Truncation error in SPH

Truncation error in SPH
Nathan Quinlan, Mihai Basa, Ruairi Nestor, Marty Lastiwka
First SPHERIC Workshop
10-12 May 2006, University of Rome La Sapienza

Statement of the problem
What is the error in the approximation
First SPHERIC Workshop, Rome, 2006
a
( )
∂A
∂W x − x
≅ −∑ A( x )
Δx
∂x ∂x
b a
b b
x = x b
b
a
b
( x )
∇A ≅ − A ∇ W V
x= x
∑
b b a b
(nD)
(1D)

Methods
Analysis
Taylor series expansions → truncation error in 1D
Numerical Experiments
Evaluation of an analytical test function
for:
• uniform and non-uniform particle distribution
• 1D and 3D
• standard SPH and consistency-corrected kernels
First SPHERIC Workshop, Rome, 2006

The SPH evaluation of a gradient
( )
a
∂A
x
∂x
Eulerian
∂ u
1
+ ( u ⋅ ∇ ) u + ∇ p = 0
∂t
ρ
≅ −
x −2h
( )
First SPHERIC Workshop, Rome, 2006
a
( )
∂W x − x
xa + 2h
A x
a
dx
∫
∂x
smoothing discretisation
( )
( )
∂W x − x
≅ − A x Δx
∑
Lagrangian
Du
1
+ ∇ p = 0
Dt ρ
b a
b
b
∂xb
b

The dimensionless kernel
Wˆ ( s)
= hW ( x − x )
a
0.7
0.6
0.5
0.4
0.3
0.2
0.1
First SPHERIC Workshop, Rome, 2006
0
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
s
=
x − x
h
a

Smoothing error
x + 2h
a
∂A ∂W
≅ − A dx
∂x ∫ ∂x
x = x x −2h
a a
Expand A(x) as a Taylor series and integrate by parts:
xa + 2h xi + 2h 3 xa + 2h
∂A dW ⎛ ⎞ 1 ∂ A 3 ∂W
= − A dx + A′ a ⎜1− Wdx ⎟ + 3 ( x − xa ) dx +
∂x x x dx ⎜ ⎟
= 6
x
a xa −2h ⎝ xa −2h ⎠
∂x
∂
a xa −2h
in dimensionless form:
∫ ∫ ∫ …
xa + 2h 2
( )
2 3
a 1
a a
x = x 2 6
x −2h
∂A ∂W
ˆ h ˆ h
= − A dx + A′ − Wds + A′′ s W ′ ds + A′′′ s Wˆ ′ ds +
∂x ∂x
a a
∫ ∫ ∫ ∫ …
First SPHERIC Workshop, Rome, 2006

Discretisation error – uniform particle spacing
The 1D SPH discretisation is a
numerical integration of the form
xn +Δx
2
n
x −Δx
1
∫
2
b=
1
( )
f ( x) dx ≅ Δx∑ f x
The Second Euler-MacLaurin
Formula:
1
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b
A(x)
W(x)
xn +Δx 2 n
∞
2k
B2k Δx
∫
( ) ∑ ( b ) ∑ 1 2
n
2
b 1 k 1 ( 2 k ) !
x −Δx
= =
− 2k + 1 ( )
( ) ( + )
x b
x a
Δx b
( 2k −1 2k −1
)
1 2 1 2
f x dx = Δx f x − − f − f
( )
( )
x

Kernel boundary smoothness
Boundary smoothness of the kernel
is the highest integer β for which
the β th derivative and all lower
derivatives are zero at the
boundaries of the compact support.
cubic B-spline kernel: β=2
Gaussian kernel: β→∞ in infinite domain
polynomial kernel: arbitrary β
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ˆ
W
ˆ
W ′
ˆ
W ′′
ˆ
W ′′′
1
0.5
0
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
1
0
-1
-2
2
-1.5 -1 -0.5 0 0.5 1 1.5 2
0
-2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
5
0
-5
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
s = ( (x-x x − )/h
i x ) h
a

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

Results – uniform particle spacing
sinusoidal test
function A(x),
wavelength λ
10 th order
polynomial
kernel, β=4
non-dimensionalised L 2 norm error
First SPHERIC Workshop, Rome, 2006
10 0
10 -1
10 -2
10 -3
10 -4
10 -5
10 -6
h/λ=
0.2239
0.0111
0.0018
10 -1
Δx/h
(b)
analysis
computation
1
6
10 0

Results – uniform particle spacing, 1D
sinusoidal test
function A(x),
wavelength λ
10 th order
polynomial
kernel, β=4
non-dimensionalised L 2 norm error
First SPHERIC Workshop, Rome, 2006
10 2
10 0
10 -2
10 -4
10 -6
10 -4
Δx/h=
4/3
4/7
4/19
4/21
10 -3
analysis, Δx/h=4/(2n+1)
computation, Δx/h=4/(2n+1)
10 -2
h/λ
(a)
10 -1
1
2
10 0

Results – uniform particle spacing, 1D
sinusoidal test
function A(x),
wavelength λ
10th order
polynomial
kernel, β=4
non-dimensionalised L 2 norm error
First SPHERIC Workshop, Rome, 2006
10 2
10 0
10 -2
10 -4
10 -6
10 -4
Δx/h=
4/3
0.9
4/7
0.5
0.3
4/19
4/21
0.1
10 -3
analysis, Δx/h=4/(2n+1)
computation, Δx/h=4/(2n+1)
computation, Δx/h≠4/(2n+1)
10 -2
h/λ
(a)
10 -1
1
2
10 0

Non-uniform particle distribution
x b is the location of particle b
xb
A(x)
is the centroid of a region associated with particle b
xb
x b
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Δx b
x
x − x
δ b =
Δx
b b
b

