An Asymptotic-Induced Scheme for Nonstationary Transport Equations in the Diffusive Limit.
An asymptotic-induced scheme for nonstationary transport equations with the diffusion scaling is developed. The scheme works uniformly for all ranges of mean-free paths. It is based on the asymptotic analysis of the diffusion limit of the transport equation.A theoretical investigation of the behavior of the scheme in the diffusion limit is given and an approximation property is proven. Moreover, numerical results for different physical situations are shown and the uniform convergence of the scheme is established numerically.
Introduction
. Transport equations are used to describe many physical phe-
nomena. Some of the best known examples are neutron transport, radiative transfer
equations, semiconductors or gas kinetics. The situation for small mean free paths
is mathematically described by an asymptotic analysis. Depending on the transport
equation and on the kind of scaling, different limit equations are obtained. For example
the gas kinetic equations may lead to Euler or (in)compressible Navier Stokes
equations. The limit equation for small mean free paths of radiative transfer, neutron
transport, or semiconductor equations is the diffusion and the drift-diffusion equation,
respectively. We refer to [3, 4, 12, 18, 20, 28] and [2, 6, 8, 10].
The main problem for numerical work on transport equations in these regimes
is the stiffness of the equations for small mean free paths. For standard numerical
schemes one has to use a very fine and expensive discretization with a discretization
size depending on the mean free path. Moreover, in general a full resolution of the
relaxation process is not necessary. The general aim is to develop numerical schemes
working uniformly for different regimes. In particular, the discretization size should
be independent of the mean free path. In recent years there has been a lot of work
on numerical methods for kinetic equations in stiff regimes. For example, stationary
transport equations in the diffusion limit have been considered, e.g., in [14, 15, 22, 21].
Nonstationary kinetic equations with a scaling leading to first order hydrodynamic
equations like the Euler equation are treated in [7, 9]. Usually for the latter case a
fractional step method with a semi-implicit treatment of the equations is used. For
general work on implicit methods for transport equations we refer to [27] and references
therein. We mention here also work on implicit methods for the full Boltzmann
equation, see [5]. Moreover, the relaxation limit of transport equations may be used
to develop schemes for the hydrodynamic equations themselves. These schemes have
been developed by many authors. For a recent general approach to these so called
relaxed or kinetic schemes we refer to [16].
The present work considers a scheme for nonstationary transport equations with
a scaling leading to the diffusion equation as the limit equation. The different space
time scalings involved in the problem are treated in a proper way. We use the standard
perturbation procedure leading from the transport to the diffusion equation.
FB Mathematik, University of Kaiserslautern, 67663 Kaiserslautern, Germany,
(klar@mathematik.uni-kl.de).
A. KLAR
Essentially the problem is transformed into a system of equations of relaxation form
and then a fractional step method is used. The analysis of the resulting problem is
based on ideas developed in [7]. Including the results of a boundary layer analysis in
the scheme, kinetic boundary layers are also treated in a correct way. Sections 2 and
3 contain a description of the results of the standard asymptotic procedure and the
presentation of the time discretization in our scheme. In Section 4 the diffusion limit
of the scheme is considered. In Section 5 the fully discretized equations are presented.
An approximation property for different ranges of the mean free path is proven in 6.
Section 7 contains numerical results for several examples and a numerical comparison
with other schemes.
Finishing the introduction we mention that the ideas developed in this paper can
be transfered to the gas kinetic and the semiconductor case, where the above scaling
leads in the limit to the incompressible Navier-Stokes equation and the drift-diffusion
equation respectively. In particular, in the gas dynamic case a more careful use has
to be made of the perturbation procedure leading from the Boltzmann equation to
the incompressible Navier-Stokes equation. This problem will be treated in a separate
paper.
2. The Equations. We consider transport equations of the following form
where S is assumed to be the unit sphere around 0 in R 3 . The collision operator Q is
defined by
with are some constants and the scattering cross
section oe is independent of v. K is an integral operator
Z
s symmetric in v and v 0 , rotationally invariant, are
some constants, and
R
1. K is compact. The collision operator
has as collision invariants only constants and is negative in a suitable function space.
The source term G(x) - 0 is assumed to be independent of v. Initial and boundary
conditions are given by
and
where @D is the boundary
of\Omega and the outer normal of
@\Omega at the point
