The Particle-in-Cell Algorithm¶
Section author: Axel Huebl
For now, please refer to the textbooks [BirdsallLangdon], [HockneyEastwood], our latest paper on PIConGPU and [Huebl2014] (chapters 2.3, 3.1 and 3.4).
System of Equations¶
for multiple particle species \(s\). \(\mathbf{E}(t)\) represents the electic, \(\mathbf{B}(t)\) the magnetic, \(\rho_s\) the charge density and \(\mathbf{J}_s(t)\) the current density field.
Except for normalization of constants, PIConGPU implements the governing equations in SI units.
Relativistic Plasma Physics¶
The 3D3V particle-in-cell method is used to describe many-body systems such as a plasmas. It approximates the Vlasov–Maxwell–Equation
with \(f_s\) as the distribution function of a particle species \(s\), \(\mathbf{x},\mathbf{v},t\) as position, velocity and time and \(\frac{q_s}{m_s}\) the charge to mass-ratio of a species. The momentum is related to the velocity by \(\mathbf{p} = \gamma m_s \mathbf{v}\).
The equations of motion are given by the Lorentz force as
Attention
TODO: write proper relativistic form
\(\mathbf{X}_s = (\mathbf x_1, \mathbf x_2, ...)_s\) and \(\mathbf{V}_s = (\mathbf v_1, \mathbf v_2, ...)_s\) are vectors of marker positions and velocities, respectively, which describe the ensemble of particles belonging to species \(s\).
Note
Particles in a particle species can have different charge states in PIConGPU. In the general case, \(\frac{q_s}{m_s}\) is not required to be constant per particle species.
Electro-Magnetic PIC Method¶
Fields such as \(\mathbf{E}(t), \mathbf{B}(t)\) and \(\mathbf{J}(t)\) are discretized on a regular mesh in Eulerian frame of reference (see [EulerLagrangeFrameOfReference]).
The distribution of Particles is described by the distribution function \(f_s(\mathbf{x},\mathbf{v},t)\). This distribution function is sampled by markers (commonly referred to as macro-particles). The temporal evolution of the distribution function is simulated by advancing the markers over time according to the Vlasov–Maxwell–Equation in Lagrangian frame (see eq. (1) and [EulerLagrangeFrameOfReference]).
Markers carry a spatial shape of order \(n\) and a delta-distribution in momentum space. In most cases, these shapes are implemented as B-splines and are pre-integrated to assignment functions \(S\) of the form:
PIConGPU implements these up to order \(n=4\). The three dimensional marker shape is a multiplicative union of B-splines \(S^n(x,y,z) = S^n(x) S^n(y) S^n(z)\).
References¶
[EulerLagrangeFrameOfReference] | (1, 2) Eulerian and Lagrangian specification of the flow field. https://en.wikipedia.org/wiki/Lagrangian_and_Eulerian_specification_of_the_flow_field |
[BirdsallLangdon] | C.K. Birdsall, A.B. Langdon. Plasma Physics via Computer Simulation, McGraw-Hill (1985), ISBN 0-07-005371-5 |
[HockneyEastwood] | R.W. Hockney, J.W. Eastwood. Computer Simulation Using Particles, CRC Press (1988), ISBN 0-85274-392-0 |
[Huebl2014] | A. Huebl. Injection Control for Electrons in Laser-Driven Plasma Wakes on the Femtosecond Time Scale, Diploma Thesis at TU Dresden & Helmholtz-Zentrum Dresden - Rossendorf for the German Degree “Diplom-Physiker” (2014), https://doi.org/10.5281/zenodo.15924 |