Accuracy and conservation properties in numerical integration: the case of the Korteweg-de Vries equation |
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Authors: | J de Frutos JM Sanz-Serna |
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Institution: | (1) Departamento de Matemática Aplicada y Computación, Universidad de Valladolid, Valladolid, Spain; e-mail: frutos@cpd.uva.es and sanzserna@cpd.uva.es, ES |
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Abstract: | Summary. When numerically integrating time-dependent differential equations, it is often recommended to employ methods that preserve
some of the invariant quantities (mass, energy, etc.) of the problem being considered. This recommendation is usually justified
on the grounds that conservation of invariant quantities may ensure that the numerical solution possesses some important qualitative
features. However there are cases where schemes that preserve invariants are also advantageous in that they possess favourable
error propagation mechanisms that render them superior from a quantitative point of view. In the present paper we consider
the Korteweg-de Vries equation as a case study. We show rigorously that, for soliton problems and at leading order, the error
of conservative schemes consists of a phase error that grows linearly with time plus a complementary term that is bounded
in the norm uniformly in time. For ‘general’, nonconservative schemes the error involves a linearly growing amplitude error, a
quadratically growing phase error and a complementary term that grows linearly in the norm. Numerical experiments are presented.
Received November 21, 1994 / Revised version received July 17, 1995 |
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Keywords: | Mathematics Subject Classification (1991):65M12 |
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