1 Idea and Derivation of the Method
9
0 =
u i+1 − u i
X
−
u i − u i−1
X
,
(1.27)
which corresponds to the centered difference scheme as given in Table 1.1.
1.1 Supplementary Problems
1.1 Forward difference approximation of the first order derivative
Derive the finite difference approximation for the forward difference scheme of the
first-order derivative where the truncation error is of order X
2 . The final result is
given in Table 1.1.
1.2 Centered difference approximation of the third order derivative
Derive the finite difference approximation for the centered difference scheme of
the third-order derivative where the truncation error is of order X
2 based on the
expressions for the first- and second-order derivatives. These expressions and the
final result are given in Table 1.1.
References
1. de Allen DN, G (1955) Relaxation methods. McGraw-Hill, New York
2. Bathe K-J (1996) Finite element procedures. Prentice-Hall, Upper Saddle River
3. Brebbia CA, Telles JFC, Wrobel LC (1984) Boundary element techniques: theory and applications. Springer, Berlin
4. Collatz L (1966) The numerical treatment of differential equations. Springer, Berlin
5. Forsythe GE, Wasow WR (1960) Finite-difference methods for partial differential equations.
Wiley, New York
6. Lau PCM, Brebbia CA (1978) The cell collocation method in continuum mechanics. Int J Mech
Sci 20:83–95
7. Mitchell AR, Griffiths DF (1980) The finite difference method in partial differential equations.
Wiley, New York
8. Öchsner A (2014) Elasto-plasticity of frame structure elements: modeling and simulation of
rods and beams. Springer, Berlin
9. Öchsner A, Merkel M (2018) One-dimensional finite elements: an introduction to the FE
method. Springer, Cham
10. Öchsner A (2020) Computational statics and dynamics: an introduction based on the finite
element method. Springer, Singapore
11. Southwell RV (1946) Relaxation methods in theoretical physics. Clarendon Press, Oxford
9
0 =
u i+1 − u i
X
−
u i − u i−1
X
,
(1.27)
which corresponds to the centered difference scheme as given in Table 1.1.
1.1 Supplementary Problems
1.1 Forward difference approximation of the first order derivative
Derive the finite difference approximation for the forward difference scheme of the
first-order derivative where the truncation error is of order X
2 . The final result is
given in Table 1.1.
1.2 Centered difference approximation of the third order derivative
Derive the finite difference approximation for the centered difference scheme of
the third-order derivative where the truncation error is of order X
2 based on the
expressions for the first- and second-order derivatives. These expressions and the
final result are given in Table 1.1.
References
1. de Allen DN, G (1955) Relaxation methods. McGraw-Hill, New York
2. Bathe K-J (1996) Finite element procedures. Prentice-Hall, Upper Saddle River
3. Brebbia CA, Telles JFC, Wrobel LC (1984) Boundary element techniques: theory and applications. Springer, Berlin
4. Collatz L (1966) The numerical treatment of differential equations. Springer, Berlin
5. Forsythe GE, Wasow WR (1960) Finite-difference methods for partial differential equations.
Wiley, New York
6. Lau PCM, Brebbia CA (1978) The cell collocation method in continuum mechanics. Int J Mech
Sci 20:83–95
7. Mitchell AR, Griffiths DF (1980) The finite difference method in partial differential equations.
Wiley, New York
8. Öchsner A (2014) Elasto-plasticity of frame structure elements: modeling and simulation of
rods and beams. Springer, Berlin
9. Öchsner A, Merkel M (2018) One-dimensional finite elements: an introduction to the FE
method. Springer, Cham
10. Öchsner A (2020) Computational statics and dynamics: an introduction based on the finite
element method. Springer, Singapore
11. Southwell RV (1946) Relaxation methods in theoretical physics. Clarendon Press, Oxford