Truncation error in SPH
Truncation error in SPH
Truncation error in SPH
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Smooth<strong>in</strong>g <strong>error</strong><br />
x + 2h<br />
a<br />
∂A ∂W<br />
≅ − A dx<br />
∂x ∫ ∂x<br />
x = x x −2h<br />
a a<br />
Expand A(x) as a Taylor series and <strong>in</strong>tegrate by parts:<br />
xa + 2h xi + 2h 3 xa + 2h<br />
∂A dW ⎛ ⎞ 1 ∂ A 3 ∂W<br />
= − A dx + A′ a ⎜1− Wdx ⎟ + 3 ( x − xa ) dx +<br />
∂x x x dx ⎜ ⎟<br />
= 6<br />
x<br />
a xa −2h ⎝ xa −2h ⎠<br />
∂x<br />
∂<br />
a xa −2h<br />
<strong>in</strong> dimensionless form:<br />
∫ ∫ ∫ …<br />
xa + 2h 2<br />
( )<br />
2 3<br />
a 1<br />
a a<br />
x = x 2 6<br />
x −2h<br />
∂A ∂W<br />
ˆ h ˆ h<br />
= − A dx + A′ − Wds + A′′ s W ′ ds + A′′′ s Wˆ ′ ds +<br />
∂x ∂x<br />
a a<br />
∫ ∫ ∫ ∫ …<br />
First <strong>SPH</strong>ERIC Workshop, Rome, 2006