x. See [3] for a thorough theoretical investigation of this equation. Extensions of the
following to other cases like, e.g., v- dependence of oe and G are possible.
Introducing the usual diffusion space-time scaling x ! x
the mean free path and scaling G(x) one obtains the scaled equations
ASYMPTOTIC-INDUCED SCHEME FOR TRANSPORT EQUATIONS 3
With the standard perturbation procedure, see e.g. [3, 4, 12, 20] the limit equation
for (2.2) as ffl tends to 0 can be derived by writing f as
One obtains
and up to a constant
be the solution of
R
3. Since by assumption s is rotationally invariant, it follows [3] that 8i; j 2
R
is the solution of the diffusion equation
oe
Doing a boundary layer analysis, one observes that the correct zeroth order boundary
conditions for the diffusion equation are given by a kinetic half space problem: Let
be the bounded solution of the following halfspace problem at x
Then
Here - x (\Gamma1; t) is independent of v.
Remark: In the absorbing case the scaled equation (2.2) is changed into
where oe A is the absorption cross section. The diffusion equation turns into
oe
3. The Numerical Scheme. For a numerical scheme for the transport equation
in the small mean free path limit it is desirable that varying mean free paths ffl can be
treated with a fixed discretization such that it is not necessary to adapt the time step
once the mean free paths tend to 0. Moreover, it is also desirable that the scheme is
in the limit ffl ! 0 a good discretization of the diffusion equation.
These points are obviously not fullfilled for a simple explicit time discretization
of (2.2) like
since, as ffl tends to 0, the time step must be shrinked due to stability considerations
in order to treat the advection term (the CFL condition has to be fulfilled) and the
4 A. KLAR
collision term properly. Therefore, large computation times are needed for small mean
path for such a scheme. In contrast, for a fully implicit discretization
there is no restriction on the time step due to stability considerations. However, one
has to solve a stationary equation in every time step, which is again time consuming.
We mention that, due to the development of fast multigrid algorithms [19, 24, 25, 26],
for the stationary equation, computation times for a fully implicit scheme are strongly
reduced. A numerical comparison of these types of algorithms with the one developed
here is presented in Section 7.
The aim in this work is to develop a semi-implicit scheme treating only such terms
in an implicit way for which it is necessary to do so in order to obtain a scheme working
uniformly in ffl. In particular, due to the different advection ( 1
scales, it is in the original formulation (2.2) not clear whether the advection has to be
treated implicitely or not. One may nevertheless discretize the original equations in a
straightforward way by treating the advection explicitely and the scattering term in
an implicit way:
This simple type of discretization has several drawbacks compared to the scheme
developed below, we discuss them at the end of Section 4.
We suggest to use the standard perturbation procedure to transform equation
(2.2) into two equations. A fractional step scheme with a semi-implicit procedure is
then used for the resulting equations. The idea is to follow the expansion procedure,
collect suitable terms together, such that only terms on
the scale 1
are involved.
be the solution of the set of equations
We take the inital and boundary values
and
One observes that f 0 fulfills the original equation (2.2) and the initial and
boundary conditions. It is therefore the desired solution of the original problem.
The results of the boundary layer analysis, see, e.g. [3], are included in the scheme
by choosing h in the following way:
be the solution of the halfspace problem (2.3). Since the outgoing
ASYMPTOTIC-INDUCED SCHEME FOR TRANSPORT EQUATIONS 5
function at the boundary for the kinetic problem (2.2) is the same as the outgoing
solution of the half space problem for ffl tending to 0, we define
In the limit ffl tending to 0 we obtain in this way the correct boundary value. For
independent of v we get q(x; v; It is obviously not
reasonable to determine the outgoing function by solving the halfspace problem. This
would need too much computing time. Here a fast approximate scheme as in [11] or
[17] is needed to determine the outgoing function. For example a first approximation
is given by choosing simply an approximation ~
of the asymptotic value
of the halfspace problem as the outgoing function:
The simplest approximation of - x (\Gamma1; t)is given by equalizing the half range fluxes
of the halfspace problem at 0 and 1:
~
R
R
A more sophisticated approximation for q, see [17], is given by
~
R
R
(3.9)D4-
Z
(v
R
R
dv
and
Z
(w
We remark that a correct treatment of the boundary conditions is important, in
particular, if zeroth order kinetic boundary layers are present and one is using a
coarse spatial grid not resolving the layer. See Section 7 for some examples. Using
the approximations above one obtains a good approximation of the solution with a first
order boundary layer even if only a very coarse grid is used. The first approximation
yields in general already very good results as can be seen in the numerical experiments
