Interpolation basis

The basis $\Phi(\xi)$ is the 1-D dimensionless interpolation function; the $D$-dimensional basis values are tensor products of $\Phi$ along each axis. A basis b must implement:

support_radius(b) → half-width in grid spacings (integer).
eval_phi(b, ξ) → $\Phi(\xi)$.
eval_phi_prime(b, ξ) → $\Phi'(\xi)$.

MultilevelSummation.CubicC1 — Type

CubicC1{T}()

Piecewise-cubic interpolation basis with C¹ continuity, support |ξ| ≤ 2, from Hardy et al. 2015 (paper, just after eq. 13):

Φ(ξ) = (1 - |ξ|)·(1 + |ξ| - (3/2) ξ²)   for 0 ≤ |ξ| ≤ 1
     = -(1/2)·(|ξ| - 1)·(2 - |ξ|)²       for 1 ≤ |ξ| ≤ 2
     = 0                                  otherwise

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MultilevelSummation.eval_phi — Function

eval_phi(basis, ξ::T) -> T

The 1-D interpolation basis function Φ(ξ).

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MultilevelSummation.eval_phi_prime — Function

eval_phi_prime(basis, ξ::T) -> T

The derivative of the 1-D interpolation basis: Φ'(ξ).

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MultilevelSummation.support_radius — Function

support_radius(basis) -> Int

Half-width of the basis support in grid spacings. Each particle scatters to (2·support_radius)^D grid points; each grid point gathers from the same number of source points during prolongation / restriction.

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