Paul von Dohlen, Patrick D. Miller
Department of Mathematics
William Paterson University
Wayne, NJ 07470, USA
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 representation for the manifold. We use a local Hermite interpolation to define a globally 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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