Adaptive simulation of two dimensional hyperbolic problems by Collocated Discrete Least Squares Meshless method

An adaptive refinement technique is presented in this paper and used in conjunction with the Collocated Discrete Least Squares Meshless (CDLSM) method for the effective simulation of two-dimensional shocked hyperbolic problems. The CDLSM method is based on minimizing the least squares functional calculated at collocation points chosen on the problem domain and its boundaries. The functional is defined as the weighted sum of the squared residuals of the differential equation and its boundary conditions. A Moving Least Squares (MLS) method is used here to construct the meshless shape functions. An error estimator based on the value of functional at nodal points used to discretize the problem domain and its boundaries is developed and used to predict the areas of poor solutions. A node moving strategy is then used to refine the predicted zones of poor solutions before the problem is resolved on the refined distribution of nodes. The proposed methodology is applied to some two dimensional hyperbolic benchmark problems and the results are presented and compared to the exact solutions. The results clearly show the capabilities of the proposed method for the effective and efficient solution of hyperbolic problems of shocked and high gradient solutions.