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Abstract
The different computational methods for 2D parabolic boundary problems have been considered.
It is assumed the coefficients of the equations are discontinuous and the lines of
discontinuity don’t coincide with coordinate ones. It results in the necessity of using
the nonorthogonal grids. The fully implicit, additive and regularized schemes have been
developed. The specific method of regularization taking into account discontinuity of
coefficient allows constructing the schemes more than first order of time approximation,
which are easily parallelized.
Contents
1. Introduction
2. Fully implicit scheme
and support operator method
3. Regularized scheme
4. Finite difference flux method
5. Variational difference flux
method
6. Variational difference
locally onedimensional method
7. Results of calculations
8. Conclusions
Reference
Introduction
Now there are many
difference schemes for the heat conductivity parabolic equations with
discontinuous coefficients, if the surfaces and lines of discontinuity being
not coincident with coordinate ones. The discontinuity of coefficients in a
heat conduction equation means that normal (to a discontinuity surface)
derivatives are discontinuous. In this case, attempts of approximating the
equations by means of the difference schemes without the taking into account
the position of discontinuities may reduce the accuracy of approximation. For
the best method of describing the phenomena it is expedient to use grids
adapted to the media structure. Besides the schemes should satisfy the
requirements of stability, monotonicity and conservativity. At the same time
because of the usage of multiprocessing systems, it needs to provide the
possibility of effective parallelezation of the difference schemes. But the
explicit schemes, most simply parallelized, are not considered because of hard
limitation of a timestep for schemes stability.
In the present work some methods of solution of the posed problem
for the 2D equation are considered
_{}, (1.1)
where
_{}, (1.2)
in domain _{}. It is supposed that the coefficient _{} is a piecewise continuous
function. Boundary conditions may be general. The examined examples correspond
to zero boundary conditions of the second type
_{}, (1.3)
where _{} is the boundary of
domain. Further, for simplicity we shall consider square
domain _{}.
The cells of examined grids have a
form of tetragons, though it should be noticed, that the proposed schemes might
be generalized easy for grids containing triangles. The wellknown support
operator method has been used for a spatial approximation of the problem. The
feature of this method is a definition of the divergence and gradient
difference operators by using the difference analogy of the known identity:
_{}. (1.4)
If the operators of the divergence and gradient are in accordance with
the difference approximation of (1.4), the approximation orders of both
operators are matched.
Let _{}. The equality (1.4)
allows a definition of the divergence operator if the operator of gradient (or
the flux operator _{}) is being determined. It is the reason of the method
denomination. Really, the support operator method approximates the system
(1.1), (1.2) instead of the approximation of the equation _{} only. It is analogous to the ideas
developed in the known article (Glowinski R., Wheeler M.F., 1988).
Five difference schemes were
compared: fully implicit scheme on the basis of a of the support operators
method (Samarskii A.A., Koldoba A.V., Poveshchenko Yu.A., Tishkin V.V.,
Favorskii A.P., 1996; Koldoba A.V., Poveshchenko Yu.A., Popov Yu.P., 1985), the
scheme with regularizator (Samarskii A.A., 1989), two modifications for
curvilinear grids of the scheme proposed by Yu.B. Radvogin, and also a variant
of the locally onedimensional scheme based on a variationaldifference
principle (Goloviznin V.M., Korshunov V.K., Samarskii A.A., Chudanov V.V.,
1985). The results of comparison in midpoint of the process evolution and in a
final are represented in the tables.
The main aim of the article is to
create the effective methods of the 2D and 3D problems solution for the
parabolic equations with piecewise continuous coefficients. There are many
methods for rectangular grids but usually the surfaces and lines of the
coefficients discontinuity don’t coincide with coordinate ones. The parabolic
equations have standard forms of the balance equations connecting the time
derivatives of value with the flow divergence. The evident method is fully
