IJPAM: Volume 71, No. 2 (2011)


Paul von Dohlen$^1$, Patrick D. Miller$^2$
$^1$Department of Mathematics
William Paterson University
Wayne, NJ 07470, USA
$^2$Department of Mathematical Sciences
Stevens Institute of Technology
Hoboken, NJ 07030, USA

Abstract. Computer simulations that track the flow of particles under the action of a time-dependent velocity field are often used to visualize the dynamics of phase-space transport. When the velocity field has two space variables, it is often sufficient to track the behavior of a curve of initial conditions, rather than a cloud of particles. Tracking a closed curve of initial particles can be used to accurately follow the evolution of a closed region in the phase space. The work presented here investigates methods for performing particle-tracking simulations that are 1) more rigorous with respect to accuracy and 2) computationally more efficient in the way in which the manifold (curve) is represented. A novel feature is to use the linear variational flow to track the first derivatives of the manifold, making it possible to construct a $C^1$ representation for the manifold. We use a local Hermite interpolation to define a globally $C^1$ curve. Error estimates for the interpolating polynomials are used as criteria to determine where additional nodes are needed (refinement) and where nodes can be removed (coarsening).

Received: June 20, 2011

AMS Subject Classification: 37M05, 65D99

Key Words and Phrases: curve tracking, flow simulation

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2011
Volume: 71
Issue: 2