On the numerical solution of the three-dimensional advection-diffusion equation
A new approach is proposed for the numerical solution of three-dimensional advection-diffusion equations, which arise, among others, in air pollution modelling. The technique is based on directional operator splitting, which results in one-dimensional advection-diffusion equations. Then upstream-t...
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Інститут програмних систем НАН України
2006
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| Cite this: | On the numerical solution of the three-dimensional advection-diffusion equation / V. Prusov, A. Doroshenko, I. Faragó, Á. Havasi // Проблеми програмування. — 2006. — N 2-3. — С. 641-647. — Бібліогр.: 42 назв. — англ. |
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irk-123456789-15702008-08-29T12:01:25Z On the numerical solution of the three-dimensional advection-diffusion equation Prusov, V. Doroshenko, A. Faragó, I. Havasi, Á. Прикладне програмне забезпечення A new approach is proposed for the numerical solution of three-dimensional advection-diffusion equations, which arise, among others, in air pollution modelling. The technique is based on directional operator splitting, which results in one-dimensional advection-diffusion equations. Then upstream-type difference approximations are applied for the first-order derivatives and non-standard difference approximations for the second-order derivatives. This approach leads to significant qualitative improvements in the behaviour of the numerical solutions. 2006 Article On the numerical solution of the three-dimensional advection-diffusion equation / V. Prusov, A. Doroshenko, I. Faragó, Á. Havasi // Проблеми програмування. — 2006. — N 2-3. — С. 641-647. — Бібліогр.: 42 назв. — англ. 1727-4907 http://dspace.nbuv.gov.ua/handle/123456789/1570 004.75 en Інститут програмних систем НАН України |
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Прикладне програмне забезпечення Прикладне програмне забезпечення Prusov, V. Doroshenko, A. Faragó, I. Havasi, Á. On the numerical solution of the three-dimensional advection-diffusion equation |
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A new approach is proposed for the numerical solution of three-dimensional advection-diffusion equations, which arise, among others, in
air pollution modelling. The technique is based on directional operator splitting, which results in one-dimensional advection-diffusion
equations. Then upstream-type difference approximations are applied for the first-order derivatives and non-standard difference
approximations for the second-order derivatives. This approach leads to significant qualitative improvements in the behaviour of the
numerical solutions. |
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Article |
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Prusov, V. Doroshenko, A. Faragó, I. Havasi, Á. |
| author_facet |
Prusov, V. Doroshenko, A. Faragó, I. Havasi, Á. |
| author_sort |
Prusov, V. |
| title |
On the numerical solution of the three-dimensional advection-diffusion equation |
| title_short |
On the numerical solution of the three-dimensional advection-diffusion equation |
| title_full |
On the numerical solution of the three-dimensional advection-diffusion equation |
| title_fullStr |
On the numerical solution of the three-dimensional advection-diffusion equation |
| title_full_unstemmed |
On the numerical solution of the three-dimensional advection-diffusion equation |
| title_sort |
on the numerical solution of the three-dimensional advection-diffusion equation |
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Інститут програмних систем НАН України |
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2006 |
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Прикладне програмне забезпечення |
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http://dspace.nbuv.gov.ua/handle/123456789/1570 |
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On the numerical solution of the three-dimensional advection-diffusion equation / V. Prusov, A. Doroshenko, I. Faragó, Á. Havasi // Проблеми програмування. — 2006. — N 2-3. — С. 641-647. — Бібліогр.: 42 назв. — англ. |
| work_keys_str_mv |
