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Nonstandard methods for the convective transport equation with nonlinear reactions
Author(s) -
Kojouharov Hristo V.,
Chen Benito M.
Publication year - 1998
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/(sici)1098-2426(199807)14:4<467::aid-num3>3.0.co;2-i
Subject(s) - mathematics , nonlinear system , truncation error , partial differential equation , interpolation (computer graphics) , convection–diffusion equation , truncation (statistics) , grid , mathematical analysis , numerical analysis , variable (mathematics) , partial derivative , classical mechanics , geometry , physics , quantum mechanics , motion (physics) , statistics
A new nonstandard Lagrangian method is constructed for the one‐dimensional, transient convective transport equation with nonlinear reaction terms. An “exact” time‐stepping scheme is developed with zero local truncation error with respect to time. The scheme is based on nonlocal treatment of nonlinear reactions, and when applied at each spatial grid point gives the new fully discrete numerical method. This approach leads to solutions free from the numerical instabilities that arise because of incorrect modeling of derivatives and nonlinear reaction terms. Algorithms are developed that preserve the properties of the numerical solution in the case of variable velocity fields by using nonuniform spatial grids. Effects of different interpolation techniques are examined and numerical results are presented to demonstrate the performance of the proposed new method. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 467–485, 1998