Truncation error in SPH

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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
Page 1: Truncation error in SPH Nathan Quin
Page 5 and 6: The dimensionless kernel Wˆ ( s) =
Page 7 and 8: Discretisation error - uniform part
Page 9 and 10: Overall error - uniform particle sp
Page 11 and 12: Results - uniform particle spacing,
Page 13 and 14: Non-uniform particle distribution x
Page 15 and 16: Non-uniform spacing - results Norma
Page 17 and 18: First-order consistent kernel (CSPH
Page 19 and 20: 3D SPH Normally distributed random
Page 21 and 22: 3D standard and corrected SPH σ/Δ
Page 27 and 28: 3D standard SPH σ/Δx = 0.02 (low
Page 29 and 30: Conclusions Error depends on • h

Truncation error in SPH

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