6
Numerical Solutions of
Advection-Dominated Problems
6.1 Advection Dominated Problems
6.1.1 Fourier Analysis of Numerical Errors
When the advection term dominates in the advection-dispersion equation,
most traditional numerical methods will encounter difficulties. In Chapter 4,
we have stated that when the FDM is used for solving advection-dominated
problems, two kinds of errors, numerical dispersion and overshoot, will occur. As a result, oscillations will appear around concentration fronts and
steep concentration fronts cannot be accurately calculated. In Chapter 5, we
pointed out that the Galerkin FEM cannot avoid the above two kinds of
errors, either. Consequently, improvement of the accuracy and stability of
numerical solutions has become an important subject in current research.
In order to further analyze the origins of numerical errors discussed in
Chapter 4, let us again consider the simplest one-dimensional advectiondispersion equation:
(6.1.1)
where the dispersion coefficient D and average velocity V are assumed to be
constant. Using Fourier's analysis method, the general solution of Eq. (6.1.1)
may be expressed by the following Fourier's series:
(6.1.2)
n=-oo
where ßn is the time frequency of the nth component; O'n the wave number or
spatial frequency; i = .j=t is the imaginary unit. Since Eq. (6.1.1) is linear,
each component of the series in Eq. (6.1.2) may be individually considered
according to the principle of superposition. For instance, consider the nth
component, and let
(6.1.3)
149
Numerical Solutions of
Advection-Dominated Problems
6.1 Advection Dominated Problems
6.1.1 Fourier Analysis of Numerical Errors
When the advection term dominates in the advection-dispersion equation,
most traditional numerical methods will encounter difficulties. In Chapter 4,
we have stated that when the FDM is used for solving advection-dominated
problems, two kinds of errors, numerical dispersion and overshoot, will occur. As a result, oscillations will appear around concentration fronts and
steep concentration fronts cannot be accurately calculated. In Chapter 5, we
pointed out that the Galerkin FEM cannot avoid the above two kinds of
errors, either. Consequently, improvement of the accuracy and stability of
numerical solutions has become an important subject in current research.
In order to further analyze the origins of numerical errors discussed in
Chapter 4, let us again consider the simplest one-dimensional advectiondispersion equation:
(6.1.1)
where the dispersion coefficient D and average velocity V are assumed to be
constant. Using Fourier's analysis method, the general solution of Eq. (6.1.1)
may be expressed by the following Fourier's series:
(6.1.2)
n=-oo
where ßn is the time frequency of the nth component; O'n the wave number or
spatial frequency; i = .j=t is the imaginary unit. Since Eq. (6.1.1) is linear,
each component of the series in Eq. (6.1.2) may be individually considered
according to the principle of superposition. For instance, consider the nth
component, and let
(6.1.3)
149