Particleclaw
Particleclaw is a characteristic particle method for scalar conservation laws.
Instead of considering a fixed grid, particles follow the the characteristic curves.
This yields an accurate approximation while the solution is smooth. On the occurrence
of shocks, particles are merged. Similarly, into large gaps, particles are inserted.
This particle management mimics the behavior of characteristic curves, and shocks
and rarefaction waves arise naturally. Shocks can be located accurately, without
being explicitly tracked. Our method is designed to perform the merging and insertion
in such a fashion that correct shock speeds are obtained. It is based on local similarity
solutions between neighboring particles, which yield an explicit definition of area.
Thus, the method conserves area exactly.
Program Code
Below is the method as a fully vectorized Matlab code.
Current version: 1.0
The Method
Particles are moved according to their characteristic equations. Between two neighboring
particles, an interpolation is defined, such that the evolution of the interpolant is an
actual solution of the conservation law. As shown in the left figure below, the
interpolating solution is defined as either coming from a discontinuity (rarefaction wave),
or going into a discontinuity (compression wave).
If two particles depart, a new particle is inserted once their distance becomes too large.
As shown in the middle figure above, the newly inserted particle is placed on the
interpolation. At the moment two particles share the same position (but different function
values), they are merged into a single particle. As shown in the right figure above, the
merged particle is placed, so that the total area in conserved. It can be shown that the
new particle's function value lies between the merged particles' function values, hence the
method is total variation diminishing, and shocks move at correct speeds.
Examples
Below examples show the particle method in comparison with
Clawpack solutions
of a similar resolution.
Quartic flux function with smooth initial data 

Videos:
animated gif (500kB),
divx codec (1MB),
msvc codec (4.3MB)

While the solution is smooth, it is represented exactly by the particles. After shocks have
occurred, the particles near jumps approximate the solution.



Burgers' equation Riemann problem 

Videos:
animated gif (500kB),
divx codec (1MB),
msvc codec (3.6MB)

A shock is always represented by one particle and it stretches one interval to the left and
one to the right. Any particle arising from a merge indicates a shock. A simple postprocessing
step can recover a sharp shock. With this step, for the simple case of a Riemann problem, the
method reproduces the exact solution.


BuckleyLeverett equation 

Videos:
animated gif (500kB),
divx codec (1MB),
msvc codec (4.4MB)

Nonconvex flux functions appear in many practical applications, such as the
BuckleyLeverett equation in oil recovery. In the particle method, they are
treated using special particles, that are carried at inflection points of the
flux function.


Flow over obstacle 

Videos:
animated gif (750kB),
divx codec (1.5MB),
msvc codec (6.4MB)

The method is based on a splitting approach:
Source terms are incorporated fully into the characteristic formulation.
However, particle management is based on the similarity solutions of the
sourcefree equation.



Current Research
The particle method preserves area exactly by construction. In addition, it gives a direct
control on entropy, namely entropy is only decreased when particles are merged. We are currently
applying the method to a version of Burgers' equation with a global source term. The physical
questions arising with this equation require an exact conservation of area, and an exact
control on the energy (which is an entropy here). Traditional numerical schemes do not allow
a controlled treatment of the energy, due to the presence of numerical dissipation.
Related Publications
Y. Farjoun, B. Seibold,
An exact particle method for scalar
conservation laws and its application to stiff reaction kinetics
Meshfree methods for Partial Differential Equations V,
Lecture Notes in Computational Science and Engineering, Vol. 79,
Griebel, M. and Schweitzer, M.A. (ed.), Springer, 2011, pp. 105124.

Y. Farjoun, B. Seibold,
A rarefactiontracking method for
conservation laws,
J. Eng. Math, Vol. 66, No. 1, 2010, pp. 237251.

Y. Farjoun, B. Seibold,
An exactly conservative
particle method for one dimensional scalar conservation laws,
J. Comput. Phys., Vol. 228, No. 14, 2009, pp. 52985315.

Y. Farjoun, B. Seibold,
Solving one dimensional scalar
conservation laws by particle management,
Meshfree methods for Partial Differential Equations IV,
Lecture Notes in Computational Science and Engineering, Vol. 65,
Griebel, M. and Schweitzer, M.A. (ed.), Springer, 2009, pp. 95109.

B. Seibold,
MMatrices
in meshless finite difference methods,
PhD thesis, Department of Mathematics, University of Kaiserslautern, 2006.
