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Numerical Simulation of Conservation Laws with Moving Grid Nodes: Application to Tsunami Wave Modelling

1
Institute of Computational Technologies of Siberian Branch of the Russian Academy of Sciences, 630090 Novosibirsk, Russia
2
Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA, 73000 Chambéry, France
3
LAMA UMR 5127 CNRS, Université Savoie Mont Blanc, Campus Scientifique, 73376 Le Bourget-du-Lac CEDEX, France
4
School of Mathematics, Statistics and Operations Research, Victoria University of Wellington, P.O. Box 600, Wellington 6140, New Zealand
5
Department of Radiology, Medical Physics, Medical Center, Faculty of Medicine, University of Freiburg, 79106 Freiburg, Germany
*
Author to whom correspondence should be addressed.
Geosciences 2019, 9(5), 197; https://doi.org/10.3390/geosciences9050197
Received: 22 March 2019 / Revised: 18 April 2019 / Accepted: 24 April 2019 / Published: 30 April 2019
(This article belongs to the Special Issue Interdisciplinary Geosciences Perspectives of Tsunami Volume 2)
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Abstract

In the present article, we describe a few simple and efficient finite volume type schemes on moving grids in one spatial dimension combined with an appropriate predictor–corrector method to achieve higher resolutions. The underlying finite volume scheme is conservative, and it is accurate up to the second order in space. The main novelty consists in the motion of the grid. This new dynamic aspect can be used to resolve better the areas with large solution gradients or any other special features. No interpolation procedure is employed; thus, unnecessary solution smearing is avoided, and therefore, our method enjoys excellent conservation properties. The resulting grid is completely redistributed according to the choice of the so-called monitor function. Several more or less universal choices of the monitor function are provided. Finally, the performance of the proposed algorithm is illustrated on several examples stemming from the simple linear advection to the simulation of complex shallow water waves. The exact well-balanced property is proven. We believe that the techniques described in our paper can be beneficially used to model tsunami wave propagation and run-up. View Full-Text
Keywords: conservation laws; finite volumes; conservative finite differences; moving grids; adaptivity; advection; shallow water equations; wave run-up conservation laws; finite volumes; conservative finite differences; moving grids; adaptivity; advection; shallow water equations; wave run-up
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Khakimzyanov, G.; Dutykh, D.; Mitsotakis, D.; Shokina, N.Y. Numerical Simulation of Conservation Laws with Moving Grid Nodes: Application to Tsunami Wave Modelling. Geosciences 2019, 9, 197.

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