implicit but it results in the solution of the linear system of the large
order. One possibility is determine the flow components at the intermediate
moments of the time and following divergent closure of the system. This idea
had been realized by Radvogin Yu.B. only for rectangular grids. The problem of
the mixed derivatives for nonrectangular grid had not been discussed in the
article. In this work it was shown that the contravariant component of flux
(DFM) or covariant components (VDFM) are sufficient to be calculated. Then
using the balance equation for each cell the divergence expression may be
obtained. The closed equations for the flux components may be constructed if
the mixed derivatives being omitted. The flux components equations are of the
order O(1). However, it is sufficient for the final approximation of the second
order for rectangular grid and of the first order for nonorthogonal grid.
Similar principles are used for VLODM. It is a sample of the additive scheme,
i.e. there is no approximation of each stage but only total.
The method considered is a
regularization method based on Samarskii idea analogous to preconditioning the
algebraic linear system. The difference approximation of the equations (1.1),
(1.2) may be rewritten _{}, where _{}, _{}, _{} and _{} are finite difference
approximation of the operators by support operators method, _{} and _{} are positive
onedimensional operators, described below. It is essential the scheme is
effective because it does not need to solve the complicated linear algebraic
equations, as the elliptic operator is determined on the explicit layer.
Besides, this scheme is easily parallelized.
The calculations for heat
conductivity coefficients with different analytical properties have been
realized for tasks on rectangular and curvilinear grids when the analytical
solution being known.
It is evident the proposed schemes may be generalized on a 3D
case.
Fully implicit scheme
and support operator method
Now we describe the most obvious
way of approximation: a fully implicit scheme and a method of operator _{} approximation. Let's mark, that
hereinafter the temperature is considered in cellcenters and corresponding
fluxes  in celledges. This method of discretisation allows the natural approximation
of boundary conditions of Neumann type. Besides, it simplifies the construction
of difference scheme near the surfaces or lines of coefficients discontinuities,
since the fluxes normal to these lines are continuous (it is supposed that
lines of discontinuities coincide with grid lines).
The fully implicit scheme has a
form
_{} (2.1)
where _{} is a value _{} on an implicit
timelayer, _{}  difference approximation
of the operator _{} by the support operator method.
Support operators method is a version of the finite volume method. The
last one is based on the known identity _{} for a vector _{}. Thus it needs to know components of vector _{} in the same points of cell
edges, but it is inconvenient, since the values of gradient different
components are usually determined in different edges. It is offered to average
the component values by usage of four neighbor bases (in a 2D case). Besides
the integral identity (1.4) will be used instead of the standard formulation of
the Gauss–Ostrogradskii theorem for calculating the divergence if the gradient
vector being known. The last one allows constructing a difference scheme not on
an initial grid, but on arbitrary one providing the best approximation.
The operator _{} was considered as the support
operator. Let _{} and _{} be curvilinear
coordinates, under level lines passing through cellcenters or their corresponding
edges, and the distance between cellcenters for coordinates _{} coincides with the
distance in coordinates _{}. Besides it is
supposed, that under transformation _{} the domain _{} is mapped into unit
square (see fig. 2.1). The numbers of a grid points along coordinate lines are
designated with _{} and _{} accordingly. Then the covariant
components of a flux _{} in local basis _{} look like
_{}, (2.2)
_{}, (2.3)
_{}, (2.4)
_{} (2.5)
and
_{
} are equal to the parts of segment
_{} and _{}, lying inside a cell _{}, _{} and _{} are the lengths of these segments.
Fig.
2.1 Fig. 2.2
Suppose, that the grid lines are orthogonal to the boundary of the
domain (Fig. 2.1). Then we have according the boundary conditions (1.3)
_{} (2.6)
For approximating the second term of the left hand part of (1.4) the
additional grid _{} has been constructed.
The edges of the grid _{} connect the cell
centers of the grid _{}. For covering of
