Geodesic
On the advantages of
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 25 (2022) no. 3, pp. 269-287.

Voir la notice de l'article provenant de la source Math-Net.Ru

The goal of this work is to highlight the advantages of using NonStandard Finite Differences (NSFD) numerical schemes for the resolution of Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs) of which some properties of the exact solution are a-priori known, such as positivity. The main reference considered is Mickens' work [14], in which the author derives NSFD schemes for ODEs and PDEs that describe real phenomena, and therefore widely used in applications. We rigorously demonstrate that NSFD methods can have a higher order of convergence than the related classical ones, deriving also the conditions that guarantee the stability of the analyzed schemes. Furthermore, we carry out in-depth numerical tests comparing the classical methods with the NSFD ones proposed by Mickens, evaluating when the latter are decidedly advantageous. 
@article{SJVM_2022_25_3_a3,
     author = {D. Conte and N. Guarino and G. Pagano and B. Paternoster},
     title = {On the advantages of},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {269--287},
     publisher = {mathdoc},
     volume = {25},
     number = {3},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2022_25_3_a3/}
}
TY  - JOUR
AU  - D. Conte
AU  - N. Guarino
AU  - G. Pagano
AU  - B. Paternoster
TI  - On the advantages of
JO  - Sibirskij žurnal vyčislitelʹnoj matematiki
PY  - 2022
SP  - 269
EP  - 287
VL  - 25
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJVM_2022_25_3_a3/
LA  - ru
ID  - SJVM_2022_25_3_a3
ER  - 
%0 Journal Article
%A D. Conte
%A N. Guarino
%A G. Pagano
%A B. Paternoster
%T On the advantages of
%J Sibirskij žurnal vyčislitelʹnoj matematiki
%D 2022
%P 269-287
%V 25
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJVM_2022_25_3_a3/
%G ru
%F SJVM_2022_25_3_a3
D. Conte; N. Guarino; G. Pagano; B. Paternoster. On the advantages of. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 25 (2022) no. 3, pp. 269-287. http://geodesic.mathdoc.fr/item/SJVM_2022_25_3_a3/

[1] R. Anguelov, J. M.S. Lubuma, “Contributions to the mathematics of the nonstandard finite difference method and applications”, Numer. Methods Partial Differ. Equ., 17:5 (2001), 518–543 | DOI | MR

[2] M. A. Budroni, G. Pagano, B. Paternoster et al, “Synchronization scenarios induced by delayed communication in arrays of diffusively coupled autonomous chemical oscillators”, Physical Chemistry Chemical Physics, 23:32 (2021), 17606–17615 | DOI

[3] M. A. Budroni, G. Pagano, B. Paternoster et al., “A model for coupled Belousov-Zhabotinsky oscillators with delay”, Proc. 14th WCCM-ECCOMAS Congress 2020, 2021 | DOI

[4] I. M. Bulai, R. Cavoretto, B. Chialva, D. Duma, E. Venturino, “Comparing disease-control policies for interacting wild populations”, Nonlinear Dyn., 79:3 (2015), 1881–1900 | DOI

[5] D. Conte, R. D'Ambrosio, M. Moccaldi, B. Paternoster, “Adapted explicit two-step peer methods”, J. Numer. Math., 27:2 (2018), 69–83 | DOI | MR

[6] D. Conte, F. Mohammadi, L. Moradi, B. Paternoster, “Exponentially fitted two-step peer methods for oscillatory problems”, Comput. Appl. Math., 39:3 (2020) | DOI | MR

[7] L. Eigentler, J. A. Sherratt, “Metastability as a coexistence mechanism in a model for dryland vegetation patterns”, Bull. Math. Biol., 81 (2019), 2290–2322 | DOI | MR

[8] J. H. Ferziger, M. Peric, Computational Methods for Fluid Dynamics, Springer, 1996 | MR

[9] N. Ganegoda, T. Götz, K. Putra Wijaya, “An age-dependent model for dengue transmission: Analysis and comparison to field data”, Appl. Math. Comput., 388 (2021) | DOI | MR

[10] P. D. Giamberardino, D. Iacoviello, F. Papa, C. Sinisgalli, “Dynamical evolution of COVID-19 in Italy with an evaluation of the size of the asymptomatic infective population”, IEEE J. Biomed., 25:4 (2021), 1326–1332

[11] L. Ixaru, G. Berghe, Exponential Fitting, Springer, 2004 | DOI | MR

[12] W. O. Kermack, A. G. McKendrick, “Contributions to the mathematical theory of epidemics-I”, Bull. Math. Biol., 53 (1991), 33–55

[13] C. Koroglu, “Exact and nonstandard finite difference schemes for the generalized KdV-Burgers equation”, Adv. Differ. Equ., 2020 | DOI | MR

[14] R. E. Mickens, “Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition”, Numer. Methods Partial Differ. Equ., 23 (2007), 672–691 | DOI | MR

[15] R. E. Mickens, “Dynamic consistency: a fundamental principle for constructing nonstandard finite difference schemes for differential equations”, J. Differ. Equ. Appl, 11 (2005), 645–653 | DOI | MR

[16] R. E. Mickens, “Analysis of a new finite-difference scheme for the linear advection-diffusion equation”, J. Sound Vib, 146:2 (1991), 342–344 | DOI | MR

[17] R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific Publishing, 1993 | MR

[18] R. E. Mickens, “Exact solutions to a finite-difference model of a nonlinear reaction-advection equation: implications for numerical analysis”, Numer. Methods Partial Differ. Equ, 5 (1989), 313–325 | DOI | MR

[19] J. D. Murray, Mathematical Biology, Springer, 1993 | MR

[20] Previdi F. Esempio, “Modelli compartimentali epidemiologici per la descrizione della diffusione delle infezioni”, University Lecture Notes, 2020, 1–35

[21] A. Markus, R. E. Mickens, “Suppression of numerically induced chaos with nonstandard finite difference schemes”, J. Comput. Appl. Math., 106 (1999), 317–324 | DOI | MR

[22] T. S. Shaikh, N. Fayyaz, N. Ahmed et al, “Numerical study for epidemic model of hepatitis-B virus”, Eur. Phys. J. Plus, 136 (2021)

[23] R. Ud Din, K. Shah, I. Ahmad, T. Abdeljawad, “Study of transmission dynamics of novel COVID-19 by using mathematical model”, Adv. Differ. Equ., 2020, 323 | DOI | MR

[24] A. Viguerie, A. Veneziani, G. Lorenzo et al, “Diffusion-reaction compartmental models formulated in a continuum mechanics framework: application to COVID-19, mathematical analysis, and numerical study”, Comput. Mech., 66 (2020), 1131–1152 | DOI | MR