Nonstandard finite difference method by nonlocal approximation
Introduction
In this paper, we are interested in designing some powerful finite difference schemes for a class of ordinary and partial differential equations. To fix ideas, we consider the initial value problem for an autonomous first-order ordinary differential equation where the function is unknown, and the function being given. We assume that t0 is finite, but the limit time T might be ∞ as for dynamical systems. We assume that problem (1) has a unique solution. As a matter of fact, this is true when the function f satisfies the standard Lipschitz condition, which for finite T guarantees well-posedness of the problem (see, for example, [6], p. 5, p. 31).
For the numerical approximation of (1), we replace the continuous interval [t0,T) by the mesh of discrete points {tk:=t0+kh}k≥0, where the parameter h>0 is the step-size. We denote by yk an approximation to the solution y(tk) at the point tk: yk≈y(tk). The sequence (yk) is obtained as solution of a finite difference equation of the form
Nonstandard finite difference schemes for Eq. (1) were introduced by Mickens in the 1980s as powerful numerical methods that preserve significant properties of exact solutions of the involved differential equations (see [7] and the references therein). Schemes were empirically developed using a collection of rules set by Mickens. In [2], [3], the authors provided some mathematical justifications for the success of these empirical procedures. In particular, nonstandard finite difference schemes were unambiguously defined as follows by using two of Mickens’ rules: Definition 1 The scheme (2) is called a nonstandard finite difference method if at least one of the following conditions is met: In the first-order discrete derivative that occurs in (2), the traditional denominator, h is replaced by a non-negative function, φ(h), such that Nonlinear terms in f(y) are approximated in a nonlocal way, i.e. by a suitable function of several points of the mesh (e.g. see , ).
Definition 2
Assume that the solutions of Eq. (1) satisfy some property . The numerical scheme (2) is called (qualitatively) stable with respect to property (or -stable), if for every value of h>0 the set of solutions of (2) satisfies property .
This paper focuses on the use of nonlocal approximation strategy (second item in Definition 1) for the construction of nonstandard difference schemes. Two new stability properties are investigated as particular cases of elementary stability with the additional capability of describing qualitatively fixed points when the above constraint is relaxed. The said properties and the related qualitative stabilities are discussed in detail in Section 2. Section 3 deals with the construction of nonstandard schemes with prescribed qualitative stability for the logistic equation and the combustion model. The obtained schemes are assembled in a numerical scheme for the reaction-diffusion equation (Section 4).
Section snippets
Some qualitative stability properties
We will assume that function F(h;y) in (2) has continuous derivatives with respect to both variables for h>0, and satisfies
We also assume that the difference scheme (2) is consistent with the differential equation (1) (see [6], [10]). Let us note that consistency implies that (4) is satisfied when y is the solution of (1).
Some nonstandard schemes
The nonlinear terms in the right-hand side f of Eq. (1) can be approximated nonlocally in many different ways. For example More generally, any linear combination of the expressions listed in (14) or (15) with the sum of the coefficients equal to 1, approximates y2 or y3, the error being of order O(h) for sufficiently smooth y(t). In this way, the function f in Eq. (1) may be approximated by an expression, which contains
The reaction–diffusion equation
To illustrate the strong connection of our theory to partial differential equations, we consider the reaction–diffusion equation In the time-independent (stationary) case, this equation reduces to while the associated space-independent equation is
From the physical point of view, an important property of Eq. (26) is the conservation of energy V(x,u)=(1/2)(ux)2+(1/3)u3−(1/4)u4. A general procedure for deriving energy preserving difference schemes is
Summary and discussion
This paper was motivated by two facts. On the one hand, the linear stability analysis developed in [1], [2], [3], [7], [9] fails to describe the qualitative properties of nonhyperbolic fixed points. On the other hand, there is a certain ambiguity in the way the few existing nonstandard schemes for nonhyperbolic fixed points where designed in [8], [9]. To address the first fact, we have in this paper introduced two types of monotonic properties of solutions of differential equations, which turn
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