AP-Cloud: Adaptive Particle-in-Cloud method for optimal solutions to Vlasov–Poisson equation

Abstract

We propose a new adaptive Particle-in-Cloud (AP-Cloud) method for obtaining optimal numerical solutions to the Vlasov–Poisson equation. Unlike the traditional particle-in-cell (PIC) method, which is commonly used for solving this problem, the AP-Cloud adaptively selects computational nodes or particles to deliver higher accuracy and efficiency when the particle distribution is highly non-uniform. Unlike other adaptive techniques for PIC, our method balances the errors in PDE discretization and Monte Carlo integration, and discretizes the differential operators using a generalized finite difference (GFD) method based on a weighted least square formulation. As a result, AP-Cloud is independent of the geometric shapes of computational domains and is free of artificial parameters. Efficient and robust implementation is achieved through an octree data structure with 2:1 balance. We analyze the accuracy and convergence order of AP-Cloud theoretically, and verify the method using an electrostatic problem of a particle beam with halo. Simulation results show that the AP-Cloud method is substantially more accurate and faster than the traditional PIC, and it is free of artificial forces that are typical for some adaptive PIC techniques.

Introduction

The Particle-in-Cell (PIC) method [1] is a popular method for solving the Vlasov–Poisson equations for a class of problems in plasma physics, astrophysics, and particle accelerators, for which electrostatic approximation applies, as well as for solving the gravitational problem in cosmology and astrophysics. In such a hybrid particle–mesh method, the distribution function is approximated using particles and the Poisson problem is solved on a rectangular mesh. Charges (or masses) of particles are interpolated onto the mesh, and the Poisson problem is discretized using finite differences or spectral approximations. On simple rectangular domains, FFT methods are most commonly used for solving the Poisson problem. In the presence of irregular boundaries, finite difference approximations are often used, complemented by a cut-cell (a.k.a. embedded boundary) method [2] for computational cells near boundaries, and fast linear solvers (including multigrid iterations) for the corresponding linear system. The computed force (gradient of the potential) on the mesh is then interpolated back to the location of particles. For problems with irregular geometry, unstructured grid with finite element method is often used.

The traditional PIC method has several limitations. It is less straightforward to use for geometrically complex domains. The aforementioned embedded boundary method, while maintaining globally second order accuracy for the second order finite difference approximation, usually results in much larger errors near irregular boundaries [3]. It is also difficult to generalize to higher order accuracy.

Another major drawback of the PIC method is associated with highly non-uniform distribution of particles. As shown in Section 2, the discretization of the differential operator and the right hand side in the PIC method is not balanced in terms of errors. The accuracy is especially degraded in the presence of non-uniform particle distributions. The AMR-PIC [4], [5] improves this problem by performing block-structured adaptive mesh refinement of a rectangular mesh, so that the number of particles per computational cell is approximately the same. However, the original AMR-PIC algorithms suffered from very strong artificial self-forces due to spurious images of particles across boundaries between coarse and refined mesh patches. Analysis of self-force sources and a method for their mitigation was proposed in [6].

In this paper, we propose a new adaptive Particle-in-Cloud (AP-Cloud) method for obtaining optimal numerical solutions to the Vlasov–Poisson equation. Instead of a Cartesian grid as used in the traditional PIC, the AC-Cloud uses adaptive computational nodes or particles with an octree data structure. The quantity characterizing particles (charge in electrostatic problems or mass in gravitational problems) is assigned to computational nodes by a weighted least squares approximation. The partial differential equation is then discretized using a generalized finite difference (GFD) method and solved with fast linear solvers. The density of nodes is chosen adaptively, so that the error from GFD and that from Monte Carlo integration are balanced, and the total error is approximately minimized. The method is independent of geometric shape of computational domains and free of artificial self-forces.

The remainder of the paper is organized as follows. In Section 2, we analyze numerical errors of the traditional PIC method and formulate optimal refinement strategy. The AP-Cloud method, generalized finite differences, and the relevant error analysis are presented in Section 3. Section 4 describes some implementation details of the method. Section 5 presents numerical verification tests using 2D and 3D problems of particle beams with halo and additional tests demonstrating the absence of artificial self-forces. We conclude this paper with a summary of our results and perspectives for the future work.