Truncation error in SPH
Truncation error in SPH
Truncation error in SPH
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Overall <strong>error</strong> – uniform particle spac<strong>in</strong>g<br />
<strong>SPH</strong> estimate<br />
∑<br />
∂A<br />
x<br />
− A W<br />
b bΔ xb<br />
=<br />
∂<br />
′<br />
b x = x<br />
2<br />
h 3<br />
A′′′ ˆ<br />
a s W ′ ds<br />
+ + …<br />
6<br />
( ) ( ) −β −<br />
( )<br />
β + 2 !<br />
First <strong>SPH</strong>ERIC Workshop, Rome, 2006<br />
∫<br />
a<br />
exact derivative<br />
smooth<strong>in</strong>g <strong>error</strong><br />
β + 2<br />
⎛ Δx<br />
⎞ Bβ<br />
+ 2 1 ( ˆ β + 2 ( 1)<br />
( ) ˆ β + ) 2<br />
− 1 2 ⎡A ′<br />
a 4Ws = 2 2 β 1 Ws = 2 O ( h ) ⎤<br />
⎜ − + + + −<br />
h<br />
⎟ …<br />
⎝ ⎠<br />
⎣ ⎦<br />
discretisation <strong>error</strong>
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