Waves and nonclassical shocks in a scalar conservation law with nonconvex flux

Abstract

The propagation of waves in the nonlinear equation¶¶\(\frac{\partial V}{\partial t} + \left(\Gamma V + \frac{\Lambda V^{2}}{2}\right) \frac{\partial V}{\partial\hat{x}} = \hat{\nu}\frac{\partial^{2}V}{\partial\hat{x}^{2}} + \hat{\beta}\frac{\partial^{3}V}{\partial\hat{x}^{3}}, \quad \hat{\nu}>0, \quad\hat{\beta}\Lambda>0,\)¶¶generates undercompressive shocks in the hyperbolic limit with dispersion and dissipation balanced. These shocks are undercompressive in type and the diversity of phenomena possible is illustrated for three different initial conditions: a propagating shock through a wave fan, a square pulse and a periodic pulse constructed from constant states. The rich variety of wave phenomena exhibited:- shocks which emanate rather than absorb characteristics, compound shocks and shock fan combinations produce waves that have no counterpart in classical shock theories. A mechanism for the formation of a nonclassical shock from a classical shock by wavefan interaction is presented.