Chapter 1
Idea and Derivation of the Method
The first widely known approximation method for partial differential equations is
the finite difference method [1, 11] which approximates the governing differential
equations of a field problem using local expansions for the variables, generally truncated Taylor’s series. Comprehensive descriptions of the method can be found, for
example, in [4, 5, 7]. Other classical approximation methods are the finite element
method [9, 10], the finite volume method, and the boundary element method.
As a first step to solve a differential equation for a one-dimensional problem, it
is assumed that the numerical solution u(X ) is to be determined only at a set of n
points within the domain X ∈ [0, L] including the ends, cf. Fig. 1.1.
These n nodes or grid points will be used to derive approximations to the derivatives of the function u(X ). To simplify the approach, let us assume in the following
that these n nodes are equally spaced, at a distance equal to
1
X =
L
n−1
. As in the
case of the finite element method, high gradients would require a smaller node spacing in order to meet the requirements on the accuracy of the approximate solution.
However, the general idea of the finite difference method will be here introduced
based on an equi-spaced division of the domain 0 ≤ X ≤ L. A typical inner node
of the domain is denoted in the following by i and the two neighboring nodes are
called i − 1 on the left-hand and i + 1 on the right-hand side. For sufficient smooth
functions u(X ), a Taylor’s series expansion around node i gives:
u i+1 = u i +
du
dX
i
X +
d
2 u
dX 2
i
X
2
2
+ · · · +
d
k u
dX k
i
X
k
k!
,
(1.1)
u i−1 = u i −
du
dX
i
X +
d
2 u
dX 2
i
X
2
2
− · · · +
d
k u
dX k
i
X
k
k!
.
(1.2)
1 It should be noted here that the first node in this derivation is denoted by ‘1’ and not as in some
references as ’0’.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. Öchsner, Structural Mechanics with a Pen,
https://doi.org/10.1007/978-3-030-65892-2_1
1
Idea and Derivation of the Method
The first widely known approximation method for partial differential equations is
the finite difference method [1, 11] which approximates the governing differential
equations of a field problem using local expansions for the variables, generally truncated Taylor’s series. Comprehensive descriptions of the method can be found, for
example, in [4, 5, 7]. Other classical approximation methods are the finite element
method [9, 10], the finite volume method, and the boundary element method.
As a first step to solve a differential equation for a one-dimensional problem, it
is assumed that the numerical solution u(X ) is to be determined only at a set of n
points within the domain X ∈ [0, L] including the ends, cf. Fig. 1.1.
These n nodes or grid points will be used to derive approximations to the derivatives of the function u(X ). To simplify the approach, let us assume in the following
that these n nodes are equally spaced, at a distance equal to
1
X =
L
n−1
. As in the
case of the finite element method, high gradients would require a smaller node spacing in order to meet the requirements on the accuracy of the approximate solution.
However, the general idea of the finite difference method will be here introduced
based on an equi-spaced division of the domain 0 ≤ X ≤ L. A typical inner node
of the domain is denoted in the following by i and the two neighboring nodes are
called i − 1 on the left-hand and i + 1 on the right-hand side. For sufficient smooth
functions u(X ), a Taylor’s series expansion around node i gives:
u i+1 = u i +
du
dX
i
X +
d
2 u
dX 2
i
X
2
2
+ · · · +
d
k u
dX k
i
X
k
k!
,
(1.1)
u i−1 = u i −
du
dX
i
X +
d
2 u
dX 2
i
X
2
2
− · · · +
d
k u
dX k
i
X
k
k!
.
(1.2)
1 It should be noted here that the first node in this derivation is denoted by ‘1’ and not as in some
references as ’0’.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. Öchsner, Structural Mechanics with a Pen,
https://doi.org/10.1007/978-3-030-65892-2_1
1