Above is only a small sample of the numerous techniques that can be used to solve the advective-diffusion equation. Others can be found in Kalf (1981), Hogarth et al. (1990), Cox and Nishikawa (1991) and Nixon and Noye (1991).
The scheme described by Kalf (1981) is worth mentioning briefly. Kalf (1981) describes the
Truncation Cancellation Procedure of Laumbach (1975). The implicit based scheme is designed so that the discretization error associated with the temporal term, dc/dt in the one-dimensional advective-diffusion equation cancels part of the error associated with the convection term. Numerical tests by Kalf (1981) suggest that although the scheme is unconditionally stable, the scheme maintains its accuracy only when Cr < 1. Since this restriction increases the number of time steps required and it is an implicit scheme, it may be computationally more expensive
be be n d2c Ax3(l - 2Cr2) l b4c
bt bx bx2 12 PeCr dx4—bx4 + 0 ( Ax4) = 0
which indicates that the scheme is third-order accurate.
The flux for the high-order scheme is given by Morrow and Noye (1992) as
Numerical Methods fo r Solving the One-Dimensional Advective-Diffusion Equation 2.48
Ö 80.
O 40.
.0 25. 50. 75. 100.
d is t a n c e
Figure 2.16 Analytical and flux corrected transport solution of the advection equation
than explicit schemes.
There are also a number of alternative techniques that could be used to solve the advective- diffusion equation. These include; (/) spectral (see, for example Gottlieb and Orszag [1977] Gottlieb et al. [1984] and Goblet and Cordier [1993]), (if) finite elements (see, for example Guymon [1970], Gray and Pinder [1976], van Genuchten [1977], Donea [1984], Donea et al.
[1984], Tezduyar and Gunjoo [1986] and Putti et al. [1990]) and (Hi) Laplace transform (see, for example Celia et al. [1989], Sudicky [1989], Moridis and Reddell [1991], Li et al. [1992], Yates [1992] and Deng et al. [1993]).
Many of these techniques have been known for many years. However, they have not gained the prominence of finite difference schemes. A reason for this is that many of these schemes have restrictions on the type of problem that they can solve, such as restricted boundary conditions, which do not exist for finite difference schemes. Alternatively, they may be difficult to implement or they do not provide significant advantages over finite difference schemes.
Recent developments with the use of Laplace transform have shown that this may be a viable alternative for the solution of both the advection and advective-diffusion equation for practical problems. Discussion of the Laplace transform technique is left to Chapter 4.
In general, finite element methods are more suited than finite-difference schemes to multi dimensional problems involving irregular boundaries.
A major criticism for the application of finite elements for the solution of the advective-diffusion equation is the computational effort required to solve large systems of simultaneous equations at each time step. For large simulation times or for small time steps this may be prohibitive.
For example, the solution of equation (1.2) by finite element methods leads to the following system of first-order differential equations
[M]{c} + [K]{c} = {f} (2'43)
in which {c} is a vector of unknown nodal concentrations, {d} is the vector of temporal derivatives, dcldt, [A] is known as the conductivity matrix, [M] is the capacity matrix and the vector {/} includes the effect of the source-sink term and boundary conditions..
Both [A] and [M] are N x AT symmetric and positive definite matrices for the diffusion equation, where N is the number of computational nodes in the problem. For the advective-diffusion equation [M] is symmetric and positive definite, however [A] is composed of a symmetric matrix [AD] containing the discrete form of the diffusion process and an asymmetric matrix [AJ from the advection process.
These equations are usually descretized in time by a finite difference scheme and solved for the unknown concentrations. For example, the Crank-Nicolson scheme gives
[M] + {c}' {c}n + A t { f } n+m. (2.44)
The solution of this equation involves solving a system of simultaneous equations each time step. For problems with a large number of computational nodes or for long simulation times using small time steps this may require considerable computer resources.