in Section 7. However, in certain situations the use of the second approximation might
be necessary to obtain an improved accuracy, compare Figure 7.6 in Section 7.
The system of equations (3.4,3.5) will be solved with a fractional step scheme:
Step 2:
6 A. KLAR
For Step 1 an explicit discretization will be used, Step 2 is discretized implicitely to
treat the stiffness of the equations in a correct way.
Let \Deltat denote the time step and f k
\Deltat the time iterations
approximating f 0 (x; v; k\Deltat); f 1 (x; v; k\Deltat). The initial and boundary values are given
as above. Introducing the notation
Z
the time discretization is then given by the following:
Step 2:
\Deltat
Rewriting (3.13,3.14) we obtain
\Deltat
\Deltat
and
\Deltat
\Deltat
This leads to
Step 2:
oe
where the operator A is defined by
\Deltatoe
\Deltatoe I
and
\Deltat
\Deltatoe I
ASYMPTOTIC-INDUCED SCHEME FOR TRANSPORT EQUATIONS 7
Here I denotes the identity. The operator
\Deltatoe I
is positive and invertible with a bounded inverse in a suitable function space for all
since K is compact and I \Gamma K is positive having only the constants as
collision invariants.
For example in the case of one-group transport with K =!? we obtain
and
In this case the semi-implicit scheme reduces to a fully explicit one. In general, in
each time step we have to solve in Step 2 two linear Fredholm integral equation of the
\Deltatoe I
This may be achieved by standard methods [1, 13].
Remark: In the absorbing case one proceeds as described above treating the
absorption in an explicit way. Then one obtains
Step 2 is unchanged.
4. The Diffusion Limit. We start with the investigation of the behaviour of
the time discretized scheme as ffl tends to 0 for fixed \Deltat.
As the operators A and B have the following behaviour: Introducing
suitable spaces, e.g. L 2 (S), we have for \Deltat fixed and ffl small, since
positive the following:
\Deltatoe
and
\Deltatoe
Using these estimates we get that the scheme reduces in the diffusion limit, ffl
tending to 0, to the following
8 A. KLAR
Step 2:
oe
Moreover, we have
and
where h was defined in Section 2. This yields
Step 2:
oe
Considering Step 2 and Step 1 together we obtain for
oe
oe
or
\Theta
oe
This is the simplest explicit time discretization for the diffusion equation. The
boundary conditions for the diffusion equation that are given in the limit by the
solution of the halfspace problem (2.3) fit to the boundary conditions for the kinetic
scheme as defined in the last section.
We finish this section by comparing the above scheme with the scheme (3.3) in
Section 3. Doing the standard asymptotic analysis [21] we get for (3.3) as ffl ! 0
oe
This means we obtain an explicit discretization of the diffusion equation as for the
above scheme, but due to the ! f
it is not the usual one. This type of
discretization of the diffusion equation is worse in terms of accuracy and stability than
(4.4). For example, doing a stability analysis one observes that only time steps are
allowed which are half the size of those that can be used in (4.4). This is essentially
due to the fully explicit treatment of the advection term in (3.3). Moreover, the
scheme developed in Section 3 gives the possibility to treat for example the collision
terms in a semi-implicit way as given in (3.13,3.14). This is at least for one-group
transport with K =!? a decisive advantage, since the semi-implicit scheme presented
here reduces in this case to a fully explicit one. If one would be trying to do the same
thing based on the scheme (3.3) it would turn out that the limit equation is not any
more the diffusion equation.
ASYMPTOTIC-INDUCED SCHEME FOR TRANSPORT EQUATIONS 9
5. The Fully Discretized Equations. We restrict from now on for notational
simplicity to the case, where f 0 and f 1 depend only on the first space coordinate: The
domain under consideration is [0; L]. Moreover, we consider only the space discretiza-
tion. The velocity space can be treated by using standard discretizations, see, e.g.,
[23].
We define a staggered grid x
\Deltax
, and x i\Gamma 1=
We use the notation
and
Moreover the operators D + and D \Gamma are defined as
The discretization of the initial values is straightforward. The boundary conditions
are discretized by
Discretizing the f 0 derivative in (3.16) with D \Gamma and the f 1 derivative in (3.11) with
D+ yields the following scheme
Step 2:
oe
A. KLAR
In the limit for small ffl we obtain the space discretized diffusion equation
oe
\Deltat
oe
or
\Theta
\Deltat
oe
This is a standard explicit discretization of the diffusion equation. In particular,
we obtain independent of the size of the discretization \Deltax a good discretization of
the limit equation for all ranges of the mean free path. The discretization possesses
all diffusion limits, the so called thin, intermediate and thick diffusion limit, see [22].
We observe, that we need in the limit a relation like
\Deltat - (\Deltax) 2oe
as for the diffusion equation, to obtain positivity and stability of our scheme. This
condition may be relaxed for ffl large.
6. A Uniform Approximation Property. In this section we prove a uniform
approximation property of our scheme. We give an estimate for the consistency error,
