Biology Reference
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where
v
p
(
t
) is the velocity of particle
p
at time
t
. The dynamics of
the particles are completely defined by the functions
K
and
F
,
which represent the model being simulated. In pure particle methods,
K
and
F
emerge from integral approximations of differential opera-
tors (cf. Sec. 5.2.1); in hybrid particle-mesh (PM) methods, they
entail solutions of field equations that are discretized on a superim-
posed mesh (cf. Sec. 5.2.2). The sums on the right-hand side of
Eq. (1) correspond to quadrature
84
(numerical integration) of some
functions.
In order to situate continuum particle methods on the map of
numerical analysis, we consider the different strategies to numerically
solve a differential equation as outlined in Fig. 4 for the example of a sim-
ple PDE, the Poisson equation, which we wish to solve for the intensive
field quantity
u
. One way consists of discretizing the equation onto a
computational mesh with resolution
h
using FD, FE, or FV, and then
numerically solving the resulting system
A
u
f
of linear algebraic equa-
tions. The discretization needs to be done consistently in order to ensure
that the discretized equations model the same system as the original
PDE, and the numerical solution of the resulting linear system is subject
to stability criteria. An alternative route is to solve the PDE analytically
=
Fig. 4.
Strategies to numerically solve a differential equation, illustrated on the
example of the two-dimensional (2D) Poisson equation: (1) discretization of the
PDE on a mesh with resolution
h
, followed by numerical solution of the discretized
equations for the intensive property
u
; or (2) integral solution for the extensive prop-
erty that is numerically approximated by quadrature.
