Modified mixed Lagrangian-Eulerian method based on Numerical
framework of MT3DMS on Cauchy boundary
Appendix S1: Assembling the global matrix equation
The process of assembling Equations 7 and 8 into a global matrix equation, and the
corresponding manipulation of the right-hand-side vector, are discussed here. The global
matrix equation can be written by combining Equations 7 and 8 as follows:
HF CFn1  RF  ZF 
(S1)
where
CFn 1
HF   MR
BE
t
C1n 1 
 n 1 
C 7 
C8n 1 
 n 1 
C 2 
C n 1 
 9n 1 
C 
  10n 1 
C 3 
C11n 1 
 n 1 
C12 
C n 1 
 4n 1 
C 5 
C n 1 
 6 
(S2a)
  SR  ER  MR   SR  VR 
δt x 
IE
BE
BE
IE
*
IE
(S2b)
RF  MR
BE
t
ZF   FR BE BR IE 
CF   MR CF 
δt x 
IE
B n
I *
(S2c)
*

 

FF11,k1
FF11,k1


1,k 1
1,k 2  
 FF7
0
 FF7
 

 FF81,k 2  FF81,k 3  

0




1,k 3
 BF22 ,k1   FF21,k 3  BF22 ,k1 
 FF2
 BF 2 ,k1  BF 2 ,k 2  

0
9
 92 ,k 2



 2 ,k 3
0
 BF10  BF10  

  2 ,k 3

3 ,k 1   
 BF3
0
 BF3
 

 BF113,k1  BF113,k 2  

0
 3,k 2

3 ,k 3  
0
 BF12  BF12  

 BF 3,k 3  BF 4 ,k1  

0
4
 44 ,k1



 BF55 ,k1  
0
 BF5





5 ,k 1
5 ,k 1
BF6
BF6

 

(S2d)
In Equation S2d, except for the first, fourth, and last rows, all other terms are zero
because FFi e ,k1 and FFi e ,k 2 of the two adjacent elements, ( k1 and k 2 ), contiguous to
interior node i , cancel each other out in a finite element approach (Istok 1989). Similarly,
BFi e ,k 1 and BFi e ,k 2 are also cancelled out. However, FF21,k 3 and BF22 ,k1 will not be
cancelled out because FF21,k 3 is obtained by applying an FEM for fine-mesh element k 3
in global element (1), and BF22 ,k1 is obtained by applying an ELM for fine-mesh element
k1 in global element (2), as shown in Figure 1. This can be directly illustrated by including
FF21,k 3 and BF22 ,k1 in Equation S3:
C 
 C 
ZF4  FF21,k 3  BF22 ,k1  VC  D
 D
 VC  x x


x  x x 
x  x x

2
2
2
(S3)
Here, it should be noted that because the velocity computation in an FEM is continuous
at the elemental interface between global elements (1) and (2) (Yeh 1981; Park and Aral
2007), and the dispersion coefficient is a function of the velocity, the dispersion coefficient is
continuous. Therefore, Equation S3 holds. Moreover,
VC  x x
on the right side of
2
Equation S3 can be rewritten as
VC  x x
2
   x2 V  x2   C  x2 
(S4)
where V x2  is already known because the flow equation is usually solved in advance.
However, Cx2  is unknown and must be solved. Therefore, a fourth row of the global
matrix HF  and a fourth element ZF  should be modified as follows:
HF j , j1C8n1  HFj, j C2n1  HF j , j1C9n1  RF j  ZFj , j  4
(S5)
where
HF j, j  HF j , j   x2 V x2  ,
j4
(S6a)
ZF j  ZF j  VC  xx  0 ,
j4
(S6b)
2
It should be noted that ZF  is modified using Equations S3–S6, so that the fourth row
of the load vector becomes zero. So, the load vector modified from ZF  is as follows:

C  

VC


D

 

 FF1  
x  x 0 

 
0

 0  


  
 and it can be calculated by using Cauchy boundary 5a and
 0  

0

 

 BF65 ,k1   D C 

 
x  x  L 
1,k 1
Neumann boundary 5b.
References
Istok, J.D. 1989. Groundwater modeling by the finite element method. Water Resources
Monograph no. 13. Washington (DC): American Geophysical Union.
Park, C.-H., and M.M. Aral. 2007. Sensitivity of the solution of the Elder problem to density,
velocity and numerical perturbations. Journal of Contaminant Hydrology 92, no. 1-2:
33-49.
Yeh, G.T. 1981. On the computation of Darcian velocity and mass balance in the finite
element modeling of groundwater flow. Water Resources Research 17, no. 5: 1529–1534.