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Hyperbolic conservation laws with stiff reaction terms of monostable type


Author: Haitao Fan
Journal: Trans. Amer. Math. Soc. 353 (2001), 4139-4154
MSC (2000): Primary 35L65, 35B40, 35B25
DOI: https://doi.org/10.1090/S0002-9947-01-02761-1
Published electronically: June 1, 2001
MathSciNet review: 1837224
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Abstract:

In this paper the zero reaction limit of the hyperbolic conservation law with stiff source term of monostable type

\begin{displaymath}\partial _{t} u + \partial _{x} f(u) = \frac {1}{\epsilon } u(1-u)\end{displaymath}

is studied. Solutions of Cauchy problems of the above equation with initial value $0\le u_{0}(x)\le 1$ are proved to converge, as $\epsilon \to 0$, to piecewise constant functions. The constants are separated by either shocks determined by the Rankine-Hugoniot jump condition, or a non-shock jump discontinuity that moves with speed $f'(0)$. The analytic tool used is the method of generalized characteristics. Sufficient conditions for the existence and non-existence of traveling waves of the above system with viscosity regularization are given. The reason for the failure to capture the correct shock speed by first order shock capturing schemes when underresolving $\epsilon >0$ is found to originate from the behavior of traveling waves of the above system with viscosity regularization.


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Additional Information

Haitao Fan
Affiliation: Department of Mathematics, Georgetown University, Washington, D.C. 20057
Email: fan@math.georgetown.edu

DOI: https://doi.org/10.1090/S0002-9947-01-02761-1
Keywords: Conservation law with source term, reaction-convection-diffusion equation, zero reaction time limit
Received by editor(s): October 28, 1999
Received by editor(s) in revised form: June 19, 2000
Published electronically: June 1, 2001
Additional Notes: Research supported in part by NSF grant No. DMS 9705732
Article copyright: © Copyright 2001 American Mathematical Society

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