144
5. Finite Element Methods for Solving Hydrodynamic Dispersion Equations
Assume that the concentration distribution C r at time t has been obtained (at
t = 0 using the initial condition), then, Cr+Ar at the next time (t + L\t) can be
solved by Eq. (5.4.4). Thus, a step-by-step process can be formed. Using the
FDM terminology, when () = 0, the relevant discrete scheme is an explicit
one; when () = 1/2, it is a Crank-Nicolson scheme; and when () = 1, it is an
implicit one.
We know that the coefficient matrix of the finite element system derived
from flow equations is sparse, symmetric and positive definite. This type of
system of equations can be solved by a special LU decomposition method. In
contrast, owing to the existence of the advection term, the coefficient matrix
[T] of Eq. (5.4.4) is definitely asymmetric. Both the computer memory for this
matrix and the solution of this system of equations require special treatments. These two aspects are not separable: the arrangement of the memory
must be adaptive to the solution method.
A lot of studies have been contributed to direct solutions for large, sparse,
and asymmetrie equation systems. We know that if the general method
of principal element elimination is used, the sparse coefficient matrix will
be "filled in." That is to say, non-zero elements will appear in the original positions ofthe zero elements during the elimination.lfwe wish to reduce
the "filling in," there will be no choiee but to write more complex programs.
Gupta and Tanji (1977) put forward both a method and relevant programs, whieh are applicable to the solution of the finite element systems of
water quality. It is known from our experience that direct solutions will be of
lower efficiency than iterative solutions when the number of nodes exceeds
500.
5.4.2 Point Iteration Methods
The point iteration method is the simplest among the iteration methods,
because it does not depend on whether the coefficient matrix is symmetric or
not. Therefore, it can be used for both groundwater flow problems and mass
transport problems. Another advantage of this method is that it is relatively
easy to program.
In order to store the coefficient matrix [T], we can create a two-dimensional array (MCPN) to store the numbers of the neighboring nodes of each
node. In addition, we can use another array (MA) to store the total number
of the neighboring nodes of each node. Coefficients A ij and B ij relating to
each node (i) with its surrounding nodes (j) can then be stored into onedimensional arrays (CF A) and (CFB) by the control of (MA) and (MCPN).
Using this method to store coefficient matrices [A] and [B] ofEq. (5.4.1), the
storage of all zero elements can be eliminated. The coefficient matrix [T] can
also be stored in a one-dimensional array with no zero elements. It is formed
by multiplying the elements of (CF A) by (), dividing the elements of (CFB) by
M, and summing these products element by element.
5. Finite Element Methods for Solving Hydrodynamic Dispersion Equations
Assume that the concentration distribution C r at time t has been obtained (at
t = 0 using the initial condition), then, Cr+Ar at the next time (t + L\t) can be
solved by Eq. (5.4.4). Thus, a step-by-step process can be formed. Using the
FDM terminology, when () = 0, the relevant discrete scheme is an explicit
one; when () = 1/2, it is a Crank-Nicolson scheme; and when () = 1, it is an
implicit one.
We know that the coefficient matrix of the finite element system derived
from flow equations is sparse, symmetric and positive definite. This type of
system of equations can be solved by a special LU decomposition method. In
contrast, owing to the existence of the advection term, the coefficient matrix
[T] of Eq. (5.4.4) is definitely asymmetric. Both the computer memory for this
matrix and the solution of this system of equations require special treatments. These two aspects are not separable: the arrangement of the memory
must be adaptive to the solution method.
A lot of studies have been contributed to direct solutions for large, sparse,
and asymmetrie equation systems. We know that if the general method
of principal element elimination is used, the sparse coefficient matrix will
be "filled in." That is to say, non-zero elements will appear in the original positions ofthe zero elements during the elimination.lfwe wish to reduce
the "filling in," there will be no choiee but to write more complex programs.
Gupta and Tanji (1977) put forward both a method and relevant programs, whieh are applicable to the solution of the finite element systems of
water quality. It is known from our experience that direct solutions will be of
lower efficiency than iterative solutions when the number of nodes exceeds
500.
5.4.2 Point Iteration Methods
The point iteration method is the simplest among the iteration methods,
because it does not depend on whether the coefficient matrix is symmetric or
not. Therefore, it can be used for both groundwater flow problems and mass
transport problems. Another advantage of this method is that it is relatively
easy to program.
In order to store the coefficient matrix [T], we can create a two-dimensional array (MCPN) to store the numbers of the neighboring nodes of each
node. In addition, we can use another array (MA) to store the total number
of the neighboring nodes of each node. Coefficients A ij and B ij relating to
each node (i) with its surrounding nodes (j) can then be stored into onedimensional arrays (CF A) and (CFB) by the control of (MA) and (MCPN).
Using this method to store coefficient matrices [A] and [B] ofEq. (5.4.1), the
storage of all zero elements can be eliminated. The coefficient matrix [T] can
also be stored in a one-dimensional array with no zero elements. It is formed
by multiplying the elements of (CF A) by (), dividing the elements of (CFB) by
M, and summing these products element by element.