### Abstract

The visualization of stationary and time-dependent flow is an important and chaltenging
topic in scientific visualization. lts aim is 10 represent transport phenomena govemed
by vector fjelds in an intuitively understandable way. In this paper. we review the use of methods based on partial differential equations (PDEs) to post-process flow datasets for the purpose of visualization. This conneets flow visualization with image processing and mathematical muhi-scale modeIs. We introduce the coneepts of flow operators and scale-space and explain their usc in modeling post processing methods for flow data. 8ascd on this framework, we present several classes of PDE-based visualization methods: anisotropic linear diffusion for stationary flow; transport and diffusion for non-slalionary flow; continuous clustering based on phase-separation; and an algebraic clustering of a matrix-encoded flow operator. We illustrale the presented classes of methods with results obtained from concrete flow applications, using datasets in 20, flows on curved surfaces, and volumetrie 3D fields.

Original language | English |
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Title of host publication | Mathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration |

Editors | T. Möller, B. Hamann, R.D. Russell |

Place of Publication | Berlin |

Publisher | Springer |

Pages | 157-189 |

ISBN (Print) | 978-3-540-25076-0 |

DOIs | |

Publication status | Published - 2009 |

### Publication series

Name | Mathematics and Visualization |
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ISSN (Print) | 1612-3786 |

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## Cite this

Preusser, T., Rumpf, M., & Telea, A. C. (2009). Flow visualization via partial differential equations. In T. Möller, B. Hamann, & R. D. Russell (Eds.),

*Mathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration*(pp. 157-189). (Mathematics and Visualization). Springer. https://doi.org/10.1007/b106657_9