64 research outputs found
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
Asymptotics of the Euler number of bipartite graphs
We define the Euler number of a bipartite graph on vertices to be the
number of labelings of the vertices with such that the vertices
alternate in being local maxima and local minima. We reformulate the problem of
computing the Euler number of certain subgraphs of the Cartesian product of a
graph with the path in terms of self adjoint operators. The
asymptotic expansion of the Euler number is given in terms of the eigenvalues
of the associated operator. For two classes of graphs, the comb graphs and the
Cartesian product , we numerically solve the eigenvalue problem.Comment: 13 pages, 6 figure, submitted to JCT
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
Exhaustion of Nucleation in a Closed System
We determine the distribution of cluster sizes that emerges from an initial
phase of homogeneous aggregation with conserved total particle density. The
physical ingredients behind the predictions are essentially classical:
Super-critical nuclei are created at the Zeldovich rate, and before the
depletion of monomers is significant, the characteristic cluster size is so
large that the clusters undergo diffusion limited growth. Mathematically, the
distribution of cluster sizes satisfies an advection PDE in "size-space".
During this creation phase, clusters are nucleated and then grow to a size much
larger than the critical size, so nucleation of super-critical clusters at the
Zeldovich rate is represented by an effective boundary condition at zero size.
The advection PDE subject to the effective boundary condition constitutes a
"creation signaling problem" for the evolving distribution of cluster sizes
during the creation era.
Dominant balance arguments applied to the advection signaling problem show
that the characteristic time and cluster size of the creation era are
exponentially large in the initial free-energy barrier against nucleation, G_*.
Specifically, the characteristic time is proportional to exp(2 G_*/ 5 k_B T)
and the characteristic number of monomers in a cluster is proportional to
exp(3G_*/5 k_B T). The exponentially large characteristic time and cluster size
give a-posteriori validation of the mathematical signaling problem. In a short
note, Marchenko obtained these exponentials and the numerical pre-factors, 2/5
and 3/5. Our work adds the actual solution of the kinetic model implied by
these scalings, and the basis for connection to subsequent stages of the
aggregation process after the creation era.Comment: Greatly shortened paper. Section on growth model removed. Added a
section analyzing the error in the solution of the integral equation. Added
reference
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