Nonclassical solutions of hyperbolic conservation laws.
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Abstract
This dissertation studies the nonclassical shock waves which appears as limits of certain type
diffusive-dispersive regularisation to hyperbolic of conservation laws. Such shocks occur very often
when the
ux function lacks the convexity especially when the initial conditions for Riemann problem
belong to different region of convexity. They have negative entropy dissipation. They do not verify
the classical Oleinik entropy criterion. The cubic function is taken as a
ux function. The existence
and uniqueness of such shock waves are studied. They are constructed as limits of traveling-wave
solutions for diffusive-dispersive regularisation. A kinetic relation is introduced to choose a unique
nonclassical solution to the Riemann problem.
The numerical simulations are investigated using a transport-equilibrium scheme to enable computing
the nonclassical solution at the discrete level of kinetic function. The method is composed
of an equilibrium step containing the kinetic relation at any nonclassical shock and a transport step
advancing the discontinuity with time.
Description
M. Sc. University of KwaZulu-Natal, Pietermaritzburg 2015.
Keywords
Differential equations, Hyperbolic., Theses -- Applied mathematics.