Hyperbolic conservation laws with stiff reaction terms of monostable type

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.