A comprehensive study is presented regarding the stability of  the forward explicit integration technique with generalized finite  difference spatial discretizations,  free of numerical diffusion, applied  to the advection-diffusion equation. The modified  equivalent partial differential equation approach is used to  demonstrate that the approximation is free of numerical  diffusion.  Two-dimensional results are obtained using the von  Neumann method of stability analysis. Numerical results are  presented showing the accuracy obtaine

Benito Muñoz, Juan José

Gavete Corvinos, Luis Antonio

Gavete, Lucia

Ureña Prieto, Francisco

Archivo Digital UPM

Modelling of the advection-diffusion equation with a meshless method without numerical diffusion

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

Proceedings of the 10th International Conference 
on Computational and Mathematical Methods 
in Science and Engineering, CMMSE2010 
Almeria, Spain, 27-30 June 2010  
 
Modelling of the advection-diffusion equation with a 
 
 meshless method without numerical diffusion 
 
Luis Gavete
1
, Francisco Ureña
2
, Juan José Benito
3  
and 
María Lucía Gavete
4 
1 
Universidad Politécnica de Madrid, Spain  
2 
Universidad Castilla-la Mancha, Ciudad Real, Spain  
3
U.N.E.D., Madrid, Spain  
4
 Universidad Rey Juan Carlos, Madrid, Spain 
 
emails: lu.gavete@upm.es, francisco.urena@uclm.es, 
jbenito@ind.uned.es, lucia.gavete@urjc.es  
Abstract 
A comprehensive study is presented regarding the stability of 
the forward explicit integration technique with generalized finite 
difference spatial discretizations, free of numerical diffusion, 
applied to the advection-diffusion equation. The modified 
equivalent partial differential equation approach is used to 
demonstrate that the approximation is free of numerical 
diffusion. Two-dimensional results are obtained using the von 
Neumann method of stability analysis. Numerical results are 
presented showing the accuracy obtained. 
 
Key words: advection-diffusion, generalized finite difference 
MSC2000: AMS Codes (optional) 
 
1. Introduction 
 
With the development of modern industry, various pollulants discharge in the air 
rivers, lakes and oceans.  The changes of pollulants in the air or in the water 
consist of the physical, chemical and biochemical process  and so on. The 
physical changes of pollution involve two main important processes, that is, 
advection and diffusion. The mathematical model describing these two processes 
is the well known advection-diffusion equation. In two dimensions this equation 
is as follows 
 
MODELLING OF THE ADVECTION-DIFFUSION  
 
 
 
 
 
 
(1) 
 
 
 
 
 
being             and         two known functions, a, b are constants ,  is the 
boundary of ,  ( , , )U x y t is a transported (advected and diffused) scalar variable,     
0 ,  0x y  are constant speeds of advection and >0 , 0x y  are constant 
diffusivities in the x-, and y- direction respectively.  
Various numerical techniques can be used to solve this partial differential 
equation with the associated initial and boundary conditions [1]. An active field of 
research is the use of meshless methods. An evolution of the method of finite 
differences has been the development of generalized finite difference method 
(GFDM) that can be applied as a meshless or meshfreee method to irregular grids 
or clouds of points. Benito, Ureña and Gavete have made interesting contributions 
to the development of this method [2-7]. This paper shows the application of the 
generalized finite difference method to solve the advection-diffusion equation by 
an explicit method. 
The paper is structured in six sections. In section 2, we describe briefly the 
GFDM. In section 3 we describe the explicit scheme used to approximate the 
advection-diffusion equation. In section 4 we study the truncation error and the 
stability. In Section 5 an error analysis is done comparing with a test case. Section 
6 contains concluding remarks. 
 
 
2. The Generalized finite difference method 
 
In the GFDM the intention is to obtain explicit linear expressions for the 
approximation of partial derivatives in the points of a domain. First of all, an 
irregular grid or cloud of points is generated in the domain . On defining 
the central node with a set of nodes surrounding that node, the star of nodes then 
refers to a group of established nodes in relation to a central node. Each node in 
the domain has an associated star assigned to it. 
We define the following function based in the approximation of second order in 
Taylor series 
 
2 2
2 2
0
0
  ; 0, x, y
with the initial condition:
, 0
and the boundary conditions:
, t
, t
T
x y x y
U U U U U
t
t x y x y
U f
U
aU b g t in
n
x
x x
x
x
2
2
0 0 0 0
0
1
1
,
2
N
j j j j j j j
j
U U U U
B u u u h k h k w h k
x y x y
f x ( )g t
MODELLING OF THE ADVECTION-DIFFUSION  
(2) 
where 
0u  is the approximated value of the function at the central node of the star, 
0 0( , )x y , ju  are the function values of the rest of the nodes, 0j jh x x , 
0j jk y y  and   ( , )j jw h k  is the  denominated weight function. 
If the function (2) is minimized with respect to the partial derivatives, the 
following linear equation system is obtained  
  
 
(3) 
 
 
 
 
 
on solving the system (3) the explicit finite difference formulae are obtained. 
 