AT prusovv onthenumericalsolutionofthethreedimensionaladvectiondiffusionequation AT doroshenkoa onthenumericalsolutionofthethreedimensionaladvectiondiffusionequation AT faragoi onthenumericalsolutionofthethreedimensionaladvectiondiffusionequation AT havasia onthenumericalsolutionofthethreedimensionaladvectiondiffusionequation |
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2025-07-02T04:58:55Z |
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Прикладне програмне забезпечення
© K. Georgiev, E. Donev, 2006
ISSN 1727-4907. Проблеми програмування. 2006. №2-3. Спеціальний випуск 641
UDC 004.75
ON THE NUMERICAL SOLUTION OF THE THREE-DIMENSIONAL
ADVECTION-DIFFUSION EQUATION
Vitaliy Prusov
Ukrainian Research Hydrometeorological Institute,
Science prosp. 37, 03650, Kiev, Ukraine
prusov@uhmi.org.ua
Anatoliy Doroshenko
Institute of Software Systems of the National Academy of Sciences of Ukraine,
Acad. Glushkov prosp., 40, block 5, 03187 Kiev, Ukraine,
dor@isofts.kiev.ua
István Faragó
Department of Applied Analysis, Eötvös Loránd University,
Budapest, Pázmány P. s. 1/C, H-1117, Hungary
faragois@cs.elte.hu
Ágnes Havasi
Department of Meteorology, Eötvös Loránd University,
Budapest, Pázmány P. s. 1/A, H-1117, Hungary
hagi@nimbus.elte.hu
A new approach is proposed for the numerical solution of three-dimensional advection-diffusion equations, which arise, among others, in
air pollution modelling. The technique is based on directional operator splitting, which results in one-dimensional advection-diffusion
equations. Then upstream-type difference approximations are applied for the first-order derivatives and non-standard difference
approximations for the second-order derivatives. This approach leads to significant qualitative improvements in the behaviour of the
numerical solutions.
Introduction
The investigation of advection-diffusion equations in higher dimensions is of great importance. The
atmospheric flows and heat transfer processes as well as the concentration changes of pollutants are commonly
described by a set of partial differential equations, which are mathematical formulations of one or more of the
conservation laws of physics. These include the equations of momentum, mass and energy conservation, which involve
advection and diffusion terms as a main constituent. Advection-diffusion equations take the form
F
x
v
x
v
x
v
t
=
∂
∂+
∂
∂+
∂
∂+
∂
∂
3
3
2
2
1
1
ξξξξ
∂
∂
∂
∂+
∂
∂
∂
∂+
∂
∂
∂
∂+
3
3
32
2
21
1
1 xxxxxx
ξµξµξµ , (1.1)
where iµ and iv are the eddy viscosity and the velocity component of the fluid, respectively, in the direction
( )3,2,1=ixi , and F is the source/sink of the quantity ξ . The terms ( )ii xv ∂∂ξ are usually called advection (or
sometimes convection) terms, and describe the transportation of the quantity ξ by the velocity field. The terms
( ) iii xx ∂∂∂∂ ξµ are called diffusion (or sometimes viscous) terms, and express the spreading of quantity ξ by the
process of turbulent diffusion. The equation is usually provided with appropriately defined initial and boundary
conditions.
Since analytical solutions of advection-diffusion problems cannot usually be found, we have to solve them
numerically. However, the numerical treatment of equations of the form (1.1) is a highly complicated task. The main
reasons for this are the following:
• the equations are nonlinear;
• due to changes in the input parameters the equations can change the type (hyperbolic, parabolic or elliptic);
• the size of the discretized problem can be very large in a real-life physical model.
Therefore, choosing a sufficiently accurate as well as efficient numerical method for solving advection-diffusion
problems is not an easy task. In this case the application of operator splitting seems a good alternative. In [1] Prusov et
Прикладне програмне забезпечення
al. suggest a new finite-difference method for the one-dimensional convection-diffusion equation. Our aim is to extend
this method to higher dimensions by using operator splitting based on directional decomposition.
The structure of the paper is as follows. In Section 2 we propose possible splitting algorithms for problems of
the form (1.1). In Section 3 we deal with the numerical solution of the sub-problems obtained by splitting. We close the
paper with some useful remarks and a summary of the results.
