Positivity-preserving nonstandard finite difference schemes for cross-diffusion equations in biosciences

dc.contributor.authorChapwanya, Michael
dc.contributor.authorLubuma, Jean M.-S.
dc.contributor.authorMickens, Ronald E.
dc.contributor.emailjean.lubuma@up.ac.zaen_ZA
dc.date.accessioned2015-08-14T06:45:23Z
dc.date.available2015-08-14T06:45:23Z
dc.date.issued2014-11
dc.description.abstractWe design nonstandard finite difference (NSFD) schemes which are unconditionally dynamically consistent with respect to the positivity property of solutions of cross-diffusion equations in biosciences. This settles a problem that was open for quite some time. The study is done in the setting of three concrete and highly relevant cross-diffusion systems regarding tumor growth, convective predator–prey pursuit and evasion model and reaction–diffusion–chemotaxis model. It is shown that NSFD schemes used for classical reaction–diffusion equations, such as the Fisher equation, for which the solutions enjoy the positivity property, are not appropriate for cross-diffusion systems. The reliable NSFD schemes are therefore obtained by considering a suitable implementation on the crossdiffusive term of Mickens’ rule of nonlocal approximation of nonlinear terms, apart from his rule of complex denominator function of discrete derivatives. We provide numerical experiments that support the theory as well as the power of the NSFD schemes over the standard ones. In the case of the cancer growth model, we demonstrate computationally that our NSFD schemes replicate the property of traveling wave solutions of developing shocks observed in Marchant et al. (2000).en_ZA
dc.description.embargo2015-11-30en_ZA
dc.description.librarianhb2015en_ZA
dc.description.sponsorshipSouth African Research Chairs Initiative of the Department of Science and Technology and National Research Foundation : SARChI Chair in Mathematical Models and Methods in Bioengineering and Biosciences.en_ZA
dc.description.urihttp://www.elsevier.com/locate/camwaen_ZA
dc.identifier.citationChapwanya, M, Lubuma JM-S & Mickens RE 2014, 'Positivity-preserving nonstandard finite difference schemes for cross-diffusion equations in biosciences', Computers and Mathematics with Applications, vol. 68, no. 9, pp.1071-1082.en_ZA
dc.identifier.issn0898-1221 (print)
dc.identifier.issn1873-7668 (online)
dc.identifier.other10.1016/j.camwa.2014.04.021
dc.identifier.urihttp://hdl.handle.net/2263/49316
dc.language.isoenen_ZA
dc.publisherElsevieren_ZA
dc.rights© 2014 Elsevier Ltd. All rights reserved. Notice : this is the author’s version of a work that was accepted for publication in Computers and Mathematics with Applications. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Computers and Mathematics with Applications, vol. 68, pp. 1071-1082, 2014. doi :10.1016/j.camwa.2014.04.021.en_ZA
dc.subjectDynamical consistencyen_ZA
dc.subjectCross-diffusion equationsen_ZA
dc.subjectReaction–diffusion–chemotaxis equationsen_ZA
dc.subjectTumor growthen_ZA
dc.subjectPredator–prey pursuit and evasion modelen_ZA
dc.subjectNonstandard finite difference (NSFD)en_ZA
dc.titlePositivity-preserving nonstandard finite difference schemes for cross-diffusion equations in biosciencesen_ZA
dc.typePostprint Articleen_ZA

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