Geoscience Reference
In-Depth Information
the problems in computational river dynamics are nonlinear, so the Lax equivalent
theorem is not always applicable.
In general, the tendency of convergence can be tested by successively refining the
computational grid and computing the root-mean-square error of the solution of each
dependent variable by Eq. (4.6). If
R
f
0, the numerical solution
is convergent. However, this test shows only the tendency rather than the ultimate
convergence (
R
f
→
0as
x
→
0), because the mesh cannot be infinitely refined and the round-off
errors may increase as the number of grid points increases.
=
4.1.3 Discretization methods
Widely used discretization methods include finite difference method, finite element
method, finite volume method, finite analytical method, and efficient element method.
The finite difference method discretizes a differential equation by approximating dif-
ferential operators with difference operators at each point. The finite analytical method
discretizes the differential equation using the analytical solution of its locally linearizd
form, and the efficient element method establishes difference operators using interpola-
tion schemes in local elements. Because of their similarity, the finite analytical method
and efficient element method are herein grouped with the finite difference method.
The finite volume method integrates the differential equation over each control vol-
ume, holding the conservation laws of mass, momentum, and energy. In the finite
element method, the differential equation is multiplied by a weight function and inte-
grated over the entire domain, and then an approximate solution is constructed using
shape functions and optimized by requiring the weighted integral to have a minimum
residual.
The algebraic equations resulting from the finite difference and finite volume meth-
ods usually have banded and symmetric coefficient matrices that can be handled
efficiently, whereas the algebraic equations from the finite element method often
have sparse and asymmetric coefficient matrices that require relatively tedious effort
for solution. However, the classic finite difference and finite volume methods adopt
structured, regular meshes and encounter difficulties in conforming to the irregular
domains of river flow, while the finite element method adopts unstructured, irreg-
ular meshes and can conveniently handle such irregular domains. Therefore, it has
been a trend in recent decades to develop the finite difference and finite volume meth-
ods on irregular meshes, which have the grid flexibility of the finite element method
and the computational efficiency of the classic finite difference and finite volume
methods.
The finite difference method and finite volume method are introduced in this topic.
The finite element method has also been used in many river models because of its grid
flexibility; however, it is absent from this topic due to the author's limited expertise.
Interested readers are encouraged to consult other references, such as Chung (1978),
Fletcher (1991), and Zienkiewicz and Taylor (2000).
One suggestion to new model developers and users is that any numerical method
may have its advantages and disadvantages, and subjectivity may prevent you from
becoming more successful. You should learn the basic properties — such as accuracy,
stability, convergence, and efficiency — of the method that you are going to use and
know how to take advantage of its strengths and avoid its weaknesses.