whole area _{} we shall add some cells formed
by segments of boundary and normals to it (see a fig. 2.2, the continuous lines
bound cells of a grid _{}, dashed  _{}).
Let's consider an internal cell _{} of grid _{}. Its vertices are denoted with F, G, H, I. The notation of angles is
represented in the fig. 2.3. Then the inner product of vectors _{} and _{} in the vicinity of vertex F has a form
_{} (2.7)
Similarly in vertex G:
_{} (2.8)
Fig. 2.3
Using (2.6), (2.7) and analogous expressions for vertices H and I the
quantity of integral over the cell may be represented
_{}. (2.9)
Here _{}, _{}, _{}, _{} are some nonnegative values,
the sum of which equals the square of FGHI. Let the vertices F, G, H, I have
indexes _{}, _{}, _{}, _{} respectively. Then the expression (2.9)
may be rewritten
_{}
_{}
_{}
_{} (2.10)
For boundary cell, but not for an angular one, the sum (2.10) will
include only two terms. For example, for a cell _{} we obtain the following according
to (2.6):
_{}
_{} (2.11)
For angular cells there will be only one term, which equals to 0 owing
to (2.6).
The integral over domain _{}, where summation is performed over all cells of _{}.
Let grid vector _{} be a difference gradient:
_{}. It is possible to consider, that vector _{} component normal to boundary is equal to
0, i.e. _{} _{}.
Then
_{}, (2.12)
Where _{}.
Here for _{}
_{}_{}_{}
_{} (2.13)
_{}_{}
_{}
_{} (2.14)
Remark, the sum (2.12) does not include expressions with _{}, that corresponds to boundary conditions (1.3).
Here _{} and _{} are length of some segments. For
example, they may be the segments connecting centers of gravity of cells _{}, _{} and _{}, _{} of grid _{} accordingly. Then it’s natural
to consider the expression
_{}, (2.15)
as a difference analogue of the operator _{}, where _{} are squares of some domains,
associated with nodes _{}. They may be squares of a tetragon with vertices in centers of gravity
of cells _{}, _{}, _{} and _{} of grid _{} (it is assumed that for boundary
cells these vertices coincide with centers of boundary segments, and for
angular  with its angle vertex). It is possible to lie _{} for simpler variant. In fact it
means the theorem of GaussOstrogradskii for this domain, and the grid
functions _{} and _{} are contravariant components of
a flux on cell edges of a grid _{}.
The finite volume method was used in works of Edwards M.G. and
Aavatsmark I. for investigation of singlephase and multiphase problems of
underground hydrodynamics (M.G. Edwards, C.F. Rogers, 1998; Aavatsmark
I., Barkve T., O. Boe, T. Mannseth, 1996). They define the divergence using the
Gauss therem for one cell. The feature of the support operator method is a
definition of flux _{} from an equality (1.4). It results to the
following: the order of the divergence and gradient operators approximations is
the same. This method of difference schemes construction based on reviewing of
the system (1.1), (1.2) instead of the equation _{}, that is similar to work Glowinski R., Wheeler M.F., 1988,
where the same representation is used for a mixed finite element method.
The regularized scheme
The following type of scheme is
developed as the regularized scheme:
_{}, (3.1)
where _{} operator is a
regularizator (Samarskii A.A., 1989).
As known, for an absolute stability
of the scheme it is sufficient to examine
_{} (3.2)
with boundary conditions
_{} (3.2 ')
for the heat conductivity equation with a variable coefficient on a
rectangular grid (Samarskii A.A., 1989). Here _{} is a unit operator, _{} is a max value of a
heat conductivity. The obtained scheme is a scheme of the first order of
approximation by time. Under drawing an analogy with rectangular grids it is
possible to use the regularizator of the following form for curvilinear grids
_{} (3.3),
with boundary conditions
_{} (3.3 ')
where _{} are curvilinear coordinates.
Coordinate lines of the system _{} coincide with lines of coordinates _{}, but the initial grid maps into uniform rectangular one in a
plane _{} with steps _{} and _{} (see fig. 2.1), _{} and _{} are factors depending on "irregularity"
of a grid and for a rectangular grid equaling 1.
However, the regularizator of the type mentioned above gives
unsatisfactory results of calculations for the discontinuous coefficient of the
heat conductivity, as it does not provide the necessary order of the
approximation in the vicinity of the discontinuity line.
Because of this fact the following form of the regularizator _{} is offered:
_{} (3.4)
with boundary conditions (3.3 ').
Usually regularized schemes are schemes of the first order of approximation
