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Chapter 9
Other numerical methods
9.1 OVERVIEW
In addition to the finite differences (FD) method, there are other numerical
approaches that have been developed and widely applied for the solution of differential equations. Overall, these alternate approaches have a domain discretization ability superior to the finite differences method but lack its simplicity.
In the following, a brief introduction will be provided of the more well-known
alternate numerical methods – the weighted residual method (WRM), the
finite elements method (FEM) and the boundary elements method (BEM).
9.2 WEIGHTED RESIDUAL METHODS
The weighted residual method was developed as an approximation technique for solving differential equations and served as a prelude to the finite
elements method. The basic concept of the WRM can be described as follows. Consider a linear differential operator L acting on a function f(x) to
produce a function g(x):
L[f(x)] = g(x)
(9.1)
Let the unknown function f(x) be approximated by a finite series,
f f
i i
i
n
≈ =
=
∑
α ϕ
1
(9.2)
where φ i are ‘trial’ or ‘basis’ functions selected from a linearly independent
set and α i are unknown constants. Substitution of the approximated function
(Equation 9.2) into the differential operator (Equation 9.1) results in an error:
R x L f x g x
( )
[ ( )] ( )
=
−
≠
0
(9.3)
Chapter 9
Other numerical methods
9.1 OVERVIEW
In addition to the finite differences (FD) method, there are other numerical
approaches that have been developed and widely applied for the solution of differential equations. Overall, these alternate approaches have a domain discretization ability superior to the finite differences method but lack its simplicity.
In the following, a brief introduction will be provided of the more well-known
alternate numerical methods – the weighted residual method (WRM), the
finite elements method (FEM) and the boundary elements method (BEM).
9.2 WEIGHTED RESIDUAL METHODS
The weighted residual method was developed as an approximation technique for solving differential equations and served as a prelude to the finite
elements method. The basic concept of the WRM can be described as follows. Consider a linear differential operator L acting on a function f(x) to
produce a function g(x):
L[f(x)] = g(x)
(9.1)
Let the unknown function f(x) be approximated by a finite series,
f f
i i
i
n
≈ =
=
∑
α ϕ
1
(9.2)
where φ i are ‘trial’ or ‘basis’ functions selected from a linearly independent
set and α i are unknown constants. Substitution of the approximated function
(Equation 9.2) into the differential operator (Equation 9.1) results in an error:
R x L f x g x
( )
[ ( )] ( )
=
−
≠
0
(9.3)