1. Solving the three-dimensional advection-diffusion equation by operator splitting
The aim of operator splitting is to replace an initial value problem with a sequence of simpler problems, for
which accurate as well as efficient solvers available in standard program packages exist [2, 3]. The mathematical
background of operator splitting can be sketched as follows. Let S denote a normed space and consider the abstract
initial value problem
),()()(
)(
21 twAAtAw
dt
tdw +== ],0[ Tt ∈
,)0( 0ww = (2.1)
where S∈)(tw , ],0[ Tt ∈ is the unknown function, and A is a given operator SS → , which can be decomposed
into a sum of two “simpler” operators A1 and A2. (By “simpler” we mean that the corresponding initial-value problems
are easier to treat numerically than the original problem.) The simplest kind of operator splitting is the so-called
sequential splitting, where we solve the following sequence of initial value problems:
),(
)( )1(
1
)1(
twA
dt
tdw
k
k = ],)1(( ττ kkt −∈
),)1(())1(()1( ττ −=− kwkw spk (2.2)
and
),(
)( )2(
2
)2(
twA
dt
tdw
k
k = ],)1(( ττ kkt −∈
),())1(( )1()2( ττ kwkw kk =− (2.3)
)()( )2( ττ kwkw ksp =
for k = 1, 2,…, n, where nT /=τ is the splitting time step, and .)0( 0wwsp = This scheme can be extended to more
than two sub-operators in a natural way.
Another possibility is the Marchuk-Strang splitting [4], defined by the following algorithm:
),(
)( )1(
1
)1(
twA
dt
tdw
k
k = ])5.0(,)1(( ττ −−∈ kkt
),)1(())1(()1( ττ −=− kwkw spk (2.4)
),(
)( )2(
2
)2(
twA
dt
tdw
k
k = ],)1(( ττ kkt −∈
),)5.0(())1(( )1()2( ττ −=− kwkw kk (2.5)
),(
)( )3(
1
)3(
twA
dt
tdw
k
k = ],)5.0(( ττ kkt −∈
)())5.0(( )2()3( ττ kwkw kk =− (2.6)
)()( )3( ττ kwkw ksp =
for k = 1, 2,…, n, where .)0( 0wwsp =
Obviously, there are several ways to define the sub-operators Ai in a splitting procedure. We can choose the
sub-operators on a physical base, e.g., we can separate the advection and diffusion terms in Eq. (2.1) (physical
Прикладне програмне забезпечення
decomposition) [5]. Another possibility is to separate the x1-, x2- and x3-derivatives in the equation (directional
decomposition) [6]. For the three-dimensional advection-diffusion problem the physical decomposition would lead to
advection problems, which are of hyperbolic type, and diffusion problems, which are of parabolic type. This would
cause us difficulties in defining appropriate boundary conditions for the sub-problems. Therefore, we recommend the
latter one, which results in three one-dimensional advection-diffusion problems at each time step. Particularly, if we
apply the sequential splitting, the detailed algorithm will read as follows:
,
)(
1
1
)1(
1
11
)1(
1
)1(
x
kkk F
x
x
xx
v
t
t
+
∂
∂
∂
∂+
∂
∂
−=
∂
∂ µξξ
],)1(( ττ kkt −∈
),)1(())1(()1( τξτξ −=− kk spk (2.7)
,
)(
2
2
)2(
2
22
)2(
2
)2(
x
kkk F
x
x
xx
v
t
t
+
∂
∂
∂
∂+
∂
∂
−=
∂
∂ µξξ
],)1(( ττ kkt −∈
),())1(( )1(
1
)2( τξτξ kk kk −=− (2.8)
,
)(
3
3
)3(
3
33
)3(
3
)3(
x
kkk F
x
x
xx
v
t
t
+
∂
∂
∂
∂+
∂
∂
−=
∂
∂ µξξ
],)1(( ττ kkt −∈
),())1(( )2(
1
)3( τξτξ kk kk −=− (2.9)
)()( )3( τξτξ kk ksp =
for k = 1, 2,…, n, where .)0( 0ξξ =sp If the original problem is defined over a bounded spatial domain, the equations
(2.7)-(2.9) are also provided with appropriately defined boundary conditions.