by time. The exception are the schemes on rectangular grids being the schemes
of the second order of approximation, if _{} and operators _{}. Therefore it is possible to expect that the schemes of the type mentioned
above with regularized operators similar to main one of the problem near
coefficients discontinuity lines will be by more precise, than additive schemes
described below.
It is easy to find
expressions for factors _{} and _{}, corresponding to absolute stability of the scheme (3.1)
with regularizator (3.4), for a concrete grid. Let _{} be a grid function,
obtained on _{}th time step (_{}). Under stability the following will be understood
_{}
_{} (*)
Here is an inner product in the space of grid functions, defined in
cells of a grid _{} (the summation is performed over
all cells of a grid _{}):
_{} (3.5)
as well as
_{} (3.5’)
It should be noted that it needs to introduce last terms in both parts
of (*). It is concerned with the fact the expressions in the (*) without terms
are not a norms, but just as seminorms. So the left part is a norm in solution
space and right part is a norm in space of initial data.
Let's prove, that the satisfaction of the inequality
_{}. (3.6)
is sufficient for
stability of the scheme (3.1) (in sense of (*)).
Really, having multiplied both parts of (3.1) by _{} (in inner product _{}), we have:
_{}
_{}
_{}
_{}
Thus, if (3.6) is
fair, then _{}, from which (*)
immediately follows.
Besides the inner product _{} we shall introduce
_{},
where the summation is performed over all cells of a grid _{}. It should be marked, that norms induced by inner products _{} and _{} are equivalent: _{}
Then,
_{}, (3.8)
_{}, (3.9)
_{}. (3.8)
and
_{}_{}
_{} (3.10)
Here _{}
For approximation of _{} it is assigned in (1.4) _{} just as it was done in support
operator method. Then the right part of (1.4) becomes 0 and according to the
formulas (2.13), (2.14)
_{}_{}
_{}, (3.11)
where _{}, and the summation is performed over all cells of a grid _{}.
Thus,
_{}
_{}, (3.12)
where
_{}
_{}
_{}
Since each edge appears twice in the sum (3.12) except the boundary
ones, it is sufficient to suppose
(3.13)
(3.14)
for satisfaction of _{}
Let's assume _{}. Then both introduced inner products coincide and it is necessary to
prove _{}
_{}, where _{},
_{} with boundary conditions (1.3).
Since both operators _{} are nonnegative determined
(with respect to inner product _{}, which coincides with _{} in a considered case), all their
eigenvalues _{}, corresponding to eigenfunctions _{}, are nonnegative. Consider the case _{}. Since all grid functions are decomposed by basis _{}: _{}, then
_{}.
Thus, we have proved, if having selected regularizing factors according
to the formulas (3.13) and (3.14) the scheme (3.1) will be absolutely stable.
As several tests without analytical solution have shown it will be also stable
for variable _{}. In other cases the regularizator (3.3) should be used.
It should be noted, that for the considered class of problems the scheme
(3.1) with _{} equal to squares of tetragons
with vertices in center of gravity of cells _{}, _{}, _{} and _{} of grids _{}, was also absolutely stable (with some _{} and _{}). Moreover, the scheme (3.1) with regularizator of the form
_{}
_{
}
(3.15)
_{}
with boundary conditions
_{}
_{} (3.15')_{}
was absolutely stable for some
_{}
and
_{}_{.}
Let's remark, that the scheme (3.1) with regularizator (3.4) or (3.15)
is economic, since the corresponding system of linear equations may be easily
inverted by two sweep methods.
Difference flux method
Flux method is understood as the
scheme of following sort: at the first step (predictor) the fluxes on edges of
cells are calculated on "a intermediate time layer using the equations for
each component of a vector _{}. Usually they are difference approximations of the equation
for a flux _{} by some methods. The
second step (corrector) is a divergent closure due to initial equation.
Let's remind the scheme proposed by
Yu.B. Radvogin for a heat conductivity equation with variable coefficient on a
rectangular uniform grid.
Predictor
_{} (4.1)
_{} (4.2)
Corrector
_{}
(4.3)
Here _{} and _{} are grid steps, _{} and _{} are fluxes normal to
the cell edges. If _{} the scheme is
absolutely stable, i.e. the scheme with surpassing definition of fluxes is
stable.
On a step of predictor we
effectively calculate components of fluxes normal to the cell edges referred to
intermediate time layer. It should be noted, that each equation of (4.1) and
(4.2) approximates equation for a flux with zero order, i.e. it is supposed
