Shock wave admissibility for quadratic conservation laws

Thesis/Dissertation · Wed Jan 01 04:00:00 EST 1992

OSTI ID:7152811

Canic, S

This dissertation presents a new approach to analysing admissible shock wave solutions for the systems of quadratic conservation laws, i.e., systems of the form U[sub t] + F(U)[sub x] = 0, x [element of] R, t > 0, U(x,t) [element of] R[sup 2], where the flux function F is a quadratic polynomial in U. The author studies systems that change type from hyperbolic to elliptic. The viscosity admissibility criterion is used to study admissibility of shock waves. This criterion requires for a shock wave to be admissible that its end states are joined by an orbit for an associated dynamical system. The author analyzes the stability of shock waves with respect to the change of the state on the left to the shock wave, and the shock speed. The stability analysis for admissible shock waves reduces to the bifurcation analysis of the three-parameter family of planar, quadratic dynamical systems. The family of all such dynamical systems is parameterized by the fundamental wave manifold W. The region of W comprising admissible shock waves is bounded by the loci of structurally unstable dynamical systems. Explicit formulae are presented for the admissibility boundaries that include the loci associated with saddle-node, Hopf, and straight line heteroclinic connections. Using the Melnikov's integral analysis, the tangent manifold to the homoclinic part of the admissibility boundary is calculated at the Bogdanov-Takens points of W. The heteroclinic loci are explored corresponding to the curved connecting orbits and the true homoclinic locus. The author shows the region of admissible waves for a generic, two-dimensional slice of the fundamental wave manifold, and compares it with the set of shock points that comply with the classical Lax criterion. The subregions are presented where the two criteria are not equivalent.

• Research Organization: State Univ. of New York, Stony Brook, NY (United States)
• OSTI ID: 7152811
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
990200* -- Mathematics & Computers
CONSERVATION LAWS
FUNCTIONS
JACOBIAN FUNCTION
MATHEMATICAL MANIFOLDS
MATHEMATICAL MODELS
SHOCK WAVES