Structure-preserving geometric particle-in-cell algorithm suppresses finite-grid instability -- Comment on "Finite grid instability and spectral fidelity of the electrostatic Particle-In-Cell algorithm'' by Huang et al

TL;DR

This paper demonstrates that the Structure-preserving geometric particle-in-cell (SPG-PIC) algorithm effectively suppresses finite-grid instability, improving spectral fidelity in plasma simulations, as confirmed by numerical experiments.

Contribution

It provides the first numerical validation that the SPG-PIC algorithm can suppress finite-grid instability, enhancing the stability of PIC simulations.

Findings

SPG-PIC suppresses finite-grid instability

SPG-PIC maintains spectral fidelity

Numerical experiments confirm stability improvements

Abstract

A recent paper by Huang et al. [Computer Physics Communications 207, 123 (2016)] thoroughly analyzed the Finite Grid Instability(FGI) and spectral fidelity of standard Particle-In-Cell (PIC) methods. Numerical experiments were carried out to demonstrate the FGIs for two PIC methods, the energy-conserving algorithm and the momentum-conserving algorithm. The paper also suggested that similar numerical experiments should be performed to test the newly developed Structure-Preserving Geometric (SPG)-PIC algorithm. In this comment, we supply the results of the suggested numerical experiments, which show that the SPG-PIC algorithm is able to suppress the FGI.

Full text

Structure-preserving geometric algorithm, particle-in-cell, finite grid instabilities

Huang et al. recently provided an in-depth analysis of the Finite Grid Instability (FGI) and spectral fidelity of standard Particle-In-Cell (PIC) methods Huang et al. (2016). The spectral errors, especially the aliased spatial modes, from charge deposition and field interpolation schemes were rigorously quantified. Numerical experiments were carefully designed and carried out to demonstrate the FGIs of the Momentum-Conserving (MC)-PIC algorithm Birdsall and Langdon (1991) and the Energy-Conserving (EC)-PIC algorithm Lewis (1970). Simulation results using a Particle-And-Spectrum (PAS) method Evstatiev and Shadwick (2013) was also given for comparison and benchmark. The paper suggested performing similar numerical experiments to test the newly developed Structure-Preserving Geometric (SPG)-PIC algorithm Squire et al. (2012); Xiao et al. (2013, 2015); He et al. (2015, 2016); Xiao et al. (2016); Qin et al. (2016); Xiao et al. (2017); Kraus et al. (2017); Xiao et al. (2018). In this comment, we supply the results of the suggested numerical experiments using the specific implementation of the SPC-PIC algorithm reported in Ref. Xiao et al. (2015).

The parameters for the numerical experiments are the same as in Ref. Huang et al. (2016), which are listed as follows. The simulation domain is a $L\times 1\times 1$ periodic box where $L=33$, $\Delta x=1$, and $\omega_{p}=2\pi/L$. The time step is set to $\Delta t=0.2\omega_{p}^{-1}$. The numbers of sampling points per grid for both electron and ions are $300$, and the mass ratio and charge ratio between electrons and ions are $1:3672$ and $-1:1$, respectively. Initially the ions are equally spaced and their velocities are set to [math]. Electrons are equally spaced with a sinusoidal displacement $\delta x\left(x_{p_{0}}\right)=LA\cos\left(2\pi Mx_{p_{0}}/L\right)/(2\pi M)$, and their velocities are $v_{x}(x)=0.01\mathrm{c}+LA\omega_{p}\sin\left(2\pi Mx/L\right)/(2\pi M)$, where $A=0.01$ and $M=9$. Initial electric field is $E_{x}\left(x\right)=-Lq_{e}A\cos\left(2\pi Mx/L\right)/\left(2\pi M\right)$.

The simulation first is performed to $t=220\omega_{p}^{-1}$. The resulting mode spectrum, final velocity distribution, energy and momentum evolution are plotted in Figs. 1 and 2, which correspond to Fig. 3 and Fig. 4 of Ref. Huang et al. (2016), respectively. These results show that even though mode alias still exists in the SPG-PIC algorithm, the FGI is suppressed. While mode alias effect, as an error of spatial discretization, is inevitable in any spatial grid, its existence does not necessarily imply unstable numerical eigenmodes will be exited. Note that when a unstable numerical eigenmode is excited, all components of the dynamics, especially the dominant ones, of the discrete system will grow exponentially. Unfortunately, such FGIs do exist in the standard PIC methods. As demonstrated in Ref. Huang et al. (2016), in about $30$ plasma oscillation periods ($200\omega_{p}^{-1}$), the total energy error for the MC-PIC algorithm exceeds $200\%,$ and the total momentum error for the EC-PIC algorithm exceeds 70%. On the other hand, Fig. 2 shows that for the SPG-PIC algorithm, the total energy error is less than $1\%,$ and the total momentum error is less than $0.4\%.$

The observed suppressing of FGI for the SPG-PIC algorithm can be attributed to the structure-preserving nature of its spatial discretization. The charge deposition and field interpolation are derived from a variational principle using the techniques of Whitney interpolation forms Squire et al. (2012); Xiao et al. (2016, 2018) or finite element discrete exterior calculus He et al. (2016); Kraus et al. (2017), which preserves the discrete gauge symmetry and the discrete exterior calculus structure of the electromagnetic field. As a result, physical laws, such as the charge conservation and $\nabla\cdot\mathbf{B}=0$, are satisfied exactly by the discrete system. This result is consistent with Ref. Huang et al. (2016)’s conclusion that charge deposition and field interpolation can be optimally designed to suppress or reduce FGIs.

Another feature of the SPG-PIC algorithm is the preserving of non-canonical symplectic structure for time-integration, which in general bounds simulation errors on conserved quantities for a very long time. We run the simulation for $500$ longer to $t=10000\omega_{p}^{-1}$, and the result is plotted in Fig. 3. Over this long simulation time, the total energy error and total momentum error are bounded by $1\%$ and $2\%$, respectively.

We finish this comment with two footnotes. First, the SPG-PIC algorithm used is for Vlasov-Maxwell system in 3D configuration space. For the simplified geometry and simulation parameters of the present numerical experiments, the dominated modes of the discrete system are longitudinal electrostatic modes. Secondly, for the PAS methods, symplectic time-integration can also be adopted. For example, Cary and Doxas Cary and Doxas (1993); Doxas and Cary (1997) first applied a canonical symplectic algorithm to simulate the particle-and-mode Hamiltonian models Mynick and Kaufman (1978); Escande (1982, 1987, 1989, 1991); Escande et al. (2018) for the Vlasov-Poisson system.

Acknowledgments

We thank Prof. John Cary and Prof. Dominique Escande for fruitful discussions. This research is supported by the National Key Research and Development Program (2016YFA0400600, 2016YFA0400601, 2016YFA0400602 and 2018YFE0304100), the National Natural Science Foundation of China (NSFC-11775219, NSFC-11575186 and NSFC-11805273), China Postdoctoral Science Foundation (2017LH002), the Fundamental Research Funds for the Central Universities (WK2030040096) and the U.S. Department of Energy (DE-AC02-09CH11466).