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where c i,new denotes the concentration in the i 'th cell after a time step, e.g.
t +
D t , while all other concentration terms are relevant at the time t . The ratio
on the right side is a finite difference representation of the second derivative
@
2 c=@x 2 . There is an entire class of numerical methods, which is founded on
such Finite Differences , whereas the simple algorithm here adopts this meth-
odology for the simulation of diffusive fluxes only. In these equations the cell
concentrations between neighbouring cells are related. Thus one may use
them applying an explicit formula, from which the new concentrations in the
system of cells can be computed:
D D t
D x 2
c i;new ¼ c i þ
ð
c i 1
2 c i þ c 1
Þ
The coefficient, which appears in front of the brackets, the parameter
combination
D x 2 is also known as Neumann-number 3 (abbreviation: Neu ),
(see Holzbecher and Sorek 2005 ). It is the given explicit formula for c i,new
that is computed in the m-file 'diffusion.m' .
D D t
70
60
50
t
40
30
20
10
10
20
30
40
50
60
70
80
90
100
x
Fig. 4.8 The solution of 1D transport in the space (x) - time (t) diagram, visualized by filled
contours
3 John von Neumann (1903-1957), US-American mathematician, physicist, chemist and computer
scientist
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