Truncation error in SPH
Truncation error in SPH
Truncation error in SPH
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Discretisation <strong>error</strong> – uniform particle spac<strong>in</strong>g<br />
The 1D <strong>SPH</strong> discretisation is a<br />
numerical <strong>in</strong>tegration of the form<br />
xn +Δx<br />
2<br />
n<br />
x −Δx<br />
1<br />
∫<br />
2<br />
b=<br />
1<br />
( )<br />
f ( x) dx ≅ Δx∑ f x<br />
The Second Euler-MacLaur<strong>in</strong><br />
Formula:<br />
1<br />
First <strong>SPH</strong>ERIC Workshop, Rome, 2006<br />
b<br />
A(x)<br />
W(x)<br />
xn +Δx 2 n<br />
∞<br />
2k<br />
B2k Δx<br />
∫<br />
( ) ∑ ( b ) ∑ 1 2<br />
n<br />
2<br />
b 1 k 1 ( 2 k ) !<br />
x −Δx<br />
= =<br />
− 2k + 1 ( )<br />
( ) ( + )<br />
x b<br />
x a<br />
Δx b<br />
( 2k −1 2k −1<br />
)<br />
1 2 1 2<br />
f x dx = Δx f x − − f − f<br />
( )<br />
( )<br />
x
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