
236 Friction-Induced Vibration and Self-Organization
After determining the region of parameters appropriate for pattern forma-
tion from the stability analysis, we need to perform a numerical simulation.
To solve the system of Equations (8.41) and (8.42), a number of numerical
schemes have been proposed. Approximating the diffusive term by second
order backward difference, we used the second order semi-implicit back-
ward difference formula (BDF) scheme proposed by Ruth (1995):
uuuffdu
(3 4)2
nnnnn
uh
n11
−+ =−+
+−
(8.49)
vvvggdv
1
(3 4)2
nnnnn
vh
n11
−+ =−+
+−
(8.50)
where n corresponds to the current time step, and n – 1 and n + 1 show the
previo ...