using the experimental model to validate the above condition. The first experiment was done with the Rossby-Obukhov equation, and the second with the KdV equation. Just as each experiment comprises several numerical tests, we show only the most expressive results and comment on the behavior of others. It was illustrated that, the phenomenon in different linear problems (the heat equation and the Rossby-Obukhov equation) and nonlinear ones (the KdV equation), possible to obtain the solution of the problem with the speed compatible with a single direct evolution using usual PDE methods, without losing precision in the solution.