45 research outputs found
Finite difference schemes for the symmetric Keyfitz-Kranzer system
We are concerned with the convergence of numerical schemes for the initial
value problem associated to the Keyfitz-Kranzer system of equations. This
system is a toy model for several important models such as in elasticity
theory, magnetohydrodynamics, and enhanced oil recovery. In this paper we prove
the convergence of three difference schemes. Two of these schemes is shown to
converge to the unique entropy solution. Finally, the convergence is
illustrated by several examples.Comment: 31 page
Existence and stability of viscoelastic shock profiles
We investigate existence and stability of viscoelastic shock profiles for a
class of planar models including the incompressible shear case studied by
Antman and Malek-Madani. We establish that the resulting equations fall into
the class of symmetrizable hyperbolic--parabolic systems, hence spectral
stability implies linearized and nonlinear stability with sharp rates of decay.
The new contributions are treatment of the compressible case, formulation of a
rigorous nonlinear stability theory, including verification of stability of
small-amplitude Lax shocks, and the systematic incorporation in our
investigations of numerical Evans function computations determining stability
of large-amplitude and or nonclassical type shock profiles.Comment: 43 pages, 12 figure
Spectral stability of noncharacteristic isentropic Navier-Stokes boundary layers
Building on work of Barker, Humpherys, Lafitte, Rudd, and Zumbrun in the
shock wave case, we study stability of compressive, or "shock-like", boundary
layers of the isentropic compressible Navier-Stokes equations with gamma-law
pressure by a combination of asymptotic ODE estimates and numerical Evans
function computations. Our results indicate stability for gamma in the interval
[1, 3] for all compressive boundary-layers, independent of amplitude, save for
inflow layers in the characteristic limit (not treated). Expansive inflow
boundary-layers have been shown to be stable for all amplitudes by Matsumura
and Nishihara using energy estimates. Besides the parameter of amplitude
appearing in the shock case, the boundary-layer case features an additional
parameter measuring displacement of the background profile, which greatly
complicates the resulting case structure. Moreover, inflow boundary layers turn
out to have quite delicate stability in both large-displacement and
large-amplitude limits, necessitating the additional use of a mod-two stability
index studied earlier by Serre and Zumbrun in order to decide stability
L^2 stability estimates for shock solutions of scalar conservation laws using the relative entropy method
We consider scalar nonviscous conservation laws with strictly convex flux in
one spatial dimension, and we investigate the behavior of bounded L^2
perturbations of shock wave solutions to the Riemann problem using the relative
entropy method. We show that up to a time-dependent translation of the shock,
the L^2 norm of a perturbed solution relative to the shock wave is bounded
above by the L^2 norm of the initial perturbation.Comment: 17 page
Global solutions to the three-dimensional full compressible magnetohydrodynamic flows
The equations of the three-dimensional viscous, compressible, and heat
conducting magnetohydrodynamic flows are considered in a bounded domain. The
viscosity coefficients and heat conductivity can depend on the temperature. A
solution to the initial-boundary value problem is constructed through an
approximation scheme and a weak convergence method. The existence of a global
variational weak solution to the three-dimensional full magnetohydrodynamic
equations with large data is established
Global Existence and Large-Time Behavior of Solutions to the Three-Dimensional Equations of Compressible Magnetohydrodynamic Flows
The three-dimensional equations of compressible magnetohydrodynamic
isentropic flows are considered. An initial-boundary value problem is studied
in a bounded domain with large data. The existence and large-time behavior of
global weak solutions are established through a three-level approximation,
energy estimates, and weak convergence for the adiabatic exponent
and constant viscosity coefficients