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Linear advection equation with upstream scheme: analytic solutions

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The linear advection equation as discretized with a first order upstream scheme is solved analytically for the case of an initial distribution restricted to one grid point. It is shown that the evolving solution is the binomial distribution. The numerical scheme predicts the position of the maximum almost perfectly but its amplitude decays ~n– with time step index n. The numerical solution for a point source is presented as well and shown to be in good agreement with the analytic solution.

Eine analytische Lösung für die mit einem stromauf orientierten Verfahren erster Ordnung diskretisierte lineare Advektionsgleichung wird angegeben für den Fall, dass die Anfangsverteilung nur an einem Gitterpunkt von Null verschieden ist. Es wird gezeigt, dass dieser Anfangswert zu späteren Zeitschritten binomisch über die Gitterpunkte verteilt wird. Das numerische Verfahren sagt die Lage des Maximums der Verteilung de facto fehlerfrei voraus, aber die Amplitude des Maximums nimmt fälschlich ~n–1/2, ab wobei n der Index der Zeitschritte ist. Es wird gezeigt, dass die numerische Lösung für eine Punktquelle gut mit der analytischen übereinstimmt.
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Document Type: Research Article

Publication date: 2004-12-01

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  • Meteorologische Zeitschrift (originally founded in 1866) is the joint periodical of the meteorological societies of Austria, Germany and Switzerland. It accepts high-quality peer-reviewed manuscripts on all aspects of observational, theoretical and computational research out of the entire field of meteorology, including climatology. Meteorologische Zeitschrift represents a natural forum for the meteorological community of Central Europe and worldwide.
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