considering the integral form of equations (3.4,3.5) assuming that the true solution is
smooth.
Written in integral form the equations for f 0 (t) and f 1 (t) are for the Cauchy
problem and one space dimension
]ds:
Approximating the integral by an integral over step functions defined in each interval
of length \Deltat and approximating the derivative with respect to x as before, we get for
oe
oe
where we defined f (k)
ASYMPTOTIC-INDUCED SCHEME FOR TRANSPORT EQUATIONS 11
Reconsidering equations (5.1,5.2) and (5.3,5.4) for Step 1 and Step 2 of the numerical
scheme and putting the steps together, we get for one time step
or
A
or
A
oe
oe
+\Deltat
A
and a similar formula for f n
1 .
In the following we want to estimate the consistency error, i.e. the difference
between the true solution (f of the integral equation (6.1,6.2) and the value
1 ) that is obtained by introducing the true solution (f
instead of (f k
into the above formula (6.3).
This means we estimate the difference between (f 0 (t); f 1 (t)) and ( ~
~
A
oe
oe
+\Deltat
A
and a similar formula for ~
1 . Restricting in the proof for simplicity to the case
I , we concentrate in the following on proving a
pointwise estimate for
The proof is based on four lemmas.
Lemma 6.1.
where C is a constant independent of ffl.
Proof.
A. KLAR
Z (k+1)\Deltat
k\Deltat
oe
oe
oe
]ds
oe
oe
Z (k+1)\Deltat
k\Deltat
oe
\Deltax)ds
oe
oe
k\Deltat
oe
ds
where
oe
oe
oe
\Deltax
oe
used. Since the second term is 0, this is
smaller than
Lemma 6.2.
A
where C is a constant independent of ffl.
Proof.
A
A
with
1. This is equal to
C \Deltat
ASYMPTOTIC-INDUCED SCHEME FOR TRANSPORT EQUATIONS 13
since A. The estimates to prove the next Lemma can be found in [7]
Lemma 6.3.
with In particular for one has
Proof.
We have
This gives
\Deltat
Moreover, since
\Deltatoe
\Deltatoe
\Deltatoe
we get
\Deltatoe
\Deltatoe
\Deltatoe
Estimates (6.4) and (6.5) give the first assertion. To prove the second assertion we
use the first one with
However, this is smaller than C \Deltat, since, if \Deltatoe
2 we have that it is smaller than
if \Deltatoe
that it is smaller than
(noe \Deltat
(noe \Deltat
14 A. KLAR
Lemma 6.4.
\Deltat
where C is a constant independent of ffl.
Proof.
A
oe
oe
The first two terms are estimated by Lemma 2 and Lemma 3. They are smaller than
C \Deltat:
The third term is for ffl - ffl 0 smaller than
Using I the second term on the right hand side in
(6.6) is smaller than
\Deltatoe
\Deltatoe
\Deltatoe
\Deltatoe
\Deltat
\Deltat
Again due to Lemma 3 with k substituted by the first term in (6.6) is smaller
than
\Deltatoe
\Deltatoe
\Deltatoe
\Deltatoe
\Deltatoe
ASYMPTOTIC-INDUCED SCHEME FOR TRANSPORT EQUATIONS 15
since x). For \Deltatoe
smaller than
For \Deltatoe
smaller than
\Deltatoe
\Deltatoe
\Deltatoe
\Deltatoe
\Deltatoe
due to
1. This means that the first term in (6.6) is smaller
than
\Deltat
\Deltat
with C independent of ffl. Collecting all the terms the lemma is proven. All together,
using Lemma 1 and 4 we have proven for t 2 [0; T ];
\Deltat
This means, that for small \Deltat; \Deltax and fixed ffl, the estimate tends to 0 like \Deltat \Deltax.
However, also for a meshsize, that is large compared to ffl the estimate shows, that
we get convergence to 0. For example, for ffl - C \Deltat we obtain convergence to 0 like
\Deltat \Deltax.
We mention that \Deltat has to be chosen in relation to \Deltax . E.g. in the diffusion
limit we need \Deltat to be of the order of (\Deltax) 2 as we have seen in the last section.
7. Numerical Results and Examples. In this section a numerical study of
the scheme is presented and the scheme is compared with fully explicit and fully
schemes.
We restrict to the one-group transport equation in slab geometry, i.e. x 2 [0; L]
and K =!?. This yields
3 . The velocity discretization is done using in all
situations a 16 point quadrature set.
We compute the solution with the semi-implicit scheme derived above for different
space discretizations. To obtain positivity and stability of the semi-implicit scheme
in the limit ffl tending to 0 one has to take - for a fixed space discretization \Deltax - a
time step \Deltat of the size given by (5.6). As mentioned above this can be relaxed for
large ffl. In particular, this means that the size of \Deltat can be chosen independent of ffl.
Comparison with the explicit scheme (3.1):
In contrast to the above we get that the explicit discretization (3.1) of equation (2.2)
requires a time step of the order min(\Deltaxffl; ffl 2
oe
) to obtain positivity and stability. In
particular, for small ffl the step size \Deltat has to be chosen in this case of the order ffl 2 ,
in contrast to the semi-implicit scheme. A comparison of the CPU time necessary for
one time step yields that the semi-implicit scheme needs about 2 times the CPU time
of the explicit scheme. This yields a big gain in computing time for small ffl for the
semi-implicit scheme compared to an explicit one. In particular, it is reasonable to use
A. KLAR
the semi-implicit scheme, if 2 \Delta
oe
2 oe and if the desired
accuracy does not require a smaller time step, than the one that can be taken for the
semi-implicit scheme. To obtain a certain required accuracy of the solution one has to
use time steps as shown in the table below for some examples, see Table 1. Looking
at

