113 research outputs found
Conditional stability of unstable viscous shock waves in compressible gas dynamics and MHD
Extending our previous work in the strictly parabolic case, we show that a
linearly unstable Lax-type viscous shock solution of a general quasilinear
hyperbolic--parabolic system of conservation laws possesses a
translation-invariant center stable manifold within which it is nonlinearly
orbitally stable with respect to small perturbations, converging
time-asymptotically to a translate of the unperturbed wave. That is, for a
shock with unstable eigenvalues, we establish conditional stability on a
codimension- manifold of initial data, with sharp rates of decay in all
. For , we recover the result of unconditional stability obtained by
Mascia and Zumbrun. The main new difficulty in the hyperbolic--parabolic case
is to construct an invariant manifold in the absence of parabolic smoothing.Comment: 32p
On a Navier-Stokes-Allen-Cahn model with inertial effects
A mathematical model describing the flow of two-phase fluids in a bounded
container is considered under the assumption that the phase transition
process is influenced by inertial effects. The model couples a variant of the
Navier-Stokes system for the velocity with an Allen-Cahn-type equation for
the order parameter relaxed in time in order to introduce inertia.
The resulting model is characterized by second-order material derivatives which
constitute the main difficulty in the mathematical analysis. Actually, in order
to obtain a tractable problem, a viscous relaxation term is included in the
phase equation. The mathematical results consist in existence of weak solutions
in 3D and, under additional assumptions, existence and uniqueness of strong
solutions in 2D. A partial characterization of the long-time behavior of
solutions is also given and in particular some issues related to dissipation of
energy are discussed.Comment: 24 page
Nonlinear Evolution Equations: Analysis and Numerics
The qualitative theory of nonlinear evolution equations is an
important tool for studying the dynamical behavior of systems in
science and technology. A thorough understanding of the complex
behavior of such systems requires detailed analytical and numerical
investigations of the underlying partial differential equations
Absorbing boundary conditions for the Westervelt equation
The focus of this work is on the construction of a family of nonlinear
absorbing boundary conditions for the Westervelt equation in one and two space
dimensions. The principal ingredient used in the design of such conditions is
pseudo-differential calculus. This approach enables to develop high order
boundary conditions in a consistent way which are typically more accurate than
their low order analogs. Under the hypothesis of small initial data, we
establish local well-posedness for the Westervelt equation with the absorbing
boundary conditions. The performed numerical experiments illustrate the
efficiency of the proposed boundary conditions for different regimes of wave
propagation
An energy-based discontinuous Galerkin method for the nonlinear Schr\"odinger equation with wave operator
This work develops an energy-based discontinuous Galerkin (EDG) method for
the nonlinear Schr\"odinger equation with the wave operator. The focus of the
study is on the energy-conserving or energy-dissipating behavior of the method
with some simple mesh-independent numerical fluxes we designed. We establish
error estimates in the energy norm that require careful selection of a test
function for the auxiliary equation involving the time derivative of the
displacement variable. A critical part of the convergence analysis is to
establish the L2 error bounds for the time derivative of the approximation
error in the displacement variable by using the equation that determines its
mean value. Using a specially chosen test function, we show that one can create
a linear system for the time evolution of the unknowns even when dealing with
nonlinear properties in the original problem. Extensive numerical experiments
are provided to demonstrate the optimal convergence of the scheme in the L2
norm with our choices of the numerical flux
Small Collaboration: Numerical Analysis of Electromagnetic Problems (hybrid meeting)
The classical theory of electromagnetism describes the interaction of electrically charged particles through electromagnetic forces, which are carried by the electric and magnetic fields. The propagation of the electromagnetic fields can be described by Maxwell's equations. Solving Maxwell's equations numerically is a challenging problem which appears in many different technical applications. Difficulties arise for instance from material interfaces or if the geometrical features are much larger than or much smaller than a typical wavelength. The spatial discretization needs to combine good geometrical flexibility with a relatively high order of accuracy.
The aim of this small-scale, week-long interactive mini-workshop jointly organized by the University of Duisburg-Essen and the University of Twente, and kindly hosted at the MFO, is to bring together experts in non-standard and mixed finite elements methods with experts in the field of electromagnetism
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