AP-Cloud: Adaptive Particle-in-Cloud Method for Optimal Solutions to Vlasov-Poisson Equation

TL;DR

AP-Cloud is an adaptive particle method that improves accuracy and efficiency in solving the Vlasov-Poisson equation by balancing errors and using a generalized finite difference approach.

Contribution

The paper introduces a novel adaptive Particle-in-Cloud method that balances PDE and Monte Carlo errors, independent of domain shape, and improves upon traditional PIC methods.

Findings

AP-Cloud achieves higher accuracy than traditional PIC.

The method is faster and more robust in simulations.

It effectively handles highly non-uniform particle distributions.

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.