HOPE

High Order Program for kinetic Equations

Overview

A C++ library for the numerical simulation of kinetic equations, designed for plasma physics and built on structure- and asymptotic-preserving methods whose stability and convergence are established mathematically.

Developed at the Institut de Mathématiques de Toulouse. A distinctive feature of HOPE is the simulation of the Vlasov–Poisson system with a strong external magnetic field in a three-dimensional toroidal geometry with an arbitrary poloidal cross-section, a configuration close to that of tokamaks.

3D particle-in-cell simulation of a magnetized plasma in a torus with a D-shaped cross-section — Three-dimensional particle-in-cell simulation of a plasma confined by a strong magnetic field in a torus with a D-shaped poloidal cross-section. Left: charge density in the volume and in two poloidal sections. Right: charge density in a poloidal section, with the E×B drift velocity (arrows) and magnetic surfaces (white lines) (PhD thesis of K.-H. Trinh, joint work with L. M. Rodrigues, 2026).

Charge density and E×B drift in a D-shaped poloidal cross-section — Three-dimensional particle-in-cell simulation of a plasma confined by a strong magnetic field in a torus with a D-shaped poloidal cross-section. Left: charge density in the volume and in two poloidal sections. Right: charge density in a poloidal section, with the E×B drift velocity (arrows) and magnetic surfaces (white lines) (PhD thesis of K.-H. Trinh, joint work with L. M. Rodrigues, 2026).

Features

Models: Vlasov–Poisson system in two and three dimensions, including strong external magnetic fields; guiding-center and drift-kinetic models; BGK relaxation models.
Particle-in-cell methods in two and three dimensions, with asymptotic-preserving time integrators that remain accurate in the strong magnetic field (guiding-center) limit, and high-order charge deposition.
Eulerian solvers: finite difference and semi-Lagrangian schemes with high-order WENO and Hermite WENO (HWENO) reconstructions.
Spectral methods: Hermite spectral discretization in velocity, currently for problems in one space dimension.
Complex geometries on Cartesian meshes: embedded boundaries treated with ghost points and inverse Lax–Wendroff procedures; disks, D-shaped domains and three-dimensional tori with an arbitrary poloidal cross-section.
Field solvers: Poisson equation (Dirichlet, periodic and nonlinear cases) in two and three dimensions, including toroidal domains, with algebraic multigrid (AGMG) and PETSc linear solvers.
Under development: coupling with Maxwell's equations and with collision operators, extension of the spectral solvers to higher dimensions.

Simulations

Particle-in-cell simulations in a torus

Watch the video — F. Filbet, L. M. Rodrigues, K.-H. Trinh.

Hermite spectral simulations in an interval

Watch the video — A. Blaustein, M. Bessemoulin-Chatard, F. Filbet.

Publications

A modified Crank–Nicolson scheme for the Vlasov–Poisson system with a strong external magnetic field F. Filbet, L. M. Rodrigues, K.-H. Trinh — J. Comput. Phys. 552 (2026), 114693
A structure and asymptotic preserving scheme for the Vlasov–Poisson–Fokker–Planck model A. Blaustein, F. Filbet — J. Comput. Phys. 498 (2024), 112693
Asymptotically preserving particle methods for strongly magnetized plasmas in a torus F. Filbet, L. M. Rodrigues — J. Comput. Phys. 480 (2023), 112015
On the convergence of discontinuous Galerkin/Hermite spectral methods for the Vlasov–Poisson system M. Bessemoulin-Chatard, F. Filbet — SIAM J. Numer. Anal. 61 (2023), 1664–1688
On the stability of conservative discontinuous Galerkin/Hermite spectral methods for the Vlasov–Poisson system M. Bessemoulin-Chatard, F. Filbet — J. Comput. Phys. 451 (2022), 110881
Convergence analysis of asymptotic preserving schemes for strongly magnetized plasmas F. Filbet, L. M. Rodrigues, H. Zakerzadeh — Numer. Math. 149 (2021), 549–593
Asymptotically preserving particle-in-cell methods for inhomogeneous strongly magnetized plasmas F. Filbet, L. M. Rodrigues — SIAM J. Numer. Anal. 55 (2017), 2416–2443
Asymptotically stable particle-in-cell methods for the Vlasov–Poisson system with a strong external magnetic field F. Filbet, L. M. Rodrigues — SIAM J. Numer. Anal. 54 (2016), 1120–1146
Conservative and non-conservative methods based on Hermite weighted essentially non-oscillatory reconstruction for Vlasov equations F. Filbet, C. Yang — J. Comput. Phys. 279 (2014), 18–36
Inverse Lax–Wendroff method for boundary conditions of Boltzmann type models F. Filbet, C. Yang — J. Comput. Phys. 245 (2013), 43–61

All publications →

Download

A simplified public version of HOPE is available. It contains three codes, each with its input files in a runs subfolder:

DG-VP1D: discontinuous Galerkin/Hermite spectral method for the one-dimensional Vlasov–Poisson system (Landau damping, two-stream and bump-on-tail instabilities).
EULERIAN-METHOD-2D: Eulerian solver with finite differences and HWENO reconstructions for the Kelvin–Helmholtz instability.
PIC-2D: asymptotic-preserving particle-in-cell method for the two-dimensional Vlasov–Poisson system with a strong magnetic field, in a square domain.

The codes are written in C++ and require MPI, the sparse direct solver MUMPS, FFTW3, BLAS/LAPACK and OpenMP. The documentation describes the models, the numerical methods, the input and output files and the test cases.

Team

Main developer: Francis Filbet (Institut de Mathématiques de Toulouse)
Contributors: Chang Yang (co-creator of HOPE, now at Harbin Institute of Technology), Nicolas Crouseilles, Michel Mehrenberger, K.-H. Trinh (three-dimensional toroidal geometry)
Collaborations: L. M. Rodrigues, M. Bessemoulin-Chatard, A. Blaustein, H. Zakerzadeh

Availability

HOPE is hosted on PLMlab, the GitLab platform of the French mathematical community. The full version, including the three-dimensional particle-in-cell code in a torus, is shared with collaborators; a simplified public version can be downloaded above. For access to the full version or for collaboration, please get in touch.

Support: ERC Starting Grant NuSiKiMo (2010–2015), the EUROfusion Consortium (Euratom research and training programme, grant agreement No 633053; projects VeriGyro and MAGYK) and the ANR projects Muffin and COOKIE.