8,113 research outputs found
An exactly conservative particle method for one dimensional scalar conservation laws
A particle scheme for scalar conservation laws in one space dimension is
presented. Particles representing the solution are moved according to their
characteristic velocities. Particle interaction is resolved locally, satisfying
exact conservation of area. Shocks stay sharp and propagate at correct speeds,
while rarefaction waves are created where appropriate. The method is variation
diminishing, entropy decreasing, exactly conservative, and has no numerical
dissipation away from shocks. Solutions, including the location of shocks, are
approximated with second order accuracy. Source terms can be included. The
method is compared to CLAWPACK in various examples, and found to yield a
comparable or better accuracy for similar resolutions.Comment: 29 pages, 21 figure
A rarefaction-tracking method for hyperbolic conservation laws
We present a numerical method for scalar conservation laws in one space
dimension. The solution is approximated by local similarity solutions. While
many commonly used approaches are based on shocks, the presented method uses
rarefaction and compression waves. The solution is represented by particles
that carry function values and move according to the method of characteristics.
Between two neighboring particles, an interpolation is defined by an analytical
similarity solution of the conservation law. An interaction of particles
represents a collision of characteristics. The resulting shock is resolved by
merging particles so that the total area under the function is conserved. The
method is variation diminishing, nevertheless, it has no numerical dissipation
away from shocks. Although shocks are not explicitly tracked, they can be
located accurately. We present numerical examples, and outline specific
applications and extensions of the approach.Comment: 21 pages, 7 figures. Similarity 2008 conference proceeding
A characteristic particle method for traffic flow simulations on highway networks
A characteristic particle method for the simulation of first order
macroscopic traffic models on road networks is presented. The approach is based
on the method "particleclaw", which solves scalar one dimensional hyperbolic
conservations laws exactly, except for a small error right around shocks. The
method is generalized to nonlinear network flows, where particle approximations
on the edges are suitably coupled together at the network nodes. It is
demonstrated in numerical examples that the resulting particle method can
approximate traffic jams accurately, while only devoting a few degrees of
freedom to each edge of the network.Comment: 15 pages, 5 figures. Accepted to the proceedings of the Sixth
International Workshop Meshfree Methods for PDE 201
Solving One Dimensional Scalar Conservation Laws by Particle Management
We present a meshfree numerical solver for scalar conservation laws in one
space dimension. Points representing the solution are moved according to their
characteristic velocities. Particle interaction is resolved by purely local
particle management. Since no global remeshing is required, shocks stay sharp
and propagate at the correct speed, while rarefaction waves are created where
appropriate. The method is TVD, entropy decreasing, exactly conservative, and
has no numerical dissipation. Difficulties involving transonic points do not
occur, however inflection points of the flux function pose a slight challenge,
which can be overcome by a special treatment. Away from shocks the method is
second order accurate, while shocks are resolved with first order accuracy. A
postprocessing step can recover the second order accuracy. The method is
compared to CLAWPACK in test cases and is found to yield an increase in
accuracy for comparable resolutions.Comment: 15 pages, 6 figures. Submitted to proceedings of the Fourth
International Workshop Meshfree Methods for Partial Differential Equation
An exact particle method for scalar conservation laws and its application to stiff reaction kinetics
An "exact" method for scalar one-dimensional hyperbolic conservation laws is
presented. The approach is based on the evolution of shock particles, separated
by local similarity solutions. The numerical solution is defined everywhere,
and is as accurate as the applied ODE solver. Furthermore, the method is
extended to stiff balance laws. A special correction approach yields a method
that evolves detonation waves at correct velocities, without resolving their
internal dynamics. The particle approach is compared to a classical finite
volume method in terms of numerical accuracy, both for conservation laws and
for an application in reaction kinetics.Comment: 14 page, 7 figures, presented in the Fifth International Workshop on
Meshfree Methods for Partial Differential Equation
Equal-area method for scalar conservation laws
We study one-dimensional conservation law. We develop a simple numerical
method for computing the unique entropy admissible weak solution to the initial
problem. The method basis on the equal-area principle and gives the solution
for given time directly.Comment: 10 pages, 7 figure
Astrophysical Weighted Particle Magnetohydrodynamics
This paper presents applications of weighted meshless scheme for conservation
laws to the Euler equations and the equations of ideal magnetohydrodynamics.
The divergence constraint of the latter is maintained to the truncation error
by a new meshless divergence cleaning procedure. The physics of the interaction
between the particles is described by an one-dimensional Riemann problem in a
moving frame. As a result, necessary diffusion which is required to treat
dissipative processes is added automatically. As a result, our scheme has no
free parameters that controls the physics of inter-particle interaction, with
the exception of the number of the interacting neighbours which control the
resolution and accuracy. The resulting equations have the form similar to SPH
equations, and therefore existing SPH codes can be used to implement the
weighed particle scheme. The scheme is validated in several hydrodynamic and
MHD test cases. In particular, we demonstrate for the first time the ability of
a meshless MHD scheme to model magneto-rotational instability in accretion
disks.Comment: 27 pages, 24 figures, 1 column, submitted to MNRAS, hi-res version
can be obtained at http://www.strw.leidenuniv.nl/~egaburov/wpmhd.pd
Between Laws and Models: Some Philosophical Morals of Lagrangian Mechanics
I extract some philosophical morals from some aspects of Lagrangian
mechanics. (A companion paper will present similar morals from Hamiltonian
mechanics and Hamilton-Jacobi theory.) One main moral concerns methodology:
Lagrangian mechanics provides a level of description of phenomena which has
been largely ignored by philosophers, since it falls between their accustomed
levels--``laws of nature'' and ``models''. Another main moral concerns
ontology: the ontology of Lagrangian mechanics is both more subtle and more
problematic than philosophers often realize.
The treatment of Lagrangian mechanics provides an introduction to the subject
for philosophers, and is technically elementary. In particular, it is confined
to systems with a finite number of degrees of freedom, and for the most part
eschews modern geometry. But it includes a presentation of Routhian reduction
and of Noether's ``first theorem''.Comment: 106 pages, no figure
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