Abstract
Diffusion processes can take place in presence of chemical reactions. Additionally, diffusion processes can also occur in a system that has a complex geometry. This paper shows a method how to treat such kind of diffusion processes numerically.
Introduction
The reactive diffusion equation is a system
of (in general nonlinear) partial differential equations and describes the time
evolution of mixable several chemical species which can also react to new
chemical species. For distinct
chemical species it has the form:
While denotes
the index of the chemical species,
is their
concentration,
is their
diffusion flux vector and
is their
gain and loss due to chemical reactions. Since the diffusion flux and the
source term
is a
nonlinear function in terms of the concentrations of all other chemical
species, equation (1) cannot be solved analytically. This equation must be
discretized by a suitable discretization method such that a difference
equation can be obtained. The solution of this discretized equation is less accurate
than the exact solution, but the more discrete elements (e.g. lattices; denoted
by
) are used
for discretization the more accurate is the numerical solution. Moreover, the
discrepancy between exact and numerical solution depends on the geometry of the
discrete elements that are chosen to discretize the region in which the reactive
diffusion equations are solved (denoted by
).
Especially in complex geometries (e.g. porous media) it becomes difficult to
fit the geometry of the discrete elements to the geometry of the solution
region
. In this
cases there must be used more complicated models (Tompson 1992) than usual
discretization methods. Typical methods for solving the reactive diffusion
equation is the finite element method (FEM) (John 2008). Another approach for
the solution of reactive diffusion equations are used in (Liu 2005).
Description of the numerical method
In general, the numerical solution of (1)
requires that is
decomposed into
discrete
elements as:
The decomposition (2) is equivalent to the ansatz
for some coefficients and the
ansatz functions
that can
also be assumed species dependent. When giving some boundary and initial
conditions, substituting (3) into (1) and integrate over the whole region
and the
time interval
, the
discrete reactive diffusion equation can be solved by iteration such that the
set of coefficients
can be
obtained. In this paper it is illustrated, how complex geometries can be discretized
by decomposing
into
elements of a mathematical graph. Mathematical graphs consist on vertices
, edges
, but also
areas
and
volumes
that are
enclosed by the edges (or areas) of the graph. For simplicity, the time
discretization is performed by setting
for some
time scale
; to make
the ansatz functions time independent, it is obvious to make the coefficients
dependent
on time. Another simplification is the assumption of species independent ansatz
functions
(species-dependent
information is given by the coefficients
). To
characterize the ansatz functions as graph elements, the decomposition (3) has
the form:
These ansatz functions have the
value 1 when
lies on
the corresponding discrete element and 0 otherwise.
When (4) is substituted into (1), every
ansatz function has to be derived by the coordinates. Coordinate derivatives
can be performed with the aid of the Gaussian law , where
is the
unit normal vector corresponding to the surface element
and
is an
infinitesimal 3-dimensional region around the discrete element
. For
example,
corresponds
to an infinitesimal 3-dimensional ball around the vertex
. The
volume integral can be approximated to
(an
analogous approximation holds for the surface integral). To obtain the discrete
reactive diffusion equation, products of the ansatz functions must be
integrated over the entire region
. Per
definition, the product of ansatz functions satisfies the simple relation:
. Finally,
one obtains the system of difference equations:
Equation (5) is satisfied if and only if
the coefficients vanish.
Conclusions
When a numerical simulation is applied to geometries with subregions that can be approximated as vertices (e.g. very small cavities), edges (e.g. tube-like structures), etc., the numerical method described above canbe applied. Discretizing these geometries by graphs can be useful, since the geometry has a structure that is difficult to treat only with 3-dimensional volume elements like in FEM. If a small subregion is subdivided into many 3-dimensional discrete elements, then more computation time than in the case of only one single discrete element is necessary.
References
[1]Tompson, A. et al. Particle-grid methods for reacting flows in porous media with application to Fisher's equation, Applied Mathematical Modelling, vol. 16, i. 7 (1992), pp. 374-383.
[2]John,V. et al. Finite element methods for time-dependent convection–diffusion–reaction equations with small diffusion, Comput. Methods Appl. Mech. Engrg. 198 (2008) 475–494.
[3] Liu,J. et al. An Operator Splitting Method for Nonlinear Reactive Transport Equations and Its Implementation Based on DLL and COM, Current Trends in High Performance Computing and Its Applications (2005), pp. 93-102
Showing 1 Reviews
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Your paper looks very motivating. There are some genuinely innovative approaches used. Since analytical solution for arbitary geomertry might be very complicated, this method provides a numerical solution. Some simulations for known solutions would have make it more interesting. Overall, interesting read.
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