Table

one observes that using an explicit scheme is not reasonable for small ffl.
Either the semi-implicit or the implicit scheme are faster. However, this changes for
large, where the explicit scheme may be better due to the small computation times
per time step.
Comparison with the fully implicit scheme (3.2):
A fully implicit dicretization of the equation obviously allows bigger time steps, since
there is no stability restriction on the time step in this case. Nevertheless, for the
accurate simulation of the time development small time steps may be necessary. To
get an accurate resolution of the behaviour of the solution up to an error of a certain
order the size of the time step for the implicit scheme has to be chosen according to

Table

An implementation of a fully implicit scheme shows that in order to obtain a
sufficient accuracy the stationary equation has to be evaluated to a very high accuracy
approximately up to an error of the order 10 \Gamma8 . To achieve this a standard iteration
scheme using for example a diamond difference discretization needs a large number of
iteration steps (sweeps over the computational domain). A comparison of the CPU
time for one iteration step shows that one time step of the semi-implicit iteration
needs less than 2 times the CPU-time of an iteration of the stationary scheme. Table
2 shows that the semi-implicit scheme has a big advantage compared to a standard
implicit iteration in many situations.
However, of course, computation times for an implicit scheme are strongly reduced
if a multigrid algorithms as described, e.g., in [24] is used. Using the convergence
estimates in [24] one observes that in essentially two cycles an accuracy of the
one needed for the solution of the stationary equation is obtained. One V (1; 1) cycle
costs about the same CPU time as 4 sweeps over the computational domain. I.e. the
estimated costs for one time step of a fully implicit scheme with a multigrid algorithm
is about 4 times as large as the one for the semi-implicit scheme. The complexity of
the implementation of a multigrid scheme especially in higher dimensions has to be
taken into consideration as well.
We consider a situation with
conditions equal to 0 at and equal to 1 at 1. The space discretizations
are 0:01. The time steps required for the semi-implicit scheme by
stability considerations are in this case \Deltat = 0:015; \Deltat = 0:00015. We consider final
times 0:5. For a stationary state is nearly reached. The
error was calculated by taking the L 1 -norm of the difference with the 'true ' solution
computed with a very fine discretization. The table shows the time steps necessary
to obtain a certain accuracy e with the implicit scheme using a diamond difference
discretization.

