Error estimates over infinite intervals of some discretizations of evolution equations

Classical discretization error estimates for systems of ordinary differential equations contain a factor exp (Lt), whereL is the Lipschitz constant. For strongly monotone operators, however, one may prove that for aϑ-method, 0<ϑ<1/2, the errors are bounded uniformly in time and with errorO(Δt)2, ifϑ=1/2−|O(Δt)|. This was done by this author (1977), for an operator in a reflexive Banach space and includes the case of systems of differential equations as a special case.In the present paper we restate this result as it may have been overlooked and consider also the monotone (inclusive of the conservative) and unbounded cases. We also discuss cases where the truncation errors are bounded by a constant independent of the stiffness of the problem. This extends previous results in [6] and [7]. Finally we discuss a boundary value technique in the context above.

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