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A Uniformly Converging Scheme for Fractal Conservation Laws

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Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 77))

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Abstract

The fractal conservation law \(\partial _t u + \partial _x(f(u))+(-\varDelta )^{{\alpha }/2}u=0\) changes characteristics as \({\alpha }\rightarrow 2\) from non-local and weakly diffusive to local and strongly diffusive. In this paper we present a corrected finite difference quadrature method for \((-\varDelta )^{{\alpha }/2}\) with \({\alpha }\in [0,2]\), combined with usual finite volume methods for the hyperbolic term, that automatically adjusts to this change and is uniformly convergent with respect to \({\alpha }\in [\eta ,2]\) for any \(\eta >0\). We provide numerical results which illustrate this asymptotic-preserving property as well as the non-uniformity of previous finite difference or finite volume type of methods.

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Authors and Affiliations

  1. School of Mathematical Sciences, Monash University, Melbourne, VIC, 3800, Australia

    Jérôme Droniou

  2. Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491, Trondheim, Norway

    Espen R. Jakobsen

Authors
  1. Jérôme Droniou
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  2. Espen R. Jakobsen
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Corresponding author

Correspondence to Jérôme Droniou .

Editor information

Editors and Affiliations

  1. Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany

    Jürgen Fuhrmann

  2. Institute Comp. Applied Mathematics, University of Münster Center for Nonlinear Sciences (CeNoS ), Münster, Germany

    Mario Ohlberger

  3. Inst. Appl. Analysis and Num. Simulation, University of Stuttgart, Stuttgart, Germany

    Christian Rohde

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Droniou, J., Jakobsen, E.R. (2014). A Uniformly Converging Scheme for Fractal Conservation Laws. In: Fuhrmann, J., Ohlberger, M., Rohde, C. (eds) Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects. Springer Proceedings in Mathematics & Statistics, vol 77. Springer, Cham. https://doi.org/10.1007/978-3-319-05684-5_22

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