Several numerical methods have been constructed for the solution of the resulting one-dimensional advection-
diffusion problems [7-12]. Recently, finite element [13] and spectral methods are very popular [14]. There are also
many finite difference schemes that can be considered according to the number of spatial grid points involved, the
number of time-levels used, and whether they are explicit or implicit in nature [15–42].
Standard three-point finite difference methods of approximating spatial derivatives can work well for smooth
solutions, but they fail when severe gradients or discontinuities are present, which are common in the shock wave
problems [17–21]. Lower-order accurate finite difference methods, such as upstream-type finite differences, can be a
remedy for the numerical oscillations and dispersions. However, they have a large amount of “numerical viscosity” that
smoothes the solution in much the same way that physical viscosity would, but to an extent that is unrealistic by several
orders of magnitude [20]. Standard four-point finite difference methods are good in their higher-order accuracy and in
reducing numerical smearing effects [21]. But, they are plagued by their generation of spurious oscillations or
overshoots in the neighbourhood discontinuities and lack accuracy [17, 18]. Total variation stable finite difference
schemes (TVD) [10, 11] guarantee oscillation-free solutions but they are limited to second-order accuracy. Higher-order
accurate TVD schemes are attractive for problems with long computational time or with required higher accuracy
solutions [11]. But, the objection to the standard higher-order schemes comes from the additional nodes necessary to
achieve the higher-order accuracy. This precludes the use of implicit methods since the obtained matrix is not of three-
diagonal form, and it is necessary to use fictitious nodes for the boundary conditions. Also, they do not allow easily for
non-uniform grids, unless at the expense of the order of accuracy. On the other hand, the compact schemes that treat
functions and their derivatives as unknowns at the grid nodes, like the scheme [31], are fourth-order accurate, and
compact in the sense that they reduce to three-diagonal form. The compact schemes generally consist of finite
difference schemes which involve two or three grid points. The three-point schemes fall into two classes. The first class
consists of methods which are fourth-order accurate for uniform grids, such as schemes [26–28], the operator compact
implicit scheme [26–28] and the Hermite finite difference method [29]. The second class consists of methods that allow
variable grids such as the cubic spline methods [30–33], and the Hermite finite difference method [34, 35]. In [36, 37] a
compact fourth-order finite difference scheme was introduced with three nodal points for the convection-diffusion
equations. This scheme does not seem to suffer excessively from spurious oscillatory behaviour or numerical viscosity.
The disadvantage of the above higher-order compact schemes involving three nodal points is that the boundary
conditions are no longer sufficient and they do not allow easily for non-uniform grids, unless at the expense of the order
of accuracy. Another disadvantage of some compact schemes is the complexity of the resulting nonlinear finite
difference equations and the associated difficulty in solving them efficiently. On the other hand, the compact scheme
with two nodal points is fourth-order accurate even for non-uniform spatial grids, and no fictitious points, neither extra
formulas are needed for Dirichlet boundary conditions [38]. The discretization of the convective term might be done in
a number of ways [39, 40]. The Ellam scheme is probably one of the best known convective schemes [41].
Прикладне програмне забезпечення
In the next section we present a non-standard finite-difference method for the solution of the sub-problems
obtained by splitting (2.7)-(2.9).
2. A finite-difference scheme for the one-dimensional advection-diffusion problem
Consider the one-dimensional advection-diffusion equation
F
xxx
v
t
+
∂
∂
∂
∂=
∂
∂+
∂
∂ ξµξξ
, 0≥µ , lx ≤≤0 , 0>t (3.1)
with initial condition
( ) ( )xx ηξ =0, , lx ≤≤0 (3.2)
and Dirichlet boundary conditions
( )tt αξ =),0( , ( )ttl βξ =),( 0>t , (3.3)
where ( )txv , , ( )tx,µ , ( )xη , ( )tα and ( )tβ are known functions, while the function ),( txξ is unknown. Let us
divide the spatial interval ],0[ l of the problem into J equal parts with division points JJ xxxx <<<< −110 ... , and
denote the length of the j-th sub-interval by jh . Besides, divide [ ]T,0 into N equal parts by points 1−= nTNt n ,
Nn ...,,1,0= , with time step τ . We define the grid },...1,0,,...1,0),,{( NnJjtx n
j ===Ω , and denote by n
jξ
the approximation of ),( n
j txξ .
