Biology Reference
In-Depth Information
•
Particle methods are not subject to any linear convective stability
condition (CFL condition
98,99
).
27,100
When simulating flows or
waves with mesh-based methods, the CFL condition imposes a
time step limit such that the flow or wave can never travel more
than a certain number of mesh cells per time step. In particle
methods, convection simply amounts to moving the particles with
the local velocity field, and no limit on how far they can move is
imposed as long as their trajectories do not intersect.
•
Thanks to advancements in fast algorithms for
N
-body interactions
(cf. Sec. 6), particle methods are as computationally efficient as
mesh-based methods. In addition, the data structures and opera-
tors in particle simulations can be distributed across many proces-
sors, enabling highly efficient parallel simulations.
32
Since continuum applications of particle methods are far less known
than discrete ones, we focus on deterministic continuous models. For
reasons of simplicity, however, we will not cover the most recent exten-
sions of particle methods to multi-resolution and multi-scale
27-31
prob-
lems using concepts from adaptive mesh refinement,
101
adaptive global
mappings,
101
or wavelets.
100
In continuum particle methods, a particle
p
occupies a certain posi-
tion
x
p
and carries a physical quantity
ω
p
, referred to as its strength. These
particle attributes — strength and location — evolve so as to satisfy the
underlying governing equations in a Lagrangian frame of reference.
27
Simulating a model thus amounts to tracking the dynamics of all
N
com-
putational particles carrying the physical properties of the system being
simulated. The dynamics of the
N
particles are governed by sets of ordi-
nary differential equations (ODEs) that determine the trajectories of the
particles
p
and the evolution of their properties
ω
over time. Thus,
d
d
x
N
p
∑
=
v
()
t
=
K x
(
,
x
;
ωω
,
)
p
=
12
, ,
…
,
N
p
p
q
p
q
t
q
1
(1)
N
d
d
ω
p
∑
=
Fx x
(, ; ,
ωω
)
p
=…
12
, ,
,
N
,
pq p
q
t
q
1
