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October  2004, 10(4): 965-986. doi: 10.3934/dcds.2004.10.965

Computation of Riemann solutions using the Dafermos regularization and continuation

Stephen Schecter 1, , Bradley J. Plohr 2, and  Dan Marchesin 3,

1. 

Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, United States
2. 

Departments of Mathematics and of Applied Mathematics and Statistics, University at Stony Brook, Stony Brook, NY 11794-3651, United States
3. 

Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, 22460 Rio de Janeiro, RJ, Brazil

Received  September 2002 Revised  December 2003 Published  March 2004

We present a numerical method, based on the Dafermos regularization, for computing a one-parameter family of Riemann solutions of a system of conservation laws. The family is obtained by varying either the left or right state of the Riemann problem. The Riemann solutions are required to have shock waves that satisfy the viscous profile criterion prescribed by the physical model. The system is not required to satisfy strict hyperbolicity or genuine nonlinearity; the left and right states need not be close; and the Riemann solutions may contain an arbitrary number of waves, including composite waves and nonclassical shock waves. The method uses standard continuation software to solve a boundary-value problem in which the left and right states of the Riemann problem appear as parameters. Because the continuation method can proceed around limit point bifurcations, it can sucessfully compute multiple solutions of a particular Riemann problem, including ones that correspond to unstable asymptotic states of the viscous conservation laws.
Keywords: Conservation law, Dafermos regularization, viscous profile, viscous conservation law, numerical method., continuation, Riemann problem.
Mathematics Subject Classification: 35L65, 35L67, 35K5.
Citation: Stephen Schecter, Bradley J. Plohr, Dan Marchesin. Computation of Riemann solutions using the Dafermos regularization and continuation. Discrete and Continuous Dynamical Systems, 2004, 10 (4) : 965-986. doi: 10.3934/dcds.2004.10.965
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