Computation of Riemann solutions using the Dafermos regularization and continuation

We present a numerical method, based on the Dafermos regularization, for computing a one-parameter family of Riemann solutions of a system of conser- vation 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 prole 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 num- ber 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.