961 research outputs found
Pointwise error estimates for relaxation approximations to conservation laws
We obtain sharp pointwise error estimates for relaxation approximation to scalar
conservation laws with piecewise smooth solutions. We first prove that the first-order partial derivatives
for the perturbation solutions are uniformly upper bounded (the so-called Lip+ stability). A
one-sided interpolation inequality between classical L1 error estimates and Lip+ stability bounds
enables us to convert a global L1 result into a (nonoptimal) local estimate. Optimal error bounds on
the weighted error then follow from the maximum principle for weakly coupled hyperbolic systems.
The main difficulties in obtaining the Lip+ stability and the optimal pointwise errors are how to
construct appropriate “difference functions” so that the maximum principle can be applied
Pointwise Green's function bounds and stability of relaxation shocks
We establish sharp pointwise Green's function bounds and consequent
linearized and nonlinear stability for smooth traveling front solutions, or
relaxation shocks, of general hyperbolic relaxation systems of dissipative
type, under the necessary assumptions ([G,Z.1,Z.4]) of spectral stability,
i.e., stable point spectrum of the linearized operator about the wave, and
hyperbolic stability of the corresponding ideal shock of the associated
equilibrium system. This yields, in particular, nonlinear stability of weak
relaxation shocks of the discrete kinetic Jin--Xin and Broadwell models. The
techniques of this paper should have further application in the closely related
case of traveling waves of systems with partial viscosity, for example in
compressible gas dynamics or MHD.Comment: 120 pages. Changes since original submission. Corrected typos, esp.
energy estimates of Section 7, corrected bad forward references, expanded
Remark 1.17, end of introductio
Three-points interfacial quadrature for geometrical source terms on nonuniform grids
International audienceThis paper deals with numerical (finite volume) approximations, on nonuniform meshes, for ordinary differential equations with parameter-dependent fields. Appropriate discretizations are constructed over the space of parameters, in order to guarantee the consistency in presence of variable cells' size, for which -error estimates, , are proven. Besides, a suitable notion of (weak) regularity for nonuniform meshes is introduced in the most general case, to compensate possibly reduced consistency conditions, and the optimality of the convergence rates with respect to the regularity assumptions on the problem's data is precisely discussed. This analysis attempts to provide a basic theoretical framework for the numerical simulation on unstructured grids (also generated by adaptive algorithms) of a wide class of mathematical models for real systems (geophysical flows, biological and chemical processes, population dynamics)
Existence and sharp localization in velocity of small-amplitude Boltzmann shocks
Using a weighted -contraction mapping argument based on the macro-micro
decomposition of Liu and Yu, we give an elementary proof of existence, with
sharp rates of decay and distance from the Chapman--Enskog approximation, of
small-amplitude shock profiles of the Boltzmann equation with hard-sphere
potential, recovering and slightly sharpening results obtained by Caflisch and
Nicolaenko using different techniques. A key technical point in both analyses
is that the linearized collision operator is negative definite on its
range, not only in the standard square-root Maxwellian weighted norm for which
it is self-adjoint, but also in norms with nearby weights. Exploring this issue
further, we show that is negative definite on its range in a much wider
class of norms including norms with weights asymptotic nearly to a full
Maxwellian rather than its square root. This yields sharp localization in
velocity at near-Maxwellian rate, rather than the square-root rate obtained in
previous analyse
A Space-time Smooth Artificial Viscosity Method For Nonlinear Conservation Laws
We introduce a new methodology for adding localized, space-time smooth,
artificial viscosity to nonlinear systems of conservation laws which propagate
shock waves, rarefactions, and contact discontinuities, which we call the
-method. We shall focus our attention on the compressible Euler equations in
one space dimension. The novel feature of our approach involves the coupling of
a linear scalar reaction-diffusion equation to our system of conservation laws,
whose solution is the coefficient to an additional (and artificial)
term added to the flux, which determines the location, localization, and
strength of the artificial viscosity. Near shock discontinuities, is
large and localized, and transitions smoothly in space-time to zero away from
discontinuities. Our approach is a provably convergent, spacetime-regularized
variant of the original idea of Richtmeyer and Von Neumann, and is provided at
the level of the PDE, thus allowing a host of numerical discretization schemes
to be employed. We demonstrate the effectiveness of the -method with three
different numerical implementations and apply these to a collection of
classical problems: the Sod shock-tube, the Osher-Shu shock-tube, the
Woodward-Colella blast wave and the Leblanc shock-tube. First, we use a
classical continuous finite-element implementation using second-order
discretization in both space and time, FEM-C. Second, we use a simplified WENO
scheme within our -method framework, WENO-C. Third, we use WENO with the
Lax-Friedrichs flux together with the -equation, and call this WENO-LF-C.
All three schemes yield higher-order discretization strategies, which provide
sharp shock resolution with minimal overshoot and noise, and compare well with
higher-order WENO schemes that employ approximate Riemann solvers,
outperforming them for the difficult Leblanc shock tube experiment.Comment: 34 pages, 27 figure
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