Kim Han Trinh - Projects

Numerical simulation of the Vlasov-Poisson system with strong external magnetic fields.

In this project, we study a Particle-In-Cell (PIC) method based on the Crank-Nicolson time discretization for trajectory of particles characterised by the Vlasov-Poisson system with a strong and inhomogeneous external magnetic field. The purpose is to design an accurate and stable numerical scheme

for any time step which is not constraint by the magnitude of magnetic field,
preserve the structure of asymptotic limit (position, energy, magnetic moment) as the magnitude of magnetic field goes to 0.

The project is separated into many steps: Solving the problem 2Dx2D, Numerical analysis , solving the problem 3Dx3D and extend to Vlasov-Maxwell system.

References:

Plasma physics via computer simulation, C. K. Birdsall and A. B. Langdon, (1991). Link
Fundamentals of plasma physics, P. M. Bellan, (2008). Link
Asymptotically preserving Particle-In-Cell method for Inhomogeneous strongly magnetized plasma, F. Filbet, L. M. Rodrigues, (2017). Link
Asymptotic-preserving gyrokinetic implicit particle-orbit integrator for arbitrary electromagnetic fields, F. Ricketson, L. Chacon, (2020). Link
Asymtoptics of the three dimensional Vlasov equation in the large magnetic field limit, F. Filbet, L. M. Rodrigues, (2021). Link

First step: we focus on 2D Vlasov-Poisson system where the external magnetic field is fixed direction and the motion of particles in the plane orthogonal to the magnetic field (so-called poloidal directions).

The Vlasov - Poisson system:

In which: f depends on phase-space (x,v) in two dimensions. The detail is in paper .

Example: Diocontron instability

Second step (Under review): we focus on analyse the numerical error of a modified Crank-Nicolson scheme on particle trajectories.

References:

Convergence analysis of Asymptotic Preserving Schemes for strongly magnetized Plasmas, F. Filbet, L. M. Rodrigues and Hamed Zakerzadeh, (2021). Link
Asymptotic-preserving schemes for multiscale physical problems, Shi Jin, (2022). Link

The trajectories of particles are determined by the characteristic curves of the Vlasov equation:

in which (x,v) are the position and velocity of the particle. The detail is published soon.

Image: the numerical error (left) and trajectory of the particle (right) in comparison with the reference solution.

Third step (On-going): we focus on 3D Vlasov-Poisson system with high intense magnetic field in torus.

References:

Asymtoptics of the three dimensional Vlasov equation in the large magnetic field limit, F. Filbet and L. M. Rodrigues, (2020). Link
Asymptotically preserving particle methods for strongly magnetized plasmas in a torus, F. Filbet and L. M. Rodrigues, (2023). Link

The Vlasov - Poisson system: