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Flow Visualization via Partial Differential Equations

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Mathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration

Part of the book series: Mathematics and Visualization ((MATHVISUAL))

Summary

The visualization of stationary and time-dependent flow is an important and chal- lenging topic in scientific visualization. Its aim is to represent transport phenomena governed by vector fields in an intuitively understandable way. In this paper, we review the use of meth- ods based on partial differential equations (PDEs) to post-process flow datasets for the purpose of visualization. This connects flow visualization with image processing and mathematical multi-scale models. We introduce the concepts of flow operators and scale-space and explain their use in modeling post processing methods for flow data. Based on this framework, we present several classes of PDE-based visualization methods: anisotropic linear diffusion for stationary flow; transport and diffusion for non-stationary flow; continuous clustering based on phase-separation; and an algebraic clustering of a matrix-encoded flow operator. We illus- trate the presented classes of methods with results obtained from concrete flow applications, using datasets in 2D, flows on curved surfaces, and volumetric 3D fields.

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Author information

Authors and Affiliations

  1. Center for Complex Systems and Visualization, University of Bremen, Universitätsallee 29, 28359, Bremen, Germany

    Tobias Preusser

  2. Institute for Numerical Simulation, Bonn University, Nussallee 15, 53115, Bonn, Germany

    Martin Rumpf

  3. Department of Mathematics and Computer Science, Eindhoven University, Den Dolech 2, Eindhoven, Netherlands

    Alex Telea

Authors
  1. Tobias Preusser
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  2. Martin Rumpf
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  3. Alex Telea
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Editor information

Editors and Affiliations

  1. School of Computing Science, Simon Fraser University, 8888 University Drive, Burnaby, BC, V5A 1S6, Canada

    Torsten Möller  & Robert D. Russell  & 

  2. Department of Computer Science, University of California, Davis, 1 Shields Avenue, Davis, CA, 95616-8562, USA

    Bernd Hamann

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Preusser, T., Rumpf, M., Telea, A. (2009). Flow Visualization via Partial Differential Equations. In: Möller, T., Hamann, B., Russell, R.D. (eds) Mathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/b106657_9

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