Hierarchy of Charge Assignment Schemes

Section author: Klaus Steiniger

In PIConGPU, the cloud shapes \(S^n(x)\) are pre-integrated to assignment functions \(W^n(x)\).

\[\begin{split}W^n(x) = \Pi(x) \ast S^n(x) = \int\limits_{-\infty}^{+\infty} \Pi(x^\prime) S^n(x^\prime - x) dx^\prime\,, \text{ where } \Pi(x) = \left\{\begin{array}{ll} 0 & |x| \gt \frac{1}{2} \\ \frac{1}{2} & |x| = \frac{1}{2} \\ 1 & |x| \lt \frac{1}{2} \end{array}\right.\end{split}\]

is the top-hat function and \(\ast\) the convolution.

Evaluating the assignment functions at mesh points directly provides the fraction of charge from the marker assigned to that point.

The assignment functions are implemented as B-splines. The zeroth order assignment function \(W^0\) is the top-hat function \(\Pi\). It represents charge assignment to the nearest mesh point only, resulting in a stepwise charge density distribution. Therefore, it should not be used. The assignment function of order \(n\) is generated by convolution of the assignment function of order \(n-1\) with the top-hat function

\[W^n(x) = W^{n-1}(x) \ast \Pi(x) = \int\limits_{-\infty}^{+\infty} W^{n-1}(x^\prime) \Pi(x^\prime - x) dx^\prime\,.\]

The three dimensional assignment function is a multiplicative union of B-splines \(W^n(x,y,z) = W^n(x) W^n(y) W^n(z)\).

PIConGPU implements these up to order \(n=4\). The naming scheme follows [HockneyEastwood], tab. 5-1, p. 144, where the name of a scheme is defined by the visual form of its cloud shape \(S\).

Scheme	Order	Assignment function
NGP (nearest-grid-point)	0	stepwise
CIC (cloud-in-cell)	1	piecewise linear spline
TSC (triangular shaped cloud)	2	piecewise quadratic spline
PQS (piecewise quadratic cloud shape)	3	piecewise cubic spline
PCS (piecewise cubic cloud shape)	4	piecewise quartic spline

References

HockneyEastwood

R.W. Hockney, J.W. Eastwood. Computer Simulation Using Particles, CRC Press (1988), ISBN 0-85274-392-0