1,446 research outputs found
Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. Part 1: The ODE connection and its implications for algorithm development in computational fluid dynamics
Spurious stable as well as unstable steady state numerical solutions, spurious asymptotic numerical solutions of higher period, and even stable chaotic behavior can occur when finite difference methods are used to solve nonlinear differential equations (DE) numerically. The occurrence of spurious asymptotes is independent of whether the DE possesses a unique steady state or has additional periodic solutions and/or exhibits chaotic phenomena. The form of the nonlinear DEs and the type of numerical schemes are the determining factor. In addition, the occurrence of spurious steady states is not restricted to the time steps that are beyond the linearized stability limit of the scheme. In many instances, it can occur below the linearized stability limit. Therefore, it is essential for practitioners in computational sciences to be knowledgeable about the dynamical behavior of finite difference methods for nonlinear scalar DEs before the actual application of these methods to practical computations. It is also important to change the traditional way of thinking and practices when dealing with genuinely nonlinear problems. In the past, spurious asymptotes were observed in numerical computations but tended to be ignored because they all were assumed to lie beyond the linearized stability limits of the time step parameter delta t. As can be seen from the study, bifurcations to and from spurious asymptotic solutions and transitions to computational instability not only are highly scheme dependent and problem dependent, but also initial data and boundary condition dependent, and not limited to time steps that are beyond the linearized stability limit
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
Multidimensional Conservation Laws: Overview, Problems, and Perspective
Some of recent important developments are overviewed, several longstanding
open problems are discussed, and a perspective is presented for the
mathematical theory of multidimensional conservation laws. Some basic features
and phenomena of multidimensional hyperbolic conservation laws are revealed,
and some samples of multidimensional systems/models and related important
problems are presented and analyzed with emphasis on the prototypes that have
been solved or may be expected to be solved rigorously at least for some cases.
In particular, multidimensional steady supersonic problems and transonic
problems, shock reflection-diffraction problems, and related effective
nonlinear approaches are analyzed. A theory of divergence-measure vector fields
and related analytical frameworks for the analysis of entropy solutions are
discussed.Comment: 43 pages, 3 figure
Boundary Stabilization of Quasilinear Maxwell Equations
We investigate an initial-boundary value problem for a quasilinear
nonhomogeneous, anisotropic Maxwell system subject to an absorbing boundary
condition of Silver & M\"uller type in a smooth, bounded, strictly star-shaped
domain of . Imposing usual smallness assumptions in addition to
standard regularity and compatibility conditions, a nonlinear stabilizability
inequality is obtained by showing nonlinear dissipativity and
observability-like estimates enhanced by an intricate regularity analysis. With
the stabilizability inequality at hand, the classic nonlinear barrier method is
employed to prove that small initial data admit unique classical solutions that
exist globally and decay to zero at an exponential rate. Our approach is based
on a recently established local well-posedness theory in a class of
-valued functions.Comment: 22 page
SBV Regularity for Genuinely Nonlinear, Strictly Hyperbolic Systems of Conservation Laws in one space dimension
We prove that if is the entropy
solution to a strictly hyperbolic system of conservation laws with
genuinely nonlinear characteristic fields then up to a
countable set of times the function is in
, i.e. its distributional derivative is a measure with no
Cantorian part.
The proof is based on the decomposition of into waves belonging to
the characteristic families and the balance
of the continuous/jump part of the measures in regions bounded by
characteristics. To this aim, a new interaction measure \mu_{i,\jump} is
introduced, controlling the creation of atoms in the measure .
The main argument of the proof is that for all where the Cantorian part
of is not 0, either the Glimm functional has a downward jump, or there is
a cancellation of waves or the measure is positive
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