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5. Finite Element Methods for Solving Hydrodynamic Dispersion Equations
Layer iteration can be done in both directions, i.e., from bottom to top and
vice versa. First, rewrite Eq. (5.4.12) in the ascending layer order:
(5.4.14)
and get the solution for each layer (m = 1,2, ... , M). Then, rewrite Eq. (5.4.12)
as
[T ] C (k+2) - R - [T] C(k+2) - [T ] C(k+l)
1 m m
-
m
2 m (m+l)
3 m m-l'
(5.4.15)
and find the solution of each layer successively in the descending layer order
(m = M, M - 1, ... ,1). We must pay attention to the application ofboundary
conditions in setting up each system of Eqs. (5.4.14) and (5.4.15).
To find the solution for the current time step, the above procedure are
repeated until a convergence criterion is satisfied. The next time step can then
be calculated in the same way.
Exercises
5.1. Derive the weighted residual method for the following equation:
O(OC)
at = div(OD grad C) - div(OCV) - ;'BC.
5.2. What are the advantages of selecting basis functions which satisfy the
requirements presented in Section 5.1.2? Are the basis functions uniquely
determined by these requirements? Prove that any basis functions satisfying these requirements must be independent of each other.
5.3. Give the details of deriving the finite element equations (5.1.17) when
triangle elements and linear basis functions in Eq. (5.1.32) are used.
5.4. Prove that the basis functions given in Table 5.1 for the quadratic element satisfy all requirements presented in Section 5.1.2.
5.5. Suppose that an arbitrary quadrilateral is used as an element, and its
four vertices are defined as nodes. What are the basis functions associated with the nodes in this element? Give algorithms for calculating the
coefficient matrices A and B of the finite element equations.
5.6. Derive the Multiple Cell Balance Method for the following equation
oC.
.
7ft = dzv(D grad C) - dzv( CV) - ;"C,
subject to the first and second types of boundary conditions.
5.7. Using the programs given in Appendix B, reproduce the results presented in Section 5.2.4.
5.8. Design a two-dimensional problem that may be countered in the field
and obtain the solution using the program given in Appendix B.
5. Finite Element Methods for Solving Hydrodynamic Dispersion Equations
Layer iteration can be done in both directions, i.e., from bottom to top and
vice versa. First, rewrite Eq. (5.4.12) in the ascending layer order:
(5.4.14)
and get the solution for each layer (m = 1,2, ... , M). Then, rewrite Eq. (5.4.12)
as
[T ] C (k+2) - R - [T] C(k+2) - [T ] C(k+l)
1 m m
-
m
2 m (m+l)
3 m m-l'
(5.4.15)
and find the solution of each layer successively in the descending layer order
(m = M, M - 1, ... ,1). We must pay attention to the application ofboundary
conditions in setting up each system of Eqs. (5.4.14) and (5.4.15).
To find the solution for the current time step, the above procedure are
repeated until a convergence criterion is satisfied. The next time step can then
be calculated in the same way.
Exercises
5.1. Derive the weighted residual method for the following equation:
O(OC)
at = div(OD grad C) - div(OCV) - ;'BC.
5.2. What are the advantages of selecting basis functions which satisfy the
requirements presented in Section 5.1.2? Are the basis functions uniquely
determined by these requirements? Prove that any basis functions satisfying these requirements must be independent of each other.
5.3. Give the details of deriving the finite element equations (5.1.17) when
triangle elements and linear basis functions in Eq. (5.1.32) are used.
5.4. Prove that the basis functions given in Table 5.1 for the quadratic element satisfy all requirements presented in Section 5.1.2.
5.5. Suppose that an arbitrary quadrilateral is used as an element, and its
four vertices are defined as nodes. What are the basis functions associated with the nodes in this element? Give algorithms for calculating the
coefficient matrices A and B of the finite element equations.
5.6. Derive the Multiple Cell Balance Method for the following equation
oC.
.
7ft = dzv(D grad C) - dzv( CV) - ;"C,
subject to the first and second types of boundary conditions.
5.7. Using the programs given in Appendix B, reproduce the results presented in Section 5.2.4.
5.8. Design a two-dimensional problem that may be countered in the field
and obtain the solution using the program given in Appendix B.