 
(4) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
3. The advection-diffusion GFDM explicit scheme 
 
On including the explicit expressions for the values of partial derivatives (5) in 
the differential equation of problem, we obtain the star equation (explicit 
difference scheme)(6): 
T
N
j
jjj
N
j
j
j
N
j
jj
wkhuuw
h
uuwhuu
1
2
0
1
2
2
0
1
2
0
......
2
...bDA
u
2 2 2
0 0 0 0 0
2 2
, , , ,
T
u
U U U U U
x y x y x y
D
p
0 ki i
i 1 1 1
1
1
1
1
M c , 1,..,5
1
, 1,..,5
1 , , : , 1,..,5 1,..,5
1
: , 1,..,5 1,..,5
0 : , 1,..,5
ij
pN
j ki ji
j i
p k
u uk k
ikk
i
k j
ij
ij
k u u M d k
k k q k i k
q
where
q i k M k j for j i i j
M for j i i j
q
for j i i
Y
D Y D
i
1
2 2
2 2 2 2 2
1 2 4 5 6
1,..,5
 the Kronecker delta function, and:
c
; ; ; ;
2 2
ij
N
ji
j
j j
j j j j j j j j j
j
with
d
h k
d h w d k w d w d w d h k w
MODELLING OF THE ADVECTION-DIFFUSION  
 
1
0 0 0
0 0
0 0 0 0
1 1
2 2
0 0
0 0 0 02 2
1 1
       
t
; ;
x y
;
x y
n n
N N
n n n n
j j j j
j j
N N
n n n n
j j j j
j j
U u u
t
U U
u u u u
U U
m u m u u u
                                   (5) 
 
1
0 0 0 0 0 0
1 1
0 0 0 0
1 1
0 0 0 0
1 1 1 1
:   ; ; ;
N N
n n n n n n
x j j y j j
j j
N N
n n n n
x j j y j j
j j
N N N N
j j j j
j j j j
u u t u u u u
t m u m u u u
with m m
                        (6)                 
 
This scheme uses the forward-difference form for the time derivative and 
generalized finite difference forms for all spatial derivatives.    
By using the modified equivalent partial differential equation approach of 
Warming and Hyett [8], we obtain the following expansion equation 
 
 
 
 
 
 
(7) 
 
 
 
 
 
and the modified equation 
 
 
(8) 
 
 
This method incorporates numerical diffusion. 
A new GFD scheme free of numerical diffusion can be created as follows 
 
2 2 3 3 4 3
1,2 3 4 3
1
3 2 3
2, 3,3 2 3
1 1
2 3
4,2 3
1
( ) ( )
, ...
2 6 24
, ... , ...
, ... 0
N
x j j j
j
N N
y j j j x j j j
j j
N
y j j j
j
U t U t U t U U U
h k
t t t t x x
U U U U
h k h k
y y x x
U U
h k
y x
22 2 2
2 2
...... 0
2 2
yx
x y x y
U U U U U
t t
t x y x y
MODELLING OF THE ADVECTION-DIFFUSION  
 
 
 
 
(9) 
 
 
 
 
 
Then by using the modified equivalent partial differential equation approach of 
Warming and Hyett [8] we obtain the following expansion equation 
 
 
 
 
(10) 
 
 
 
 
 
and the modified equation 
 
 
 (11) 
 
 
The modified equivalent partial differential equation of this method shows that 
this GFD formula (9) is free of numerical diffusion. 
 
4. Convergence 
 
According to Lax’s equivalence theorem, if the consistency condition is satisfied, 
stability is the necessary and sufficient condition for convergence. In this section 
we study firstly the truncation error of the advection-diffusion equation, and 
secondly consistency and stability. 
We split the truncation error (TTE) in time derivative error (TEt) and space 
derivatives error (TEx). As the first order time derivative is given by 
 
 
 
 
2
21
12
, , , ,
2
U t U t t U t U tt
O t t t t t
t t t
0 0 0 0
x x x x
2 2 3 3 4 3
1,2 3 4 3
1
23 2 3
2, 3,3 2 3
1 1
2 2
4,2
1
( ) ( )
, ...
2 6 24
, ... ( ) , ...
2
( ) ,
2
N
x j j j
j
N N
x
y j j j x j j j
j j
N
y
y j j j
j
U t U t U t U U U
h k
t t t t x x
tU U U U
h k h k
y y x x
t U
h k
y
3
3
... 0
U
x
1
0 0 0 0 0 0
1 1
22
0 0 0 0
1 1
0 0 0
1 1 1
2 2
:   ; ;
N N
n n n n n n
x j j y j j
j j
N N
yn n n nx
x j j y j j
j j
N N N
j j j
j j j
u u t u u u u
t t m u m u t u u
with m m 0
1
;
N
j
j
2 2
2 2
...... 0x y x y
U U U U U
t x y x y
MODELLING OF THE ADVECTION-DIFFUSION  
 
then the truncation time error is given by 
 
2
20 0 1
12
( , , )
(( ) ),
2
t
u x y tt
TE t t t t t
t                           (12)  
 
