On the $L^2$-moment closure of transport equations: The Cattaneo approximation
T. Hillen - Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada (email)
We consider the moment-closure approach to transport equations
which arise in Mathematical Biology. We show that the negative
$L^2$-norm is an entropy in the sense of thermodynamics, and it
satisfies an $H$-theorem. With an $L^2$-norm minimization
procedure we formally close the moment hierarchy for the first two
moments. The closure leads to semilinear Cattaneo systems, which
are closely related to damped wave equations. In the linear case
we derive estimates for the accuracy of this moment approximation.
The method is used to study reaction-transport models and
transport models for chemosensitive movement. With this method
also order one perturbations of the turning kernel can be treated
- in extension of an earlier theory on the parabolic limit of
transport equations (Hillen and Othmer 2000). Moreover, this
closure procedure allows us to derive appropriate boundary
conditions for the Cattaneo approximation. Finally, we illustrate
that the Cattaneo system is the gradient flow of a weighted
Dirichlet integral and we show simulations.
Keywords: Moment closure, transport equations, Cattaneo system.
Received: June 2003; Revised: April 2004; Available Online: August 2004.
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