Page 201 - The Combined Finite-Discrete Element Method

184 TEMPORAL DISCRETISATION

i.e. is again equal to 1. For
2
h k/m > 4 (5.30)
Equation (5.24) yields

#
2 2 2
−(h k/m − 2) ± (h k/m − 2) − 4
λ 1,2 = (5.31)
2
and the spectral radius is given by

#
2 2 2
(h k/m − 2) + (h k/m − 2) − 4
max |λ 1,2 |= > 1 (5.32)
2
and is always greater than 1. A graph of spectral radius is shown in Figure 5.2.
It is evident that the spectral radius is equal to 1 for

2
h k/m ≤ 4 (5.33)
i.e. the central difference time integration scheme is stable for the time step smaller than

2
h ≤ √ (5.34)
k/m
For the time steps
2
h> √ (5.35)
k/m
the central difference time integration scheme is always numerically unstable.

3.5
3

Spectral radius 2.5
2

1.5
1

0.5
0
0 0.5 1 1.5 2 2.5
2
h k/m
Figure 5.2 Spectral radius of the recursive operator A and stability of the central difference time
integration scheme.