Table

1: Time steps required to obtain a certain accuracy e.
These accuracy requirements together with the above estimated CPU time give
the following relation between the CPU time for the explicit (E), the semi-implicit (S)
ASYMPTOTIC-INDUCED SCHEME FOR TRANSPORT EQUATIONS 170.00050.00150.0025
Fig. 7.1. Error for different values of ffl.
and the implicit scheme with multigrid (IM) and with a standard iteration procedure

Table

2: Relative CPU times
This shows that for coarse grids the semi-implicit scheme has to be prefered. For
finer grids and nearly stationary situations the advantage of a fully implicit scheme
with multigrid is clearly seen. Implicit schemes with a standard iteration procedure
are in all considered situations slower than the semi-implicit scheme.
Further investigation of the semi-implicit scheme:
To show the uniform convergence in ffl for the semi-implicit scheme numerically, we
compute the error for different values of ffl ranging from As
before, we use conditions equal to 0 and 1 at
and respectively, and the following values for the space discretization \Deltax with
the corresponding \Deltat values due to the stability condition (5.6):
Hence, each cell contains between 0:125 and 10 5 mean free paths. The error was
calculated by taking the L 1 -norm of the difference of the solutions with discretization
size 0.0125 and 0.025 (error 3), 0.025 and
tively. This results in three curves, which are plotted in Figure 7.1. Looking at the
figure, one observes that the error behaves perfectly uniform as ffl ! 0.
The solution of the kinetic equation computed by the new scheme derived in
this work is in the following computed for different physical situations. The physical
examples under consideration are
Example 1: the boundary conditions
Example 2: the boundary conditions
are equal to 0 and
A. KLAR0.30.50.70
x
semi-implicit25 true
Fig. 7.2.
Example 3: conditions equal to 0,
Example 4: boundary conditons
0:4. The solution of this problem
has a kinetic boundary layer at
Example 5: As Example 4, but with
Example A two material problem. In [0; 0:1] we consider a
purely absorbing material with 0:1. I.e. the region
has the size of one mean free path. In [0:1; 1:1] we take a purely scattering material
region). The
boundary conditions are f(0; The solution of this
problem has an interface layer at
The initial condition is always 0.
The solutions for the physical situations described above are plotted in the following
figures. In Figure 7.2 to 7.4 the situations from Example 1 to 3 are shown.
The solutions are plotted using space discretizations for the
semi-implicit scheme. We use the label 'semi-implicit10' to denote the solution with
the semi-implicit scheme with 10 spatial cells. The time discretization is chosen due
to the stability condition (5.6) for Example 2 and 3. For Example 1 the restriction
on the time step is relaxed to a CFL-type condition. The reference solution is the
solution with a very fine discretization. For this case the solution of the semi-implicit
scheme and of the other schemes are coincident. The solution of the diffusion equation
is computed by the usual triangular explicit scheme, which is the limiting scheme of
our semi-implicit scheme as ffl tends to 0, compare (4.3). The example shows that for
isotropic boundary conditions the solution is approximated with good accuracy for
different ranges of ffl.
In Figure figure layer1 Example 4 is considered. We plot the reference solution
and the solution of the diffusion equation with boundary coefficients derived from the
halfspace problem. The solutions of the semi-implicit scheme are found with
such that a discretization cell contains and the corresponding size of
the time discretization. The boundary values are found by determining approximately
the outgoing distribution of the halfspace problem (2.3) as described in Section 3.
ASYMPTOTIC-INDUCED SCHEME FOR TRANSPORT EQUATIONS 190.050.150.250
x
true
diffusion
Fig. 7.3.
x
semi-implicit25 true
Fig. 7.4.
This is done using first the approximation of the asymptotic value of the halfspace
problem by (3.7), as the outgoing function (the solution in the plot is labeled 'semi-
implicit10-1') and second an outgoing function determined by formula (3.10) labeled
'semi-implicit10-2'. In this first case the two approaches give coincident results. One
observes that even for a coarse diffusive discretization the behaviour of the solution at
the boundary is found with very good accuracy. We mention that other approaches to
obtain the correct discrete boundary conditions for the stationary equation are shown
in [14, 21].

