Efficient and Reliable Integration Methods for Particle Tracing in Unsteady Flows on Discrete Meshes

  • Christian Teitzel
  • Roberto Grosso
  • Thomas Ertl
Part of the Eurographics book series (EUROGRAPH)


In real applications the velocity field of a flow is not available in analytical but in discrete form. One goal of this paper is to analyze particle integration methods for discretized data defined on meshes with regard to numerical efficiency and accuracy. Careful error analysis of the particle tracing process relates the error of velocity interpolation in space and time to the error of the numerical integration. Hence, a fast integration routine which provides accuracy similar to that of interpolation is necessary. This leads to a robust integration routine with adaptive step size control and error monitoring. A second aspect of this work is the treatment of stiff problems. Stiffness occurs m flows with strong shear deformations or vorticity. To detect stiffness in a given flow field, the Jacobian of the velocity field is analyzed. Implicit integration methods are used to handle stiff systems of ordinary differential equations.


Velocity Field Stiff System Implicit Algorithm High Order Algorithm Implicit Integration Method 
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  1. 1.
    B. Becker, D. A. Lane, and N. L. Max. Unsteady Flow Volumes. In G.M. Nielson and Silver D., editors, Visualization ’95, pages 329–335, Los Alamitos, CA, 1995. IEEE Computer Society, IEEE Computer Society Press.Google Scholar
  2. 2.
    J. C. Butcher. Coefficients for the study of Runge-Kutta integration processes.J. Austral Math. Soc., 3: 185 – 201, 1963.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    J. C. Butcher. Order, stepsize and stiffness switching.Computing, 44: 209 – 220, 1990.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    D. L. Darmofal and R. Haimes. An Analysis of 3-D Particle Path Integration Algorithms. InProceedings of the 1995 AIAA CFD Meeting, 1995.Google Scholar
  5. 5.
    P. Deuflhard and F. Bornemann.Numerische Mathematik II: Integration gewdhn- licher Differentialgleichungen. Walter de Gruyter, Berlin, New York, 1994.Google Scholar
  6. 6.
    P. Kaps and P. Rentrop.Numerische Mathematik, 33: 55 – 68, 1979.MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    D. N. Kenwright and D. A. Lane. Optimization of Time-Dependent Particle Tracing Using Tetrahedral decomposition. In G. M. Nielson and Silver D., editors,Visualization ’95, pages 321–328, Los Alamitos, CA, 1995. IEEE Computer Society, IEEE Computer Society Press.Google Scholar
  8. 8.
    D. N. Kenwright and G. D. Mallinson. A 3-D Streamline Tracking Algorithm Using Dual Stream Functions. In A. E. Kaufman and G. M. Nielson, editors,Visualization ’92, pages 62–68, Los Alamitos, CA, October 1992. IEEE Computer Society, IEEE Computer Society Press.CrossRefGoogle Scholar
  9. 9.
    William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery.Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press, Cambridge, New York, Victoria, second edition, 1992.Google Scholar
  10. 10.
    D. Stalling and H.-C. Hege. Fast and resolution independent line integral convolution. In Computer Graphics Proceedings, Annual Conference Series, pages 249–256, Los Angeles, California, August 1995. ACM SIGGRAPH, Addison-Wesley Publishing Company, Inc.Google Scholar

Copyright information

© Springer-Verlag/Wein 1997

Authors and Affiliations

  • Christian Teitzel
    • 1
  • Roberto Grosso
    • 1
  • Thomas Ertl
    • 1
  1. 1.Computer Graphics GroupUniversity of ErlangenErlangenGermany

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