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Volume 20,     Number 3,     Fall 2012

 

ASYMPTOTICALLY CORRECT
INTERPOLATION-BASED SPATIAL ERROR
ESTIMATION FOR 1D PDE SOLVERS
TOM ARSENAULT, TRISTAN SMITH, PAUL MUIR
AND JACK PEW

Abstract. BACOL and BACOLR are B-spline Gaussian collocation method-of-lines packages for the numerical solution of systems of one-dimensional parabolic partial differential equations (PDEs). In previous studies, they were shown to be superior to other similar packages, especially for problems exhibiting sharp spatial layer regions where a stringent tolerance is imposed. A signi cant feature of these solvers is that, in addition to the temporal error control provided by the underlying time-integrator, they adapt the spatial mesh to control a high order estimate of the spatial error. In addition to computing a primary collocation solution of a given spatial order, the BACOL/BACOLR codes also compute, at a substantial cost, a secondary collocation solution of one higher order, and then the di erence between the two collocation solutions is used to give an estimate of the leading order term in the error for the lower order solution. In this paper we consider an approach in which the computation of lower order collocation solution is replaced by an inexpensive interpolant (based on evaluations of the higher order collocation solution) constructed so that the leading order term in the interpolation error agrees (asymptotically) with the leading order term in original BACOL/BACOLR error estimate. We provide numerical results to show that this interpolation-based error estimate can provide spatial error estimates of comparable accuracy to those currently computed by BACOL, but at a signi cantly lower cost.

 

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