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

Non-uniform spacing – results
Normally distributed
random perturbation
of standard deviation
σ imposed on uniform
particle distribution.
sinusoidal test
function,
wavelength λ
10 th order
polynomial kernel,
β=4
Δx/h = 0.364
non-dimensionalised L 2 norm error
10 0
10 -1
10 -2
10 -3
10 -4
First SPHERIC Workshop, Rome, 2006
10 -3
σ/Δx=0.2
σ/Δx=0.02
σ/Δx=0.002
σ/Δx=0
10 -2
1
h/λ
1
10 -1
1
2
10 0

Non-uniform spacing – results
Normally distributed
random perturbation
of standard deviation
σ imposed on uniform
particle distribution.
sinusoidal test
function,
wavelength λ
10 th order
polynomial kernel,
β=4
Δx/h = 0.364
non-dimensionalised L 2 norm error
10 0
10 -1
10 -2
10 -3
10 -4
First SPHERIC Workshop, Rome, 2006
1
0.1 0.5 1
Δx/h
(a)
3
σ/Δx=
0.2
0.02
0.002
0

First-order consistent kernel (CSPH, RKPM…)
∑
∂A
− A W
b bΔ xb
=
∂
′
b x = x
2
( ˆ )
h 2 ˆ h
3
− A′ 1
ˆ
a Wds − + A′′ a s W ′ ds + A′′′ a s W ′ ds +
2 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
⎞ ⎤
⎛ ⎞
− h ⎢ δO
+ O
2
⎜
h
⎟ ⎜
2
⎜
h
⎟ ⎟⎥
−…
⎢ ⎝ ⎠ ⎜⎝ ⎠ ⎟
⎣ ⎝ ⎠⎥⎦
First SPHERIC Workshop, Rome, 2006
x
− ⎢ ⎥
⎢⎣ ⎝⎝ ⎠ ⎠⎥⎦
a
∫ ∫ ∫ …
smoothing error
discretisation
error

First-order consistent kernel – results
First-order
convergence
observed in
discretisationlimited
error
sinusoidal test
function,
wavelength λ
10th order
polynomial
kernel, β=4
Δx/h = 0.7
non-dimensionalised L 2 norm error
10 -1
10 -2
10 -3
10 -4
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σ/Δx=0.2
σ/Δx=0.02
10 -3
1
1
σ/Δx=0, 0.002
10 -2
h/λ
(a)
10 -1
1
2
10 0

3D SPH
Normally distributed
random perturbation
of standard
deviation σ imposed
on uniform particle
distribution.
sinusoidal test
function,
wavelength λ
10 th order
polynomial kernel,
β=4
non-dimensionalised L 2 norm error
10 0
10 -1
10 -2
10 -3
First SPHERIC Workshop, Rome, 2006
σ/Δx=0.2
0.02
0
0.02
0.2
0.01 0.05
h/λ
(a)
0.1
1
0
1
Δx/h=0.50
Δx/h=0.98
1
2

3D standard SPH
σ/Δx = 0
(uniform)
log 10(h/λ)
First SPHERIC Workshop, Rome, 2006
log 10 (Δx/h)
log 10 (non-dimensionalised error)

3D standard and corrected SPH
σ/Δx = 0
(uniform)
log 10(h/λ)
First SPHERIC Workshop, Rome, 2006
log 10(Δx/h)
log 10 (non-dimensionalised error)

3D standard SPH
σ/Δx = 0.02
(low disorder)
log 10(h/λ)
First SPHERIC Workshop, Rome, 2006
log 10 (Δx/h)
log 10 (non-dimensionalised error)

3D standard and corrected SPH
σ/Δx = 0.02
(low disorder)
log 10(h/λ)
First SPHERIC Workshop, Rome, 2006
log 10(Δx/h)
log 10 (non-dimensionalised error)

3D standard SPH
σ/Δx = 0.2
(high disorder)
log 10 (h/λ)
First SPHERIC Workshop, Rome, 2006
log 10 (Δx/h)
log 10 (non-dimensionalised error)

3D standard and corrected SPH
σ/Δx = 0.2
(high disorder)
log 10(h/λ)
First SPHERIC Workshop, Rome, 2006
log 10(Δx/h)
log 10 (non-dimensionalised error)

3D standard SPH
σ/Δx = 0
(uniform)
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neighbours
Δx/h
93 81 33 19
0.7 0.8 1.0 1.2
log 10 (non-dimensionalised error)

3D standard SPH
σ/Δx = 0.02
(low disorder)
First SPHERIC Workshop, Rome, 2006
neighbours
Δx/h
93 81 33 19
0.7 0.8 1.0 1.2
log 10 (non-dimensionalised error)

3D standard SPH
σ/Δx = 0.2
(high disorder)
First SPHERIC Workshop, Rome, 2006
neighbours
Δx/h
93 81 33 19
0.7 0.8 1.0 1.2
log 10 (non-dimensionalised error)

Conclusions
Error depends on
• h (smoothing length) – second order
• Δx/h ∝ (number of neighbours) –1 – dependent on kernel smoothness
SPH estimates do not necessarily converge with h
Optimal number of neighbours depends on smoothing length
Non-uniform spacing can cause divergence of standard SPH as h is
reduced
Consistency-corrected kernels ensure convergence
First SPHERIC Workshop, Rome, 2006

Acknowledgements
Supported by Basic Research Grant
SC/2002/189 from the Irish Research
Council for Science, Engineering and
Technology under the Embark Initiative,
funded by the National Development
Plan.
First SPHERIC Workshop, Rome, 2006