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Scheme start-up and marching (2)
• For practical reasons, the code also computes the
full-step velocities:
∆t
u
= v + u
2
n + 1 n + 1/2 n + 1
• These are the velocities printed out in the listing
and visualized in post-processing
• However, the fundamental quantity in the time
integration scheme is the mid-step velocity!
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Integration Scheme Characteristics
• Central difference scheme is second-order accurate
and introduces no numerical damping
• However, it is conditionally stable (Courant):
e
L
∆t L c
e stab
≈
e /
e
e
c
e
e
∆ t = ϕ ⋅∆
t
e
stab
(with ϕ < 1)
• In highly non-linear cases, small steps are needed even
with unconditionally stable schemes, to get good accuracy
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