Integrating Eq. (3.1) at jx from nt to 1+nt yields
dtF
xxx
v
n
n
t
t j
n
j
n
j ∫
+
−
∂
∂
∂
∂−
∂
∂−=+
1
1 ξµξξξ (3.4)
Approximating the integral on the right-hand side by the mean-value theorem, we obtain
θ
ξµξτξξ
=
+
−
∂
∂
∂
∂−
∂
∂−=
t
j
n
j
n
j F
xxx
v1 , (3.2)
where 1+<< nn tt θ . For the approximation of the derivatives ( ) θξ =∂∂ t
j
x and ( )[ ] θξµ =∂∂∂∂ t
j
xx we will use
the following difference relations:
−
−
+
−
+
=
∂
∂
=
−
−+
−
−
= θθ ξξξξξ
t
j
jj
j
j
jj
j
jj
t
j h
h
h
h
hhx 1
11
1
1
1
θ
ξ
=
−
∂
∂
t
jj
x
hh
3
3
1
6
, (3.3а)
( )
−
−
+
+
=
∂
∂
∂
∂ +
+
−
=
j
jj
jj
jj
t
j
hhhxx
ξξ
µµξµ
θ
1
1
1
1
( )
θθ
ξξξ
µµ
=
−
=
−
−
−
∂
∂−
−
−
+−
t
jj
t
j
jj
jj
x
hh
h 3
3
1
1
1
1 3
(3.4а)
The unilateral difference expressions ( ) jjj hξξ −+1 and ( ) 11 −−− jjj hξξ in (3.3a) and (3.4a) will be taken at
different time levels (n and 1+n ). For construction of approximations only by two points it is natural for physical
reasons to have on the ( )1+n -th layer a point jx as central, and to select the second one from that side from where ξ
is transferred by advection to the central point. It is easy to see that the created difference scheme (3.3а) and (3.4а) has
an approximation error of the first order in τ . In this manner we gain the following form:
Прикладне програмне забезпечення
• for 0>v
•
+
−
+
−
+
≈
∂
∂
−
+
−
+
+
−
−
=
1
1
1
1
1
1
1
1
j
n
j
n
j
j
j
n
j
n
j
j
jj
t
j
h
h
h
h
hhx
ξξξξξ
θ θ
ξτ
=
∂∂
∂
t
j
xt
2
, (3.3b)
( )
−
−
+
+
≈
∂
∂
∂
∂ +
+
−
=
j
n
j
n
j
jj
jj
t
j
hhhxx
ξξ
µµξµ
θ
1
1
1
1 ( )
θ
ξτ
ξξ
µµ
=
−
+
−
+
− ∂∂
∂+
−
+
t
jj
n
j
n
j
jj xth
2
1
1
1
1
1 ; (3.4b)
• for 0<v
+
−
+
−
+
≈
∂
∂
−
−
++
+
−
−
=
1
1
11
1
1
1
1
j
n
j
n
j
j
j
n
j
n
j
j
jj
t
j
h
h
h
h
hhx
ξξξξξ
θ θ
ξτ
=
∂∂
∂
t
j
xt
2
, (3.3c)
( )
−
−
+
+
≈
∂
∂
∂
∂ ++
+
+
−
=
j
n
j
n
j
jj
jj
t
j
hhhxx
11
1
1
1
1 ξξ
µµξµ
θ
( )
θ
ξτ
ξξ
µµ
=
−
−
− ∂∂
∂+
−
+
t
jj
n
j
n
j
jj xth
2
1
1
1 (3.4c)
Substituting (3.3b), (3.4b) or (3.3c), (3.4c) in (3.2) we will receive a difference scheme for the one-dimensional
advection-diffusion problem (3.1) in the following form:
• for 0>v
−
−
+
−
+
+
−
−
+
−
+
++
−
−
+
1
1
1
1
11
1
1
1
1
j
n
j
n
jn
jj
j
n
j
n
jn
jj
jj
n
j
n
j
h
vh
h
vh
hh
ξξξξ
τ
ξξ
( )
−
−
+
+
− +
+
− j
n
j
n
jn
j
n
j
jj hhh
ξξ
µµ 1
1
1
1 ( ) 0
1
1
1
1
1
1
1 =−
−
+
−
+
−
+
+
−
+ n
j
j
n
j
n
jn
j
n
j F
h
ξξ
µµ , (3.5а)
1...,,2,1 −= Jj , ...,1,0=n ,
( )jj xηξ =0 , Jj ...,,1,0= ,
( )nn tαξ =0 , ( )nn
J tβξ = , ...,1,0=n N.
