# 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

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