β Scribed by Arindama Singh
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Singh, A., Comments on decouplings and Euler's one-step schemes, Mathematics and Computers in Simulation 33 (1991) 197-203. In this paper, effects of total and partial decouplings in conjunction with Euler's one-step implicit and explicit schemes for linear singularly perturbed systems are considered. It is shown that in some cases the equations obtained by decoupling the system of difference equations for such problems might lead to inverting ill-conditioned matrices; such cases are categorically pointed out.
Consider the singularly perturbed linear system of ordinary differential equations
A22(t) y2
(1 .I)
where 0 < t < 1, "." denotes d/dl. 9 < E: e 1 is a parameter, JJ', I' are m-dimensional real vector functions, y2, fZ are k-dimensional real vec;or functions. and A'.'(!) are real matrices of appropriate dimensions. Systems of :he type (1.1) have been quite well studied (see, e.g., ). Under the central assumption that the eigenvalues of the fast block A22( t) of the Fj,itern matrix are bounded away from the imaginary axis, the fast component r2( t, E) of a solution becomes a combination of two pieces each contained in different manifolds ; one of the pieces is stable in time t and the other is stable in the reverse time -L This supplies the motivation to decouple the system into sfow and fast modes. 1Wattheij [7]? for example, has applied decoupling transformations on the corresponding difference equations of (I .I) obtained by the use of Euler's implicit one-step scheme, provided that real parts of the eigenvalues of A2'( r) are all negative; it is also anticipated in [7] that the method would work well when these are positive. This paper envisages the possibility of solving (1.1) by decoupling the difference equations t at are obtained by applying Euler's implicit and explicit schemes.
In Section 2, two types of decoupling (total an artial) are introduced.
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