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Nonstandard finite difference method by nonlocal approximation. (English) Zbl 1015.65034
Two types of monotonic properties of solutions of differential equations are discussed and general finite difference schemes, which are stable with respect to these properties are investigated. Apart from being elementary stable, these schemes are also shown to preserve qualitative properties of nonhyperbolic fixed points of the differential equations. From the practical point of view, a systematic procedure based on nonlocal approximation is proposed for the construction of qualitatively stable nonstandard finite difference schemes for the logistic equation, the combustion model and the reaction-diffusion equation.
MSC:
65M06Finite difference methods (IVP of PDE)
65L12Finite difference methods for ODE (numerical methods)
65L20Stability and convergence of numerical methods for ODE
34A34Nonlinear ODE and systems, general
80A25Combustion, interior ballistics
65M12Stability and convergence of numerical methods (IVP of PDE)
35K57Reaction-diffusion equations
35L65Conservation laws