# UNCONDITIONALLY POSITIVE FINITE DIFFERENCE AND STANDARD EXPLICIT FINITE DIFFERENCE SCHEMES FOR POWER FLOW EQUATION

### Abstract

**Power flow equation for step-index glass optical fiber was solved using recently reported unconditionally-positive finite difference (UPFD) scheme. Solution obtained using UPFD scheme was compared with solution obtained using standard explicit finite difference (EFD) scheme. For accuracy testing both schemes were compared with analytical solution for steady state distribution of given fiber. The advantage of UPFD is reflected in stability of the scheme regardless of discretization step taken. Nevertheless EFD scheme has better concurrence with analytical solution than UPFD. This is due to the additional truncation-error terms in the approximations of the first and second derivatives with respect to θ.**

### References

Anderson, J. D. 1995. Computational fluid dynamics. New York: McGraw-Hill.

Bear, J. 2007. Hydraulics of Groundwater. Dover - Minneola.

Chen-Charpentier, B. M., & Kojouharov, H. V. 2013. An unconditionally positivity preserving scheme for advection–diffusion reaction equations. Mathematical and Computer Modelling, 57(9-10), pp. 2177-2185. doi:10.1016/j.mcm.2011.05.005

Dang, Q. A., & Ehrhardt, M. 2006. Adequate numerical solution of air pollution problems by positive difference schemes on unbounded domains. Mathematical and Computer Modelling, 44(9-10), pp. 834-856. doi:10.1016/j.mcm.2006.02.016

Djordjevich, A., & Savović, S. 2000. Investigation of mode coupling in step index plastic optical fibers using the power flow equation. IEEE Photonics Technology Letters, 12(11), pp. 1489-1491. doi:10.1109/68.887704

Djordjevich, A., & Savović, S. 2013. Solute transport with longitudinal and transverse diffusion in temporally and spatially dependent flow from a pulse type source. International Journal of Heat and Mass Transfer, 65, pp. 321-326. doi:10.1016/j.ijheatmasstransfer.2013.06.002

Drljaca, B. 2011. Modelovanje prostiranja svetlosti kroz višemodna optička vlakna sa stepenastim indeksom prelamanja primenom jednačine protoka snage. Kragujevac: PMF Kragujvac.

Garito, A. F., Wang, J., & Gao, R. 1998. Effects of Random Perturbations in Plastic Optical Fibers. Science, 281(5379), pp. 962-967. doi:10.1126/science.281.5379.962

Gloge, D. 2013. Optical Power Flow in Multimode Fibers. Bell System Technical Journal, 51(8), pp. 1767-1783. doi:10.1002/j.1538-7305.1972.tb02682.x

Hetrick, D. K. 1971. Dynamics of Nuclear Reactors. Chicago: University of Chicago.

Kevkic, T., Stojanovic, V., & Petkovic, D. 2019. Solving Schrödinger Equation for a Particle in One-Dimensional Lattice: An Homotopy Perturbations Approach. Romanian Reports in Physics, 71(101).

Liu, L., Clemence, D. P., & Mickens, R. E. 2010. A nonstandard finite difference scheme for contaminant transport with kinetic Langmuir sorption. Numerical Methods for Partial Differential Equations, 27(4), pp. 767-785. doi:10.1002/num.20551

Murray, J. D. 2002. Mathematical Biology I. Berlin: Springer-Verlag.

Petrović, M., Kontrec, N., & Panić, S. 2017. Determination of accelerated factors in gradient descent iterations based on Taylor's series. The University Thought - Publication in Natural Sciences, 7(1), pp. 41-45. doi:10.5937/univtho7-14337

Savović, S., & Caldwell, J. 2009. Numerical solution of Stefan problem with time-dependent boundary conditions by variable space grid method. Thermal Science, 13(4), pp. 165-174. doi:10.2298/tsci0904165s

Savović, S., Djordjevich, A., & Ristić, G. 2012. Numerical solution of the transport equation describing the radon transport from subsurface soil to buildings. Radiation Protection Dosimetry, 150(2), pp. 213-216. doi:10.1093/rpd/ncr397

Savović, S., Drljača, B., & Djordjevich, A. 2013. Influence of launch-beam distribution on bandwidth in step-index plastic optical fibers. Applied Optics, 52(6), p. 1117. doi:10.1364/ao.52.001117

Savović, S., & Djordjevich, A. 2013. Numerical solution for temporally and spatially dependent solute dispersion of pulse type input concentration in semi-infinite media. International Journal of Heat and Mass Transfer, 60, pp. 291-295. doi:10.1016/j.ijheatmasstransfer.2013.01.027

Savovic, S., Kovacevic, M. S., Bajic, J. S., Stupar, D. Z., Djordjevich, A., Zivanov, M., Drljaca, B., Simovic, A., & Oh, K. 2015. Temperature Dependence of Mode Coupling in low-NA Plastic Optical Fibers. Journal of Lightwave Technology, 33(1), pp. 89-94. doi:10.1109/jlt.2014.2375515

Shih, T. 1984. Numerical Heat Transfer. Berlin: Springer-Verlag.

Simović, A., Savović, S., Drljača, B., & Djordjevich, A. 2014. Influence of intermediate layer on transmission characteristics of W-type optical fibers. Optics and Laser Technology, 57, pp. 209-215. doi:10.1016/j.optlastec.2013.10.024

Urošević, V., & Nikezić, D. 2003. Radon transport through concrete and determination of its diffusion coefficient. Radiat. Prot. Dosim, 104, pp. 65-70.

Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.