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Two positivity preserving flux limited, second‐order numerical methods for a haptotaxis model
Author(s) -
Kolev Mikhail K.,
Koleva Migle.,
Vulkov Lubin G.
Publication year - 2013
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.21748
Subject(s) - mathematics , flux limiter , logarithm , convergence (economics) , rate of convergence , nonlinear system , limiting , algebraic number , limiter , space (punctuation) , flux (metallurgy) , partial differential equation , mathematical analysis , computer science , physics , mechanical engineering , computer network , channel (broadcasting) , telecommunications , quantum mechanics , engineering , economics , economic growth , operating system , materials science , metallurgy
Two numerical methods for a one‐dimensional haptotaxis model, which exploit the use of van Leer flux limiter, are developed and analyzed. Sufficient conditions time step size and flux limiting are given for such formulation to ensure the non‐negativity of the discrete solution and second‐order accuracy in space. Another advantage is that we avoid solving large nonlinear systems of algebraic equations. The discrete preservation of total conservation of cell density, concentration, and logarithmic density is also verified for the numerical solution. Numerical results concerning accuracy, convergence rate, positivity, and conservation properties are presented and discussed. Similar approach could be applied efficiently in the corresponding two‐ and three‐dimensional problems. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013