 
In order to obtain the truncation error for space GFD derivatives, Taylor’s series 
expansion including higher order derivatives is used and then higher order 
function *( )B u  is obtained 
2
2)
0 0 0 0
0
3) 4)
1
0 0 0 0
1
2
*( ) ( , )
1 1
...
6 24
i j j j j
N
j j
j
j j j j
U U U U
u u h k h k
x y x y
B u w h k
U U U U
h k h k
x y x y
 
 
If *( )B u is minimized with respect to the partial derivatives up to second order, 
the following linear equation system is defined 
2 2
1 1 1 1 12 2
T
N N N N N
j j
j j j j
j j j j j
h k
h k h kuA D
                                (13)                                   
 
where 
3) 4)
2
0 0 0 0
0
1 1
... ( , )
6 24
i j j j j j j
U U U U
u u h k h k w h k
x y x y
 
with N=8, and then 
2 2
1
( , )
1 1 1 1 12 2
T
N N N N N
j j
x y j j j j
j j j j j
h k
TE h h h kCA
           (14)
 
where 
3) 4)
2
0 0 0 01 1 ... ( , )
6 24
j j j j j j
U U U U
h k h k w h k
x y x y
 
 
MODELLING OF THE ADVECTION-DIFFUSION  
3 2
2 2 2 2 2 2 2
1 1 1 1 1
2 3
2 2 2 2 2 2
1 1 1 1
4 2 2 3
2 2 2
1 1 1
4 3
2 2
1 1
2 2 2
1
2 2
2 2
4 4 2
4 2
N N N N N
j j j
j j j j j
j j j j j
N N N N
j j j
j j j
j j j j
N N N
j j j j j
j j j
N N
j j j
j j
N
j j
j
h h k
h w h k w w w h k w
h k k
k w w w h k w
h h k h k
w w w
k h k
w w
h k w
A  
22
0
2 2
yx
x y x y
tt
C  
where 
 
2
2 ( , )j jw w h k  
Then by solving (14), we obtain 
 
3 3 3 3
( , ) 1, 2, 3, 4,3 2 2 3
1
3 3 3 3
5, 6, 7, 8,3 2 2 3
1
2 3 3 3
9, 10, 11,3 2
....
....
2
N
x y x j j j j
j
N
y j j j j
j
x
x j j j
U U U U
TE
x x y x y y
U U U U
x x y x y y
t U U
x x y
3
12,2 3
1
2 3 3 3 3
13, 14, 15, 16,3 2 2 3
1
....
.... ,
2
N
j
j
N
y
y j j j j j j
j
U U
x y y
t U U U U
h k
x x y x y y
(15)
 
where , ( , )i j j jh k
 
 are second-order rational functions and  ,j jh k  is a series 
of third- and higher-order functions.
 The total truncation error for advection-diffusion equation is 
 
                                       TTE=TEt+TE(x,y)                                                        (16)        
 
By considering bounded derivatives in TTE, we have consistency 
 
                                                                                                                           (17) 
,
0
j jt, h k 0, 0, 0                             
        lim TTE
MODELLING OF THE ADVECTION-DIFFUSION  
 
“Boundary conditions are neglected by the von Neumann method which applies 
in theory only to pure initial value problems with periodic initial data. It does 
however provide necessary conditions for stability of constant coefficient 
problems regardless of the type of boundary condition” [9]. 
From the previously obtained formula (9) 
 
1 2 2
0 0 0 0 0 0
2 2
1 1 1 1
1 ( ) ( )
2 2
( ) ( )
2 2
n n
x y x x y y
N N N N
n n n n
x j j y j j x x j j y y j j
j j j j
t t
u t m u
t t
t u u m u u
  (18)
 
 
Fort the stability analysis a harmonic decomposition is made of the approximate 
solution at grid points at a given time level n 
  
0 00 0
( ( ) )( )
0 ;
x j y jx y
i x h y ki x yn n n n
ju e u e  
where 
0 0( , )x y  are the coordinates in the central node of the star, and ( , )j jh k  are 
the coordinates of the other nodes of the star with respect to the central node. 
 
