scholarly journals Two-Dimensional Riemann Problem for Scalar Conservation Laws

2002 ◽  
Vol 183 (1) ◽  
pp. 239-261 ◽  
Author(s):  
Wancheng Sheng
Keyword(s):  
Conservation Laws ◽  
Riemann Problem ◽  
Two Dimensional ◽  
Scalar Conservation Laws ◽  
Scalar Conservation

Undercompressive shocks and Riemann problems for scalar conservation laws with non-convex fluxes

1999 ◽  
Vol 129 (4) ◽  
pp. 733-754 ◽  
Author(s):  
Brian Hayes ◽  
Michael Shearer
Keyword(s):  
Initial Data ◽  
Conservation Laws ◽  
Riemann Problem ◽  
Initial Value Problem ◽  
Inflection Point ◽  
Travelling Wave ◽  
Problem Solution ◽  
Scalar Conservation Laws ◽  
Initial Value ◽  
Scalar Conservation

The Riemann initial value problem is studied for scalar conservation laws whose fluxes have a single inflection point. For a regularization consisting of balanced diffusive and dispersive terms, the travelling wave criterion is used to select admissible shocks. In some cases, the Riemann problem solution contains an undercompressive shock. The analysis is illustrated by exploring parameter space for the Buckley–Leverett flux. The boundary of the set of parameters for which there is a physical solution of the Riemann problem for all data is computed. Within the region of acceptable parameters, the solution hasseveral different forms, depending on the initial data; the different forms are illustrated by numerical computations. Qualitatively similar behaviour is observed in Lax–Wendroff approximations of solutions of the Buckley–Leverett equation with no dissipation or dispersion.

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