Abstract
Many existing numerical schemes for evolutionary problems in partial differential equations (PDEs) can be viewed as method of lines (MOL) schemes. This paper treats the convergence of one-step MOL schemes. Our main purpose is to set up a general framework for a convergence analysis applicable to nonlinear problems. The stability materials for this framework are taken from the field of nonlinear stiff ODEs. In this connection, important concepts are the logarithmic matrix norm and C-stability. A nonlinear parabolic equation and the cubic Schrödinger equation are used for illustrating the ideas.
Zusammenfassung
Viele numerische Verfahren für Anfangswertprobleme für partielle Differentialgleichungen kann man als Linienmethoden interpretieren. Diese Arbeit behandelt solche Verfahren vom Einschriftt-Typ. Unser Ziel ist die Behandlung von Konvergenzfragen, insbesondere für nichtlineare Probleme. Unsere Hilfsmittel zum Nachweis der Stabilität entnehmen wir der stark entwickelten Theorie für nichtlineare steife gewöhnliche Differentialgleichungen. Wichtig sind hierbei die logarithmische Matrixnorm und der C-Stabilitätsbegriff. Eine nichtlineare parabolische Gleichung und die kubische Schrödingergleichung werden verwendet, um die Ideen zu illustrieren.
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Verwer, J.G., Sanz-Serna, J.M. Convergence of method of lines approximations to partial differential equations. Computing 33, 297–313 (1984). https://doi.org/10.1007/BF02242274
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DOI: https://doi.org/10.1007/BF02242274