An Algorithm for Multi-Resolution Grid Creation Applied to Explicit Finite Difference Scheme

摘要

This paper deals with the main shortcoming of the finite difference schemes: the use of a discretization grid with the same resolution over the entire problem space. We propose to avoid this problem by using a multi-resolution grid. The algorithm for the grid creation is presented, that is correct for numeric calculations and optimized for the use in program application. The algorithm is illustrated with the numerical simulation of the propagation of a light beam in a photonic lattice. It is implemented by an explicit finite difference method. An explicit method is adopted, due to the multidimensionality of the problem and the presence of nonlinearity. The efficiency of the algorithm is increased by further improving the precision of the explicit method by the use of a multidimensional generalization of the Runge-Kutta scheme.