The fourier method for nonsmooth initial data

Andrew Majda, James McDonough, Stanley Osher

Research output: Contribution to journalArticle

Abstract

Application of the Fourier method to very general linear hyperbolic Cauchy problems having nonsmooth initial data is considered, both theoretically and computationally. In the absence of smoothing, the Fourier method will, in general, be globally inaccurate, and perhaps unstable. Two main results are proven: The first shows that appropriate smoothing techniques applied to the equation gives stability; and the second states that this smoothing combined with a certain smoothing of the initial data leads to infinite order accuracy away from the set of discontinuities of the exact solution modulo a very small easily characterized exceptional set. A particular implementation of the smoothing method is discussed; and the results of its application to several test problems are presented, and compared with solutions obtained without smoothing.

Original languageEnglish (US)
Pages (from-to)1041-1081
Number of pages41
JournalMathematics of Computation
Volume32
Issue number144
DOIs
StatePublished - 1978

Fingerprint

Fourier Method
Smoothing
Exceptional Sets
Smoothing Techniques
Hyperbolic Problems
Smoothing Methods
Inaccurate
Test Problems
Modulo
Discontinuity
Cauchy Problem
Exact Solution
Unstable

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics

Cite this

The fourier method for nonsmooth initial data. / Majda, Andrew; McDonough, James; Osher, Stanley.

In: Mathematics of Computation, Vol. 32, No. 144, 1978, p. 1041-1081.

Research output: Contribution to journalArticle

Majda, Andrew ; McDonough, James ; Osher, Stanley. / The fourier method for nonsmooth initial data. In: Mathematics of Computation. 1978 ; Vol. 32, No. 144. pp. 1041-1081.
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