Optimal residuals and the Dahlquist test problem

Abstract

We show how to compute the optimal relative backward error for the numerical solution of the Dahlquist test problem by one-step methods. This is an example of a general approach that uses results from optimal control theory to compute optimal residuals, but elementary methods can also be used here because the problem is so simple. This analysis produces new insight into the numerical solution of stiff problems.Keywords stiff IVP · backward error · residual · optimal control Mathematics Subject Classification (2010) 65L04 · 65L05 · 65L20 · 49M05 IntroductionThe study of stiff differential equations and their efficient numerical solution is by now a mature field. There are several, perhaps many, efficient practical methods with freely available high quality implementations. The literature on the theory of such methods is extensive. Surprisingly, it is not yet complete: for instance, see the survey [24], which has the intriguing title Stiffness 1952-2012: Sixty years in search of a definition. That paper re-examines the fundamentals and thoroughly surveys the literature, and proposes a new stiffness indicator that they claim is useful both a priori for indicating stiffness and a posteriori for indicating varying regions of the solution that stiffness is important. This paper takes a different approach, that of optimal residuals, i.e., backward error, and uses it on the Dahlquist test problem to generate some new observations about this, the simplest of all stiff problems. Indeed, [24] calls this problem "simplistic" and with good reason, but surprisingly it still has things to teach us.Trying to study stiff problems from the point of view of backward error analysis is itself not new. For instance, there is the PhD thesis of W.L. Seward and the paper [8]. But there is an intrinsic dissonance: a stiff problem has the feature that errors are (often sharply) damped as the integration proceeds forward in time, and thus it is not obvious why one might prefer

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