# IJPAM: Volume 71, No. 2 (2011)

**AN ERROR-BASED CURVE TRACKING ALGORITHM**

FOR 2-DIMENSIONAL FLOWS

FOR 2-DIMENSIONAL FLOWS

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

**Download paper from here.**

**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