2 2
1
2 2
1
1 ( ) ( ) 1 cos( )
2 2
( ) ( ) ( )
2 2
N
x x j y y j x j y j x j y j
j
N
x x j y y j x j y j x j y j
j
t t
t m h k
t t
i t m sen h k
 
Taking into account that the stability condition is 1, then 
a)  Real( ) 1 
 
2 2
1
2 2
1
2 2
0 0 0 0
1 ( ) ( ) 1 cos( ) 1
2 2
1 1 ( ) ( ) 1 cos( ) 1
2 2
( ) ( )
2 2
N
x x j y y j x j y j x j y j
j
N
x x j y y j x j y j x j y j
j
x y x x y y
t t
t m h k
t t
t m h k
t t
t m 1                                                       (19)
 
 
b)  1 
MODELLING OF THE ADVECTION-DIFFUSION  
 
 
2
2 2 2
1
2
2 2
1
1 ( ) ( ) 1 cos( )
2 2
( ) ( ) ( ) 1
2 2
N
x x j y y j x j y j x j y j
j
N
x x j y y j x j y j x j y j
j
t t
t m h k
t t
t m sen h k
 
2
2 2
1
2 2
1
2 2
1
( ) ( ) ( )
2 2
( ) ( ) 1 cos( )
2 2
2 ( ) ( ) 1 cos(
2 2
N
x x j y y j x j y j x j y j
j
N
x x j y y j x j y j x j y j
j
N
x x j y y j x j y j x j
j
t t
t m sen h k
t t
m h k
t t
t m h )y jk
 
 
 
 
2 2
2 2
1 1
2 2
2 2
0 0
1 1
1
5 5
6 6
N N
x j y j
j j
N N
x x j y y j
j j
t
m t t
          (20)
 
 
Both formulae (19) and (20) give us the conditions for the stability. 
 
5. Numerical results 
 
In order to illustrate the application of the numerical explicit GFD scheme 
developed previously, a problem for which an exact solution is available is 
required so that approximate results obtained can be compared with an exact 
solution. The problem to be solved is 
 
 
(21) 
 
 
 
The exact solution is 
2 2
2 2
2 2
, , , , ,
0.1
0, 9 y 25
U t U t U t U t U t
t x y x y
t x
x x x x x
MODELLING OF THE ADVECTION-DIFFUSION  
 
 
 
 
 
 
(22) 
 
 
The global error is evaluated in the last step considered, using the following 
formula 
 
 
(23) 
 
 
where sol(i) is the GFDM solution at the node i, exac(i) is the exact value of 
solution at the node (i), 
maxexac  is the maximum value of the exact solution in 
the cloud of nodes considered and M is the total number of nodes of the domain. 
In this problem we consider different irregular clouds of points as given in Fig. 1.  
   
 
 
Fig. 1. Different irregular clouds of points. 
max
M
1
2
exac
M
iexacisol
errorglobal
1.8
2 2
j j
3
2 2
j j
U x, y, t
Initial conditions:   U ,0
 boundary conditions at 9 y 25
1
The weight function is:   w x , y
x  y
t x ye
Dirichlet x
x
MODELLING OF THE ADVECTION-DIFFUSION  
The influence on global error of using different number of nodes is given in Fig. 
2. 
 
 
 
Fig. 2 Global error versus the number of nodes.   
 
Also we consider for the irregular cloud of 1248 nodes the influence on global 
error versus different values of time increment in Fig.3.  
 
 
 
Fig. 3  Global error versus the time increment.  
 
As it is shown in Fig.2  and 3, the global error decreases by increasing the number 
of nodes or decreasing the time increment. 
6. Conclusions 
 
An explicit solution of advection-diffusion equation has been presented for the 
case of using the GFDM over irregular grids. We have defined the truncation 
0
0,0002
0,0004
0,0006
0,0008
0,001
0,0012
0,0014
160 336 576 1248
% GLOBAL ERROR (with t = 0,0005)
% GLOBAL ERROR (1248 nodes)
0
0.001
0.002
0.003
0.004
0.005
0.006
0.0005 0.001 0.005 0.01
MODELLING OF THE ADVECTION-DIFFUSION  
error of the scheme in the case of irregular grids of nodes. Then, we have 
established the consistency and stability (following the von Neumann stability 
analysis) criteria for this scheme. An academic test has been presented to 
illustrate the application of this method. 
The fully explicit generalized finite difference schemes are simple to 
implement and economical to use. They are very efficient and very quick. 
They are conditionally stable. 
The modified equivalent partial differential equation approach of Warming 
and Hyett[8] has been employed which permits to demonstrate that the GFD 
scheme is free of numerical diffusion.  
Acknowledgements 
The authors acknowledge the support fron Ministerio de Ciencia e Innovación 
of Spain, research project CGL2008-01757/CLI. 
References 
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[6] J.J. BENITO, F. UREÑA , L. GAVETE, Application of the Generalized Finite 
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Modelling of the advection-diffusion equation with a meshless method without numerical diffusion

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