A nonconvex scalar conservation law with trilinear flux

Hayes, Brian T.; Shearer, Michael

doi:10.1090/qam/1866551

Authors: Brian T. Hayes and Michael Shearer
Journal: Quart. Appl. Math. 59 (2001), 615-635
MSC: Primary 35L65; Secondary 35B25, 35L60
DOI: https://doi.org/10.1090/qam/1866551
MathSciNet review: MR1866551
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Abstract: The focus of this paper is on traveling wave solutions of the equation \[ {u_t} + f{\left ( u \right )_x} = \epsilon {u_{xx}} + {\epsilon ^2}\gamma {u_{xxx}}\], in which the flux function $f$ is trilinear and nonconvex. In particular, it is shown that for combinations of parameters in certain ranges, there are traveling waves that converge as $\epsilon \to 0$ to undercompressive shocks, in which the characteristics pass through the shock. The analysis is based on explicit solutions of the piecewise linear ordinary differential equation satisfied by traveling waves. The analytical results are illustrated by numerical solutions of the Riemann initial value problem, and are compared with corresponding explicit results for the case of a cubic flux function.

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