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Differenzenapproximationen Partieller Anfangswertaufgaben by Rainer Ansorge

Differenzenapproximationen Partieller Anfangswertaufgaben

by Rainer Ansorge
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Book cover type: Paperback
  • ISBN13: 9783519023470
  • Binding: Paperback
  • Subject: N/A
  • Publisher: Vieweg+teubner Verlag
  • Publisher Imprint: Vieweg+teubner Verlag
  • Publication Date:
  • Pages: 302
  • Original Price: EUR 46.68
  • Language: German
  • Edition: 1978
  • Item Weight: 359 grams
  • BISAC Subject(s): Differential Equations / General

1956 veröffentlichten Lax und Richtmyer [6 eine Arbeit, in der unter Benutzung funktionalanalytischer Hilfsmittel die Struktur des Konvergenzverhaltens von Differenzapproximationen für eine gro e Klasse linearer Anfangswertaufgaben bei partiellen Differential- gleichungen aufgeklärt werden konnte. Insbesondere konnte der Satz über die Äquivalenz der numerischen Stabilität mit der punktweisen Konvergenz eines mit der gegebenen Anfangswertaufgabe konsistenten Differenzenverfahrens weitestgehend unabhängig von dem der Aufga- benstellung zugrundeliegenden normierten Raum und unabhängig vom Typ der approximierten Aufgabe formuliert werden. Zugleich ergab sich, da unter gewissen Voraussetzungen neben den klassischen Lö- sungen der gegebenen Anfangswertaufgabe auch deren verallgemeiner- te Lösungen durch das Differenzenverfahren approximierbar sind, wenngleich Fehlerabschätzungen oder auch nur Angaben über die Kon- vergenzordnung im Falle verallgemeinerter Lösungen zunächst aus- blieben. In den seither vergangenen zwei Jahrzehnten wurden mit Erfolg zahl- reiche Versuche unternommen, diese Lax-Richtmyer-Theorie in ver- schiedenen Richtungen zu ergänzen und zu verallgemeinern. Dabei zeigte sich insbesondere, da die punktweise Konvergenz der ite- rierten Differenzenoperatoren bei der Approximation nichtlinearer Differentialgleichungen in der Regel nicht ausreicht, um die nume- rische Brauchbarkeit eines Verfahrens zu gewährleisten, jedoch ge- lang es, auch bei entsprechend verfeinerten Konvergenzbegriffen un- ter Verwendung geeigneter Stabilitätsdefinitionen Äquivalenzsätze aufzustellen und damit die Lax-Richtmyer-Theorie einschlie lich der aus ihr für konkrete Probleme in den Anwendungsgebieten resultie- renden Forderungen auf solche nichtlinearen Probleme zu erweitern. Naturgemä spielte dabei die Frage der Existenz und der numerischen Erfa barkeit verallgemeinerter Lösungen nichtlinearer Probleme ei- ne nicht unerhebliche Rolle.

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