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6. Numerical Solutions of Advection-Dominated Problems
Regarding other numerical solutions to advection-dominated problems,
the work of some experts should be mentioned: the finite analytic solution by
Hwang et al. (1985), the boundary element method by Taigbenu and Liggett
(1986), the advection controlling method by Sun and Liang (1988), and
the Laplace transform Galerkin method by Sudicky (1989). Li et al. (1992)
extended the finite analytic method to solve the transient advectiondispersion problem through the Laplace transformation. When advection
dominates dispersion, this method can produce accurate results for a wide
range of Peclet numbers. Zeitoun and Pinder (1993) used an optimal control
least squares method to solve coupled flow-transport problems. At each time
step, the solution of the discretized differential system is transformed into
an optimal control problem. A one-dimensional example shows that this
method can produce more accurate results than FEM when the Peclet number is large.
Exercises
6.1. Calculate the numerical enlargement factor when the Crank-Nicolson
finite difference scheme is used for solving Eq. (6.1.1).
6.2. Derive the third-order upstream difference formula in Eq. (6.2.9).
6.3. Use the example given in Section 5.2.4 to compare the results obtained
by FDM, FEM, or MCBM, with and without upstream weights.
6.4. Extend Eq. (6.3.10) to the two-dimensional case.
6.5. Solve the one-dimensional advection-dispersion problem presented in
Section 5.2.4 with the single step reverse method.
6.6. Write a flow chart for the hybrid moving point and characteristics FEM
given in Section 6.4.3.
6. Numerical Solutions of Advection-Dominated Problems
Regarding other numerical solutions to advection-dominated problems,
the work of some experts should be mentioned: the finite analytic solution by
Hwang et al. (1985), the boundary element method by Taigbenu and Liggett
(1986), the advection controlling method by Sun and Liang (1988), and
the Laplace transform Galerkin method by Sudicky (1989). Li et al. (1992)
extended the finite analytic method to solve the transient advectiondispersion problem through the Laplace transformation. When advection
dominates dispersion, this method can produce accurate results for a wide
range of Peclet numbers. Zeitoun and Pinder (1993) used an optimal control
least squares method to solve coupled flow-transport problems. At each time
step, the solution of the discretized differential system is transformed into
an optimal control problem. A one-dimensional example shows that this
method can produce more accurate results than FEM when the Peclet number is large.
Exercises
6.1. Calculate the numerical enlargement factor when the Crank-Nicolson
finite difference scheme is used for solving Eq. (6.1.1).
6.2. Derive the third-order upstream difference formula in Eq. (6.2.9).
6.3. Use the example given in Section 5.2.4 to compare the results obtained
by FDM, FEM, or MCBM, with and without upstream weights.
6.4. Extend Eq. (6.3.10) to the two-dimensional case.
6.5. Solve the one-dimensional advection-dispersion problem presented in
Section 5.2.4 with the single step reverse method.
6.6. Write a flow chart for the hybrid moving point and characteristics FEM
given in Section 6.4.3.