• for 0<v
•
−
−
+
−
+
+
−
−
−
++
++
−
−
+
1
1
11
11
1
1
1
1
j
n
j
n
jn
jj
j
n
j
n
jn
jj
jj
n
j
n
j
h
vh
h
vh
hh
ξξξξ
τ
ξξ
( )
−
−
+
+
−
++
+++
+
− j
n
j
n
jn
j
n
j
jj hhh
11
111
1
1
1 ξξ
µµ ( ) 0
1
1
1 =−
−
+
−
−
−
n
j
j
n
j
n
jn
j
n
j F
h
ξξ
µµ , (3.5b)
1,2...,,2,1 −−= JJj , ...,1,0=n ,
( )jj xηξ =0 , Jj ...,,1,0= ,
( )nn tαξ =0 , ( )nn
J tβξ = , ...,1,0=n N.
The templates corresponding to the scheme (3.5) are shown in Figures 1a and 1b.
Прикладне програмне забезпечення
In this manner for the solution of the advection-diffusion problem (3.1)–(3.3) we have received a clone of the
so-called "running computation scheme" usually used for the solution of one-dimensional wave equations of the first
order (see, for example [42]). Therefore, in spite of the fact that the scheme (3.5) is formally implicit, it is easily solved
in an explicit way.
3. Remarks
A thorough theoretical analysis of the finite-difference method introduced in Section 3 and some simple
numerical experiments demonstrating stability and convergence properties of the scheme can be found in [1]. It has
been shown that this scheme possesses some good properties of both the explicit and implicit difference schemes. It is
as economic as an explicit scheme and is stable on any permissible grids as an implicit scheme.
The problem of convective diffusion is an example of problems for which the application of the implicit
scheme (3.5) is really justified. As it is known, the stability condition for the explicit schemes demands that τ should
decrease as 2h . This requirement necessitates the application of a much greater number of time steps than it is dictated
by reasons of accuracy only. Besides, it can happen that the differences n
j
n
j ξξ −+1 , Jj ...,,1,0= become as small as
disturbances arising as a result of round-off errors. The implicit scheme (3.5) is free from this lack as it is
unconditionally stable and has an approximation error of equal order in τ and h . Therefore, if there is a necessity for
increasing the accuracy of the numerical solution, it is possible to achieve it at the expense of decreasing τ and h in
equal measure.
From (3.3) and (3.4) it follows that the scheme can reach almost second order of accuracy, intrinsic to central
difference schemes for spatial derivatives by using small time steps, or when the field of the gradient x∂∂ξ of the
transferred value ξ varies smoothly.
4. Summary
We considered the three-dimensional advection-diffusion problem on a bounded domain with Dirichlet
boundary conditions. A splitting scheme based on directional decomposition was proposed for the solution. This
procedure allows us to replace the three-dimensional problem with three simpler, one-dimensional advection-diffusion
problems at each time step of the numerical integration. We proposed a non-standard finite-difference method for the
solution of the one-dimensional sub-problems. This method unites the advantages of explicit and implicit schemes.
Acknowledgements
This research was supported by NATO Collaborative Research Grant ENVIR.CLG 930449. Ágnes Havasi is a
grantee of the Bolyai János Scholarship.
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Fig. 1. Templates of difference networks: a) of the scheme (3.5a); b) of the scheme (3.5b)
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