A Legendre–Fourier spectral method with exact conservation laws for the Vlasov–Poisson system
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Istituto di Matematica Applicata e Tecnologie Informatiche, Pavia (Italy)
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- KTH Royal Institute of Technology, Stockholm (Sweden)
In this study, we present the design and implementation of an L2-stable spectral method for the discretization of the Vlasov–Poisson model of a collisionless plasma in one space and velocity dimension. The velocity and space dependence of the Vlasov equation are resolved through a truncated spectral expansion based on Legendre and Fourier basis functions, respectively. The Poisson equation, which is coupled to the Vlasov equation, is also resolved through a Fourier expansion. The resulting system of ordinary differential equation is discretized by the implicit second-order accurate Crank–Nicolson time discretization. The non-linear dependence between the Vlasov and Poisson equations is iteratively solved at any time cycle by a Jacobian-Free Newton–Krylov method. In this work we analyze the structure of the main conservation laws of the resulting Legendre–Fourier model, e.g., mass, momentum, and energy, and prove that they are exactly satisfied in the semi-discrete and discrete setting. The L2-stability of the method is ensured by discretizing the boundary conditions of the distribution function at the boundaries of the velocity domain by a suitable penalty term. The impact of the penalty term on the conservation properties is investigated theoretically and numerically. An implementation of the penalty term that does not affect the conservation of mass, momentum and energy, is also proposed and studied. A collisional term is introduced in the discrete model to control the filamentation effect, but does not affect the conservation properties of the system. Numerical results on a set of standard test problems illustrate the performance of the method.
- Research Organization:
- Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
- Sponsoring Organization:
- USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA)
- Grant/Contract Number:
- AC52-06NA25396
- OSTI ID:
- 1331266
- Alternate ID(s):
- OSTI ID: 1347624
- Report Number(s):
- LA-UR-15-27359
- Journal Information:
- Journal of Computational Physics, Vol. 317, Issue C; ISSN 0021-9991
- Publisher:
- ElsevierCopyright Statement
- Country of Publication:
- United States
- Language:
- English
Web of Science
| Convergence of spectral discretizations of the Vlasov-Poisson system | text | January 2016 |
| Annotations on the virtual element method for second-order elliptic problems | preprint | January 2016 |
| Quadratic conservative scheme for relativistic Vlasov--Maxwell system | text | January 2018 |
A Semi-Lagrangian Spectral Method for the Vlasov–Poisson System Based on Fourier, Legendre and Hermite Polynomials
|
journal | May 2019 |
| On the stability of conservative discontinuous Galerkin/Hermite spectral methods for the Vlasov-Poisson system | text | January 2021 |
| Conservative discontinuous Galerkin/Hermite Spectral Method for the Vlasov-Poisson System | preprint | January 2020 |
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