The generalized conjugate residual method is often used in practice to solve large nonsymmetric indefinite problems (see, for example Fletcher [1975], Saad and Schultz [1986] and van der Vorst [1992]).
Alternatively, instead of solving equation (2.44) the system of ordinary differential equations, equation (2.43) could be solved. The main advantage of solving the system of ordinary differential equations is that solutions that are continuous in time are possible, thereby avoiding the discretization errors associated with finite difference schemes. In addition, there are a number of reduction techniques that could be used to significantly reduce the computational effort required to solve large systems of equations. This is achieved by reducing the solution of a large system of ordinary differential equations into the solution of a smaller set of equations. Discussion of a number of reduction methods can be found in Amoldi (1951), Parlett (1980), Nour-Omid et al. (1983), Hwang et al. (1984), Nour-Omid (1987), Dunbar and Woodbury (1989), Woodbury et al. (1990), Nour-Omid et al. (1991) and Gambolati (1993).
Two reduction techniques that could be employed are the classical modal decomposition or methods based on the Lanczos vector.
The basis of the modal decomposition method is to treat equation (2.43) as an eigenvalue problem in which the solution of a system of equations can be accurately described by a few "modes". A reduction in size is achieved by using a subset of the eigenvectors corresponding to the smallest eigenvalues to obtain the solution. Generally, for large problems the number of modes required to obtain an accurate solution is much smaller than the number of computational
Numerical Methods fo r Solving the One-Dimensional Advective-Diffusion Equation 2.50
nodes. Hwang et al. (1984) treated the solution of equation (2.43) as an eigenvalue problem. The major difficulty with this approach is that for large matrices convergence problems may arise in the iterative procedure used to calculate the required eigenvalues.
Instead of using eigenvalues, Lanczos or Amoldi vectors may be used (see, for example Nour- Omid et al. [1983], Nour-Omid [1987], Dunbar and Woodbury [1989], Woodbury et al. [1990] and Nour-Omid et al. [1991]).
In the Lanczos based methods the behaviour of the equation being modelled can be described with a co-ordinate transformation matrix which is a subset of the Lanczos vector (see, for example Gambolati [1993]).
The Amoldi method may be viewed as an extension of the Lanczos method to asymmetric matrices. It is less efficient than the Lanczos method for symmetric matrices.
Nour-Omid et al. (1991) compared the Lanczos and Amoldi method for the solution of the advective-diffusion equation. They concluded that the Lanczos method is preferred for diffusion dominated problems, while Amoldi is the method of choice for the advection dominated problems. As Pe increased, more Amoldi and Lanczos vectors were required to produce accurate results. This would also be the case for the eigenvector problem.
Lanczos vectors produce a more accurate solution and are easier to generate than the use of the same number of eigenvalues. The calculation of the Amoldi and Lanczos vectors are computationally more expensive than the cost of solving a single time step in a finite element scheme. However, for constant coefficient problems the Amoldi and Lanczos vectors are only computed once, resulting in considerable computational savings for long simulation times. The advantages of the Amoldi and Lanczos methods may not be revealed for time dependent boundary conditions or for variable coefficient problems where the matrix [M] and [K] and the Amoldi and Lanczos vector must be evaluated every time step. In addition, they are both sensitive to the initial estimate of the vector.
Woodbury et al. (1990) compared the computational efficiency of the Amoldi method with the Crank-Nicolson solution of the advective-diffusion equation. For the constant coefficient problem, the Amoldi method was found to be more efficient than Crank-Nicolson method. It is not known whether it is significantly more efficient than standard explicit schemes or methods based on the Laplace transform.
On the other hand, Gambolati (1993) compared the Crank-Nicolson finite difference scheme, the Lanczos method and the eigenvalue problem for the solution of the diffusion equation. Although he found that the Lanczos method was superior to the eigenvalue problem, both were inferior to the Crank-Nicolson scheme, which was more robust, simpler and more efficient. The only circumstances where the reduction algorithms were superior to the finite difference scheme were for steady state problems.