CONTRIBUTIONS TO THE THEORY OF NON-STANDARD FINITE DIFFERENCE METHODS AND APPLICATIONS TO SINGULAR PERTURBATION PROBLEMS

Abstract

We consider singular perturbation problems defined by first-order (systems of) ordinary differential equations, second-order ordinary differential equations, advection-reaction equations and reaction-diffusion equations. These problems, in which a small positive parameter E is multiplied to the highest derivative, arise in various fields of science and engineering such as fluid mechanics, fluid dynamics, quantum mechanics, chemical reactor theory, etc. The main concern with such problems is the rapid growth or decay of their solutions in one or more narrow “layer region(s)”. Often, the problems are dissipative or dispersive as the rapidly varying component of the solution decays exponentially (dissipates) or oscillates (disperses) from some points of discontinuity in the layer region(s) as E tends to zero. This singular behavior of the solution makes classical numerical methods not reliable. We provide some complements to the theory of non-standard finite difference method. We use this theory to design non-standard schemes, which replicate the above mentioned physical properties of the exact solution and which, for a class of linear problems, are €-uniformly convergent in the sense that the parameter E and the mesh step vary independently from one another. For a fixed E , the schemes obtained are elementary stable or stable with respect to the monotone dependence on initial values in the case of first-order equations and advection-reaction problems; they are stable with respect to some kind of conservation laws in the case of second-order equation and they preserve the boundedness and positivity property of the solution of reaction-diffusion problems. Several numerical simulations that support the theory are provided.