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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Hyperbolic conservation laws with stiff reaction terms of monostable type
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by Haitao Fan PDF
Trans. Amer. Math. Soc. 353 (2001), 4139-4154 Request permission

Abstract:

In this paper the zero reaction limit of the hyperbolic conservation law with stiff source term of monostable type \[ \partial _{t} u + \partial _{x} f(u) = \frac {1}{\epsilon } u(1-u)\] 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
  • 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
  • © Copyright 2001 American Mathematical Society
  • 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
  • MathSciNet review: 1837224