Figure

7.6 shows Example 5. The same as in Figure 7.5 is shown. However, in
this case one cell contains now 1000 mean free paths. The advantage of using here
an exact approximation of the outgoing fiunction of the half space problem is clearly
seen.
Finally Figure 7.7 shows Example 6. The space discretization is here
in the absorbing region and \Deltax = 0:1 in the scattering region. In particular, one cell
in the scattering region contains 100 mean free paths. The situation at the interface
A. KLAR0.250.350.450.550.650.75
x
true
semi-implicit10-2 diffusion
Fig. 7.5.
x
true
semi-implicit10-2
Fig. 7.6.
is treated in the same way as the one at the boundaries before. One observes again
a good agreement of the solution in the diffusive region with the true solution. We
mention here the work of [14, 18, 21] who treated similar problems for the stationary
equation.
ASYMPTOTIC-INDUCED SCHEME FOR TRANSPORT EQUATIONS 210.020.060.10.14
x
true
Fig. 7.7.
8. Conclusions. From the analytical and numerical results one can conclude:
ffl The semi-implicit scheme works uniformly for all ranges of the mean free
path. This is shown by numerical experiments and a consistency proof.
ffl The limiting scheme for small mean free paths is a standard explicit discretization
of the diffusion equation.
ffl By including a boundary layer analysis one obtains a suitable treatment of
the boundary conditions for coarse (diffusive) discretizations.
ffl A comparison of the scheme with fully explicit and fully implicit schemes
shows advantages and disadvantages. In particular, the semi-implicit scheme
is faster than the fully implicit scheme, if the detailed time development is
computed with a coarse discretization or with higher accuracy requirements.
However, for nearly stationary situations with a fine grid the fully implicit
scheme, if combined with a fast multigrid method as in [24], is faster.
ffl The numerical results have been generated for the one group transport case. A
further numerical treatment should include the implementation of the scheme
with other scattering ratios. Using methods as in [1, 13] this should be possible
without too much difficulties.



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A Survey of Numerical Methods for the solution of Fredholm Integral Equations of the Second Kind
Fluid dynamic limits of kinetic equations: Formal derivations
Diffusion approximation and computation of the critical size
Boundary layers and homogenization of transport processes
Implicit and iterative methods for the Boltzmann equation
The fluid dynamical limit of the nonlinear Boltzmann equation
Uniformly accurate schemes for hyperbolic systems with relaxation
The Boltzmann Equation and its Applications
Numerical passage from kinetic to fluid equations
Incompressible Navier Stokes and Euler limits of the Boltzmann equation
A numerical method for computing asymptotic states and outgoing distributions for kinetic linear half space problems
Uniform asymptotic expansions in transport theory with small mean free paths
Theorie und Numerik


The relaxation schemes for systems of conservation laws in arbitrary space dimensions

Diffusion theory as an asymptotic limit of transport theory for nearly critical systems with small mean free paths

Asymptotic solution of neutron transport problems for small mean free path
Asymptotic solution of numerical transport problems in optically thick
Asymptotic solution of numerical transport problems in optically thick
Computational Methods of Neutron Transport
A fast multigrid algorithm for isotropic transport problems I: Pure scattering
A fast multigrid algorithm for isotropic transport problems II: With absorption
Multilevel methods for transport equations in diffusive regimes
Methods of Numerical Mathematics
Diffusion approximation of the linear semiconductor equation
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asymptotic analysis;diffusion limit;transport equations;numerical methods for stiff equations