that for each layer of cells the flux through its lateral boundaries is equal
to zero. On a step of corrector using these values we calculate values of
temperature for an implicit time layer.
The natural generalizing for
curvilinear grids of the scheme proposed by Yu.B. Radvogin looks like the
following:
predictor_{}
_{}_{}
_{}_{ }(4.4)
_{}_{}
_{}_{ }(4.5)
Here _{} is defined by support
operators method for each layer along directions _{} and _{} under the following
assumption. It is supposed that flux along the same direction in other layers
and flux along other direction are equal to 0.
Corrector
_{}
(4
.6)
The operator _{} is defined by a
support operator method: using contravariant components of a flux we calculate
the covariant ones by a method depicted above. The equations (4.4) and (4.5)
also may be solved effectively.
Variational difference flux method
Let the operator _{} be employed to both parts of (1.1) and then outcome be
multiplied by _{}. Then due to (1.2) the equation for flux is obtained
_{} (5.1)
Discretising in time with the help
of the implicit scheme (_{}), we have
_{} (5.2)
Multiplying (5.2) by an arbitrary
variation _{} and integrating over
arbitrary domain _{} we obtain due to
boundary conditions (1.3), if boundary fluxes being equal to zero:
_{}. (5.3)
Thus, functional
_{} (5.4)
reaches its stationary value on a solution of the equations (1.1) and
(1.2) with boundary conditions (1.3).
Let the curvilinear coordinates _{} in domain _{} be
introduced by the same way as in the third item. For a definiteness we suppose
that a Jacobian _{}. Designating _{} and _{} number of cells along
directions _{} and _{} respectively, the
expressions in (5.4) may be rewritten in local contravariant base of a curvilinear
coordinate system _{}.
The orts of contravariant base look
like
_{}, (5.5)
where _{} are Lame coefficients
of a curvilinear coordinate system.
Since _{},
where _{} and _{} are contravariant
components of flux then having made substitution _{} we obtain
_{}, (5.6)
_{}. (5.7)
As well as in a difference flux
method, we suppose that for each layer of cells the flux through its boundary
is equal to zero. Let's consider the approximation of a functional (5.4),
having taken into account of any layer of cells along _{} or _{} as _{} (see fig. 5.1).
For a layer _{} along _{} we have:
_{}, (5.8)
where
_{}
_{} (5.9)
_{} (5.10)
_{} (5.11)
_{} (5.12)
_{} (5.13)
Fig. 5.1
Then functional _{} may be represented as
_{}
_{}, (5.14)
where
_{}, (5.15)
_{}. (5.16)
The difference analogue of the
equation (5.1) follows from a condition of a minimum of a functional (5.8): _{} or
_{}
_{} (5.17)
_{}
Using boundary conditions _{} the equation is
solved by sweep method.
For a layer _{} along _{} the reasonings are
similar, therefore we list only finite outcome  difference equation:
_{}
_{} (5.18)
_{}
and boundary conditions _{}.
For divergent closure we use the
initial equation (1.1) and formula (5.7):
_{}
_{}. (5.19)
The variational locally onedimensional method
The developed scheme is very
similar with previous one. Let's specify their differences. In a flux method
using known distribution of temperature the fluxes on an intermediate time
layer are calculated, and then divergent closure is performed. In the given
scheme using known temperature covariant component of a flux only along one
direction (for example, along _{}) is calculated, then the temperature on an intermediate time
layer is obtained under supposition that the fluxes along _{} are equal to zero.
Then using the new distribution of temperature on an intermediate time layer
covariant component of a flux along _{} is calculated.
Finally, the temperature on an implicit time layer is obtained under assumption
that of a component of the flux along _{} is zero. Since the
calculations coincide with one described above we represent only final formulas
– the difference equations:
_{}
_{} (6.1)
_{}
_{} (6.2)
_{}
_{} (6.3)
_{}
_{} (6.4)
Calculation results
All five algorithms were tested on
a rectangular grid, scalene grid (see fig. 7.1) and sine grid (see fig. 7.2).
The square _{} was divided into 10õ10, 20õ20 and 40õ40 of cells. The exact solutions were the
following
1. _{}, _{}
2. _{}, _{}
3. _{}, _{}
4. _{}, _{},
where _{} is obtained from the
equation
_{} (7.1)
Besides, the problem with sources
have been tested too. Instead of (1.1) the following equation have been
considered _{}. At the same time
the regularized and fully implicit schemes have been developed. These schemes
were also tested on solution
5. _{}
Fig. 7.1 Fig. 7.2

