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Combination of nonstandard schemes and Richardson’s extrapolation to improve the numerical solution of population models. (English) Zbl 1205.65015
Summary: We combine nonstandard finite-difference (NSFD) schemes and Richardson’s extrapolation method to obtain numerical solutions of two biological systems. The first biological system deals with the dynamics of phytoplankton-nutrient interaction under nutrient recycling and the second one deals with the modeling of whooping cough in the human population. Since both models requires positive solutions, the numerical solutions need to satisfy this property. In addition, it is necessary in some cases that numerical solutions reproduce correctly the dynamical behavior while in other cases it is necessary just to find the steady state. NSFD schemes can do this. In this paper Richardson’s extrapolation is applied directly to the NSFD solution to increase the order of accuracy of the numerical solutions of these biological systems. Numerical results show that Richardson’s extrapolation method improves accuracy.
65B10Summation of series (numerical analysis)
65N06Finite difference methods (BVP of PDE)
[1]Britton, N. F.: Essential mathematical biology, (2003)
[2]Das, K. Pada; Chatterjee, S.; Chattopadhyay, J.: Dynamics of nutrient–phytoplankton interaction in the presence of viral infection and periodic nutrient input, Math. model. Nat. phenom. 3, No. 3, 149-169 (2008)
[3]Ruan, S.: Persistence and co-existence in zooplankton–phytoplankton–nutrient models with instantaneous nutrient recycling, J. math. Biol. 31, 633-654 (1993) · Zbl 0779.92021 · doi:10.1007/BF00161202
[4]Christie, Celia; Marx, Mary L.; Marchant, Colin D.; Reising, Shirley F.: The 1993 epidemic of pertussis in cincinnati – resurgence of disease in a highly immunized population of children, N. engl. J. med. 331, No. 1, 16-21 (1994)
[5]Nelson, John D.: The changing epidemiology of pertussis in Young infants: the role of adults as reservoirs of infection, Am. J. Dis. child. 132, No. 4, 371-373 (1978)
[6]Cherry, J. D.: The epidemiology of pertussis and pertussis immunization in the united kingdom and the united states: a comparative study, Curr. probl. Pediatr. 14, No. 2, 1-78 (1984)
[7]Duncan, C. J.; Duncan, S. R.; Scott, S.: Whooping cough epidemic in London, 17011812:infection dynamics seasonal forcing and the effects of malnutrition, Proc. R. Soc. lond. B 263, 445-450 (1996)
[8]Lambert, J. D.: Computational methods in ordinary differential equations, (1973) · Zbl 0258.65069
[9]Bruggeman, Jorn; Burchard, Hans; Kooi, Bob W.; Sommeijer, Ben: A second-order, unconditionally positive, mass-conserving integration scheme for biochemical systems, Appl. numer. Math. 57, No. 1, 36-58 (2007) · Zbl 1123.65067 · doi:10.1016/j.apnum.2005.12.001
[10]Bolley, C.; Crouzeix, M.: Conservation de la positivit lors de la discrtisation des problmes d’volution parabolic, RAIRO anal. Numer. 12, No. 3, 237-245 (1978) · Zbl 0392.65042
[11]Mickens, Ronald E.: Nonstandard finite difference models of differential equations, (1994) · Zbl 0810.65083
[12]Mickens, Ronald E.: Nonstandard finite difference schemes for differential equations, J. difference equ. Appl. 8, No. 9, 823-847 (2002) · Zbl 1010.65032 · doi:10.1080/10236919021000000807
[13]Mickens, R. E.: Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition, Wiley interscience 23, No. 3, 672-691 (2006) · Zbl 1114.65094 · doi:10.1002/num.20198
[14]Mickens, R. E.: Numerical integration of population models satisfying conservation laws: NSFD methods, Biol. dyn. 1, No. 4, 427-436 (2007)
[15]Burg, C.; Erwin, T.: Application of Richardson extrapolation to the numerical solution of partial differential equations, Numer. methods partial differential equations 25, No. 4, 810-832 (2007) · Zbl 1169.65330 · doi:10.1002/num.20375
[16]Brezinski, C.; Zaglia, M. Redivo: Extrapolation methods. Theory and practice, (1991) · Zbl 0744.65004
[17]Shampine, L.; Watts, H.: Global error estimation for ordinary differential equations, ACM trans. Math. softw. 2, 172-186 (1976) · Zbl 0328.65041 · doi:10.1145/355681.355687
[18]Piyawong, W.; Twizell, E. H.; Gumel, A. B.: An unconditionally convergent finite-difference scheme for the SIR model, Appl. math. Comput. 146, 611-625 (2003) · Zbl 1026.92041 · doi:10.1016/S0096-3003(02)00607-0
[19]Anguelov, Roumen; Lubuma, Jean M. -S.: Contributions to the mathematics of the nonstandard finite difference method and applications, Numer. methods partial differential equations 17, No. 5, 518-543 (2003) · Zbl 0988.65055 · doi:10.1002/num.1025