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Scaling identities for kernel moments and derivatives
Statement
For the scaled kernels of Scaled symmetric compactly supported kernel family and every ε>0, supp W_ε⊂\overline B(0,Rε), ∫W_ε=1, ∫zW_ε(z)dz=0, and ∫|z|²W_ε(z)dz=ε²m₂. Pointwise, ∇W_ε(z)=ε^(−d−1)∇W(z/ε) and D²W_ε(z)=ε^(−d−2)D²W(z/ε); consequently their sup norms equal ε^(−d−1)M₁ and ε^(−d−2)M₂.
Facts & Assumptions
Given: The admissible kernel W and a scale ε>0.
Scaled symmetric compactly supported kernel family: W_ε(z)=ε^(−d)W(z/ε), supp W⊂\overline B(0,R), and the constants M₁, M₂, m₂ are finite.
Symmetry cancels the first kernel moment: ∫zW(z)dz=0.
A compactly supported Riemann integrand admits the global change-of-variables formula from a diffeomorphism near the relevant compact preimage: let U⊂R^d be open and T:U→R^d be injective C¹ with invertible derivative. If f:R^d→R is compactly supported Riemann integrable with supp(f)⊂T(U), then ∫{R^d}f(y)dy=∫{R^d}h(x)dx, where h(x)=f(T(x))|det DT(x)| on U and h=0 outside U. Here U=R^d and T(u)=εu is onto R^d for ε>0.
A continuous real function on a compact Jordan measurable set is Riemann integrable over that set: continuous real integrands on compact Jordan boxes are Riemann integrable.
The chain rule for total derivatives: : derivatives of compositions with the affine scaling map are given by the chain rule.
Proof
If W_ε(z)≠0 then z/ε∈supp W, so |z|≤Rε. The chain rule in [F5] applied to ε^(−d)W(z/ε) gives the displayed pointwise first and second derivative formulas; taking suprema and substituting u=z/ε gives the claimed derivative sup norms.
The functions W_ε, zW_ε, and |z|²W_ε are continuous and compactly supported, hence integrable on a bounding box by [F4]. In the integrals, the affine substitution z=εu has Jacobian determinant ε^d. Thus ∫W_ε=∫W=1, ∫zW_ε(z)dz=ε∫uW(u)du=0 by [F2], and ∫|z|²W_ε(z)dz=ε²∫|u|²W(u)du=ε²m₂.
Depends on
- Scaled symmetric compactly supported kernel family
- Symmetry cancels the first kernel moment
- A compactly supported Riemann integrand admits the global change-of-variables formula from a diffeomorphism near the relevant compact preimage
- A continuous real function on a compact Jordan measurable set is Riemann integrable over that set
- The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$
Used by
- Interior constant and affine reproduction by a symmetric kernel Example
- Explicit local Lipschitz bound for the regularized SPH vector field Lemma
- Explicit cellwise error for kernel value and gradient quadrature Theorem
- Second-order bias for the kernel particle gradient Theorem
- Second-order convolution bias for an interior C² field Theorem
Dependency tree · two levels
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Sources
- Volker Springel, Smoothed Particle Hydrodynamics in Astrophysics, Annual Review of Astronomy and Astrophysics 48 (2010), author full text (standard reference, not scraped)
- Benjamin Seibold, MIT OpenCourseWare 18.336, Numerical Methods for Partial Differential Equations, Lecture 25: Particle Methods (Spring 2009) (standard reference, not scraped)