The
middle of calculation

Final
moment

Rectang.

Scalene

Sine

Rectang.

Scalene

Sine

10

Implicit

0.002543

0.005791

0.017498

0.004267

0.004786

0.005831

Regul.

0.001028

0.005610

0.018306

0.001701

0.002891

0.004672

DFM

0.003288

0.009470

0.044702

0.005555

0.006121

0.007731

VDFM

0.003288

0.006604

0.015847

0.005555

0.005988

0.006746

VLODM

0.003288

0.006745

0.017656

0.005555

0.006006

0.006412

20

Implicit

0.001802

0.002757

0.003224

0.003002

0.003170

0.002974

Regul.

0.000255

0.002479

0.002564

0.000419

0.001269

0.001628

DFM

0.002562

0.014380

0.068315

0.004299

0.004519

0.008980

VDFM

0.002562

0.003858

0.009687

0.004299

0.004412

0.004330

VLODM

0.002562

0.003945

0.010538

0.004299

0.004419

0.004205

40

Implicit

0.001614

0.001661

0.001674

0.002684

0.002732

0.002675

Regul.

5.910e5

0.001393

0.001048

9.692e5

0.001078

0.001036

DFM

0.002378

0.037927

0.043497

0.003983

0.005954

0.006813

VDFM

0.002378

0.003501

0.003585

0.003983

0.003988

0.003924

VLODM

0.002378

0.003427

0.003743

0.003983

0.003990

0.003912

Table 7.1

The
middle of calculation

Final
moment

Rectang.

Scalene

Sine

Rectang.

Scalene

Sine

10

Implicit

0.060256

0.084513

0.052579

6.954å9

0.003246

7.405å4

Regul.

0.035828

0.066245

0.034633

1.694e9

0.004266

0.001353

DFM

0.076859

0.099951

0.093450

1.642å8

0.003478

7.715å4

VDFM

0.076859

0.100001

0.064092

1.642å8

0.003300

0.001186

VLODM

0.076859

0.100644

0.062796

1.642å8

0.003302

0.001289

20

Implicit

0.042323

0.049574

0.039917

2.493å9

7.776å4

9.130å5

Regul.

0.016587

0.024491

0.021972

4.006e10

0.001244

4.014å4

DFM

0.059831

0.069960

0.087101

6.660å9

9.922å4

1.112å4

VDFM

0.059831

0.066459

0.058264

6.660å9

8.126å4

6.367å4

VLODM

0.059831

0.066603

0.058629

6.660å9

8.135å4

6.438å4

40

Implicit

0.037857

0.039757

0.037575

1.895å9

1.930å4

4.612å6

Regul.

0.011800

0.015915

0.017438

2.436e10

6.094å4

1.101å4

DFM

0.055590

0.062290

0.072428

5.273å9

4.033å4

5.492å6

VDFM

0.055590

0.057125

0.055641

5.273å9

2.220å4

1.999å4

VLODM

0.055590

0.057186

0.055720

5.273å9

2.223å4

2.004å4

Table 7.2

The
middle of calculation

Final
moment

Rectang.

Scalene

Sine

Rectang.

Scalene

Sine

10

Implicit

0.058951

0.080214

0.038918

9.32å10

0.002992

2.058å4

Regul.

0.033464

0.060647

0.042225

1.762e10

0.004628

2.669å5

DFM

0.058421

0.077163

0.055781

9.03å10

0.003042

1.950å4

VDFM

0.058421

0.079921

0.045288

9.03å10

0.003015

2.455å4

VLODM

0.066577

0.087797

0.048211

1.468å9

0.003015

2.640å4

20

Implicit

0.043829

0.050063

0.040376

3.47å10

7.128å4

1.048å5

Regul.

0.017012

0.024025

0.020200

4.493e11

0.001601

3.104å5

DFM

0.043922

0.051174

0.065077

3.49å10

7.568å4

2.731å5

VDFM

0.043922

0.050462

0.045016

3.49å10

7.251å4

3.991å5

VLODM

0.052266

0.058096

0.050214

5.93å10

7.254å4

4.490å5

40

Implicit

0.040019

0.041671

0.039675

2.68å10

1.760å4

4.102å7

Regul.

0.012873

0.016171

0.015776

2.893e11

8.474å4

7.318å6

DFM

0.040273

0.048669

0.055690

2.72å10

2.191å4

6.596å6

VDFM

0.040273

0.042710

0.042085

2.72å10

1.856å4

9.758å6

VLODM

0.048662

0.050290

0.050019

4.69å10

1.864å4

1.131å5

Table 7.3

The
middle of calculation

Final
moment

Rectang.

Scalene

Sine

Rectang.

Scalene

Sine

10

Implicit

0.029916

0.139758

0.111295

0.068440

0.132269

0.068473

Regul.

