The Riemann problem for a system of conservation laws of mixed type with a cubic nonlinearity

Abstract

Using the viscosity-capillarity admissibility criterion for shock waves, we solve the Riemann problem for the system of conservation lawswhere σ(u) = u3 − u. This system is hyperbolic at (u, v) unless . We find that the Riemann problem has a unique solution for all data in the hyperbolic regions, except for a range of data in the same phase (i.e. on the same side of the nonhyperbolic strip). In the nonunique cases, there are exactly two admissible solutions. The analysis is based upon a formula describing all saddle-to-saddle heteroclinic orbits for a family of cubic vector fields in the plane.

Convergent and conservative schemes for nonclassical solutions based on kinetic relations. I

“…Nonclassical solutions have the distinctive feature of being dynamically driven by small-scale effects such as diffusion, dispersion, and other high-order phenomena. Their selection requires an additional jump relation, called a kinetic relation, and introduced in the context of phase transition dynamics by Slemrod [35,36,13], Truskinovsky [37,38], Abeyaratne and Knowles [1,2], LeFloch [23], and Shearer [33,34], and developed in the more general context of nonlinear hyperbolic systems of conservation laws by LeFloch and collaborators [15]- [17], [3]- [5], [28]- [30], and [27]. See [24] for a review.…”

High-Order Entropy-Conservative Schemes and Kinetic Relations for van der Waals Fluids

“…Recently, after the works by James [17], Truskinovsky [32,33], Slemrod [26], Abeyaratne and Knowles [1,2], LeFloch [19][20][21], Shearer et al [16,24], and Hayes and LeFloch [13][14][15], it became clear that nonstationary, subsonic phase interfaces (in the hyperbolic-elliptic regime) and nonclassical shock waves (in the hyperbolic, but not genuinely nonlinear regime) should be included when solving the Riemann problem. Indeed, such waves are admissible in the sense that they do arise in viscosity-capillarity limits of the system.…”

A New Entropic Riemann Solver of Conservation Law of Mixed Type Including Ziti’s δ-Method with some Experimental Tests

“…6 -= P -= 0.7 -= −0.7 R ; 6 9 = P 9 = −0.327 9 = −0.327 R (21) The system (1) change of type (hyperbolic-elliptic) following the sign of : The elliptic region is the domain ∈ ) $ ; ) ( , the hyperbolic region is outside ∈ -; ) $ ∪ ) ( ; 9 , figure 4. …”

“…Recently, after the works by James [15], Truskinovsky [25,26], Slemrod [23], Abeyaratne and Knowles [1,2], Le Floch [17][18][19], Shearer [14,21], and Hayes and LeFloch [11][12][13], it became clear that nonstationary, subsonic phase interfaces (in the hyperbolicelliptic regime) and nonclassical shock waves (in the hyperbolic, but not genuinely nonlinear regime) should be included when solving the Riemann problem.…”