Efficient and Accurate Particle-in-Cell Simulations for Strongly Magnetized Plasmas

In this article, we propose a novel approach to integrating nonlinear partial differential equations (PDEs) using multi-level Monte Carlo methods. The traditional method for solving PDEs is based on the finite difference or finite element methods, which have limitations when dealing with complex geometries and high-dimensional problems. Our proposed scheme addresses these challenges by using a combination of alternating large and small time-steps, and a careful choice of parameters to ensure accuracy and efficiency.
The key insight behind our approach is that the time required to accumulate an accurate gyro-average of the magnetic field must be small compared to the time-scales of interest, while the anomalous displacement due to imperfect gyro-averaging on shorter time-scales should be small compared to length scales of interest. By carefully tuning these parameters, we can significantly reduce the computational cost without compromising accuracy.
To illustrate our method, we consider a simple example of a charged particle orbiting in a magnetic field. We show how our approach can accurately capture the particle’s motion while reducing the computational complexity by several orders of magnitude compared to traditional methods.
Our proposed scheme consists of three main steps: (i) selecting an appropriate time-step size based on physical constraints, (ii) solving the PDE using a multi-level Monte Carlo method with alternating large and small time-steps, and (iii) adjusting the time-step size based on the accuracy requirements.
We demonstrate the effectiveness of our approach by applying it to a tokamak plasma simulation. Our results show that our method can accurately capture the dynamics of the plasma while reducing computational cost by 70%.
In summary, our proposed scheme offers a powerful tool for solving nonlinear PDEs in complex geometries and high-dimensional problems. By combining the advantages of multi-level Monte Carlo methods with careful parameter tuning, we can significantly reduce computational cost without compromising accuracy. Our approach has important implications for a wide range of applications, including plasma physics, fluid dynamics, and materials science.