Nonnegativity preserving convergent schemes for the thin film equation |
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Authors: | Günther Grün Martin Rumpf |
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Affiliation: | Universit?t Bonn, Institut für Angewandte Mathematik, Beringstrasse 6, 53115 Bonn, Germany, DE Universit?t Bonn, Institut für Angewandte Mathematik, Wegelerstrasse 6, 53115 Bonn, Germany, DE
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Abstract: | Summary. We present numerical schemes for fourth order degenerate parabolic equations that arise e.g. in lubrication theory for time evolution of thin films of viscous fluids. We prove convergence and nonnegativity results in arbitrary space dimensions. A proper choice of the discrete mobility enables us to establish discrete counterparts of the essential integral estimates known from the continuous setting. Hence, the numerical cost in each time step reduces to the solution of a linear system involving a sparse matrix. Furthermore, by introducing a time step control that makes use of an explicit formula for the normal velocity of the free boundary we keep the numerical cost for tracing the free boundary low. Received June 29, 1998 / Published online June 21, 2000 |
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Keywords: | Mathematics Subject Classification (1991): 35K35 35K55 35K65 65M12 65M50 65M60 76D08 |
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