Hyperbolic conservation laws with stiff reaction terms of monostable type

In this paper the zero reaction limit of the hyperbolic conservation law wi th stiff source term of monostable type partial derivative (t)u + partial derivative (x)f(u) = 1/epsilon u(1-u) is studied. Solutions of Cauchy problems of the above equation with initial value 0 less than or equal to u(0) (x) less than or equal to 1 are proved to converge, as epsilon --> 0, to piecewise constant functions. The constan ts are separated by either shocks determined by the Rankine-Hugoniot jump c ondition, or a non-shock jump discontinuity that moves with speed f'(0). Th e analytic tool used is the method of generalized characteristics. Sufficie nt conditions for the existence and non-existence of traveling waves of the above system with viscosity regularization are given. The reason for the f ailure to capture the correct shock speed by first order shock capturing sc hemes when underresolving epsilon >0 is found to originate from the behavio r of traveling waves of the above system with viscosity regularization.