### Abstract

This paper deals with the two main shortcomings of explicit finite difference schemes: the use of a discretization grid with the same resolution over the entire problem space, and low level of precision and stability. We present a combination of two improvements. Their application is illustrated with the numerical simulation of the propagation of a light beam in a photonic lattice. The discretization problem is avoided by using a multi-resolution grid. An algorithm for the grid creation is developed and that algorithm is optimized for software implementation and parallelization. The efficiency of the algorithm is increased by further improving the precision of the explicit method by use of a multidimensional generalization of the Runge-Kutta scheme. Due to the multidimensionality and nonlinearity of the considered problem, our improved explicit finite difference gave better results than Crank-Nicholson scheme.

Original language | English |
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Pages (from-to) | 549-558 |

Number of pages | 10 |

Journal | WSEAS Transactions on Mathematics |

Volume | 7 |

Issue number | 9 |

Publication status | Published - 1 Dec 2008 |

Externally published | Yes |

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### Keywords

- Adaptive grid algorithm
- Finite differences
- Multi-resolution
- Numerical optimization
- Simulation

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*WSEAS Transactions on Mathematics*,

*7*(9), 549-558.