Numerical recipes in C (2nd ed.): the art of scientific computing
Numerical recipes in C (2nd ed.): the art of scientific computing
Finite Elements in Analysis and Design
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A Massively Parallel Algorithm for Compact Finite Difference Schemes
ICPP '94 Proceedings of the 1994 International Conference on Parallel Processing - Volume 03
Compact finite difference method for American option pricing
Journal of Computational and Applied Mathematics
Numerical approach in quantum modelling for semiconductors
ICCOMP'09 Proceedings of the WSEAES 13th international conference on Computers
Two algorithms for numerical simulation of counter propagating matter waves in optical lattices
AMERICAN-MATH'10 Proceedings of the 2010 American conference on Applied mathematics
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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.