Truncation error in SPH

Non-uniform spacing
∑
∂A
− A W
b bΔ xb
=
∂
′
b x = x
2
( ˆ )
h
3
− A′ 1
ˆ
a Wds − + A′′′ a s W ′ ds +
6
3 4
1 ⎡ ⎛⎛ Δx ⎞ ⎞ Aa
⎛ 2 1 ⎞ ⎛⎛ Δx
⎞ ⎞⎤
− ⎢Aaδ O ⎜ + δ + O ⎥
h ⎜⎜ h
⎟ ⎟
⎜ ⎟
2
⎜
12
⎟ ⎜⎜ h
⎟
⎢ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎟
⎣ ⎝ ⎠ ⎝ ⎠⎥⎦
3
⎡ ⎛⎛ Δx
⎞ ⎞⎤
A′ aδO
⎜
⎜⎜ ⎟
h
⎟ ⎟
4
⎡ A′′ a ⎛ Δx ⎞ A′′
⎛
a ⎛ Δx
⎞ ⎞⎤
First SPHERIC Workshop, Rome, 2006
x
− ⎢ ⎥
⎢⎣ ⎝⎝ ⎠ ⎠⎥⎦
a
∫ ∫ …
− h ⎢ δO
O
2
⎜ +
h
⎟ ⎜ ⎟⎥
2 ⎜⎜ h
⎟ −…
⎢ ⎝ ⎠ ⎝ ⎠ ⎟
⎣ ⎝ ⎠⎦⎥
smoothing error
discretisation
error