Abstract:

The deficiencies of the traditional collocation method, such as the large number of meshes and collocation points, may cause the nonlinear equations to be computationally expensive and result in huge discrete errors. Due to these drawbacks of the traditional collocation method, an adaptive mesh refinement and collocation optimization method is proposed. Firstly, the optimization problem is transformed into the nonlinear programming problem in different meshes. After implementing the Lagrange interpolation method to solve the optimization problem in these meshes, the absolute and relative errors are then obtained. Secondly, according to the magnitude of the trajectory curvature, the positions of non-smooth meshes are determined. Afterwards, the number of meshes and points are further determined by the relationship between the maximum relative error and the allowable error. Meanwhile, the number of collocation points is increased to enhance the solution accuracy in the case where the smooth meshes cannot satisfy the tolerance error. Finally, several numerical simulations are conducted to compare the proposed method with some other currently existing methods. The simulation results show that the collocation points and the number of meshes of the proposed method are less than those of the traditional adaptive mesh refinement methods. Moreover, the efficiency of the proposed method is higher than that of the traditional methods.