The impact of finite precision arithmetic and sensitivity on the numerical solution of partial differential equations
type: Item
instance of: scholarly article
title: The impact of finite precision arithmetic and sensitivity on the numerical solution of partial differential equations
zbMATH Open document ID: 1060.65116
DOI: 10.1016/S0895-7177(02)00078-X
author: John A. Burns
author: D. S. Gilliam
author: Jim Hill
author: Edward J. Allen
author: V. I. Shubov
published in: Mathematical and Computer Modelling
publication date: 2004-08-20T00:00:00Z
review text: It is shown that problem sensitivity and finite precision arithmetic can combine to produce false numerical solutions to steady state problems. Although one might guess that it is possible to construct pathological examples with this property, it is somewhat remarkable that this phenomenon can occur for a ``simple'' Burgers' equation. In addition, in the space of antisymmetric \(L^2 \) functions \(L_{AS}^2 (0,1)\), the authors have shown that the steady state Burgers' equation has a unique solution \((v(x)= 0)\), and yet, discretized versions of this equation can yield nonunique solutions. More importantly, for sufficiently large initial data, marching schemes will converge to this discrete (yet false) solution. The authors also presented a sensitivity and stability analysis that provides insight into these numerical difficulties. One implication of these results is that more analysis needs to be done on recent ``numerical based proofs'' of nonuniqueness. In particular, it is established that in a perfectly reasonable model, numerical computations that yield nonunique discrete stationary solutions do not imply anything about nonuniqueness for the continuous boundary value problem. It is important to again emphasize that it is finite precision arithmetic that causes these false solutions. Therefore, even converging an algorithm to machine roundoff error will not eliminate the difficulty. Finally, it is observed that the false numerical solutions can differ by orders of magnitude from real stationary solutions. Therefore, in such cases, cascading the numerical solutions into an optimization or control algorithm can produce bad designs.
Mathematics Subject Classification ID: 65N12
Mathematics Subject Classification ID: 65N06
Mathematics Subject Classification ID: 65N15
Mathematics Subject Classification ID: 35Q53
zbMATH DE Number: 2093859
zbMATH Keywords: Burgers equation
zbMATH Keywords: steady state problems
zbMATH Keywords: marching schemes
zbMATH Keywords: nonuniqueness
zbMATH Keywords: sensitivity
zbMATH Keywords: stability
zbMATH Keywords: finite precision arithmetic
zbMATH Keywords: algorithm
zbMATH Keywords: false numerical solutions
reviewed by: Leonid B. Chubarov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2093859