### 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 |
---|---|

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 |

### Fingerprint

### 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.

**An algorithm for creation of an optimized adaptive grid for improved explicit finite difference scheme.** / Jovanovic, Raka; Tuba, Milan; Simian, Dana.

Research output: Contribution to journal › Article

*WSEAS Transactions on Mathematics*, vol. 7, no. 9, pp. 549-558.

}

TY - JOUR

T1 - An algorithm for creation of an optimized adaptive grid for improved explicit finite difference scheme

AU - Jovanovic, Raka

AU - Tuba, Milan

AU - Simian, Dana

PY - 2008/12/1

Y1 - 2008/12/1

N2 - 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.

AB - 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.

KW - Adaptive grid algorithm

KW - Finite differences

KW - Multi-resolution

KW - Numerical optimization

KW - Simulation

UR - http://www.scopus.com/inward/record.url?scp=58149381925&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=58149381925&partnerID=8YFLogxK

M3 - Article

VL - 7

SP - 549

EP - 558

JO - WSEAS Transactions on Mathematics

JF - WSEAS Transactions on Mathematics

SN - 1109-2769

IS - 9

ER -