0.027471

0.143325

0.117483

0.062569

0.126339

0.071540

DFM

0.031644

0.139261

0.118225

0.072636

0.136601

0.066644

VDFM

0.031644

0.143370

0.119056

0.072636

0.137619

0.070704

VLODM

0.031644

0.144451

0.117742

0.072636

0.137898

0.071240

20

Implicit

0.010815

0.152431

0.024321

0.022945

0.056930

0.016478

Regul.

0.008049

0.153958

0.021091

0.017029

0.051391

0.012137

DFM

0.012768

0.147452

0.068678

0.027176

0.060852

0.021627

VDFM

0.012768

0.155467

0.025705

0.027176

0.061600

0.021214

VLODM

0.012768

0.157788

0.025298

0.027176

0.061783

0.021195

40

Implicit

0.005784

0.097919

0.006320

0.012025

0.028871

0.011572

Regul.

0.002941

0.096378

0.003688

0.006122

0.023593

0.006318

DFM

0.007790

0.092611

0.042937

0.016247

0.032853

0.017442

VDFM

0.007790

0.098255

0.008891

0.016247

0.033321

0.015941

VLODM

0.007790

0.100245

0.008740

0.016247

0.033407

0.015942

Table 7.4

The
middle of calculation

Final
moment

Rectang.

Scalene

Sine

Rectang.

Scalene

Sine

10

Implicit

8.242å5

8.480å5

6.975å4

0.010368

0.010438

0.014873

Regul.

2.162e5

2.398e5

4.494e5

3.519å4

3.905å4

0.001466

20

Implicit

2.250å5

2.959å5

8.245å5

0.002562

0.002585

0.002981

Regul.

2.452å5

2.536e5

2.656e5

9.644e5

1.422å4

2.441å4

40

Implicit

4.942å6

9.133å6

6.943å6

6.071å4

6.142å4

6.127å4

Regul.

2.552e5

2.574e5

2.532e5

6.182e5

7.687e5

7.351e5

Table 7.5
The results of calculations are
represented in tables 7.1…7.5 according to number of the test (1…5). Final time
for tests 1…4 was 0.25, for the test 5 was _{}. Numbers of steps by time are 100 and 600 respectively.
Factors _{} and _{} are supposed to be
equal to 1.5 in the scheme with regularizator (3.15).
Conclusions
The computer simulation of the different
boundary and initial problem demonstrates some features of the methods
developed. All additive and almost additive schemes DFM, VDFM, VLODM have the
same accuracy and CPU times. The fully implicit scheme accuracy is better than
for additive scheme but CPU time is essentially more and there is a serious
problem of parallelelizing this scheme. The best scheme is a regularized one.
The error of the approximation and CPU time are less than for other methods
except the case of the large values of function decrements. Besides the order
of the time approximation is more than the first as it has been discussed
above.
If the spatial part of the function is _{} and the decrement is _{} the asymptotic stability of the scheme is not
satisfied. Really at the end of evolution the absolute error of the solution is
1.694e9 for rectangular grid, 0.00426 for scalene grid and 0.00135 for
sinusoidal one, if the step is about of 1/10. For N=20, 40 correspondingly the
values of errors are 0.00124, 6.094e4 for scalene grid. As for other methods
the error were less. It may be explained by the fact that regularization
operators are not strictly agreed with the approximation of the elliptic
operator by the support operator method.
Author expresses thank to Pergament
A.Kh. for proposed topic of investigation and invaluable assistance during
article preparation.
Reference
1. Samarskii
A.A. The theory of difference schemes. Moscow, “Nauka”, 1989 (in Russian).
2. Samarskii
A.A, Koldoba A.V., Poveshchenko Yu.A., Tishkin V.V., Favorskii A.P. Difference
schemes on irregular grids. Minsk, 1996. (in Russian)
3. Goloviznin
V.M., Korshunov V.K., Samarskii A.A., Chudanov V.V. The method of factorized
heat displacements for solution of 2D problems of heat conductivity on
irregular grids. Preprint of KIAM, N58, 1985 (in Russian).
4. Glowinski
R., Wheeler M.F. Domain decomposition and mixed finite element methods for
elliptic problems.  First
International Symposium on Domain Decomposition Methods for Partial
Differential Equations, Philadelphia, 1988, SIAM, pp.144171
5. Edwards
M.G., Rogers C.F. Finite volume discretisation with imposed flux continuity for
the general tensor pressure equation. Computational geosciences, vol 2, 1998,
pp 259290
6.
Aavatsmark I., Barkve T., O. Boe, T. Mannseth.
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