[Paper Review] Variational Integrators in Plasma Physics

M. Kraus|arXiv (Cornell University)|Jul 22, 2013

Numerical methods for differential equations6 references21 citations

TL;DR

This paper develops variational integrators for plasma physics models—guiding centre dynamics, Vlasov-Poisson, and magnetohydrodynamics—by discretizing the Lagrangian and variational principle first, ensuring exact conservation of energy, momentum, and other invariants up to machine precision. The method extends to systems without standard Lagrangians via Ibragimov’s integrating factors, enabling symplectic, geometric time integrators that prevent numerical dissipation and unphysical energy loss.

ABSTRACT

Variational integrators are a special kind of geometric discretisation methods applicable to any system of differential equations that obeys a Lagrangian formulation. In this thesis, variational integrators are developed for several important models of plasma physics: guiding centre dynamics (particle dynamics), the Vlasov-Poisson system (kinetic theory), and ideal magnetohydrodynamics (plasma fluid theory). Special attention is given to physical conservation laws like conservation of energy and momentum. Most systems in plasma physics do not possess a Lagrangian formulation to which the variational integrator methodology is directly applicable. Therefore the theory is extended towards nonvariational differential equations by linking it to Ibragimov's theory of integrating factors and adjoint equations. It allows us to find a Lagrangian for all ordinary and partial differential equations and systems thereof. Consequently, the applicability of variational integrators is extended to a much larger family of systems than envisaged in the original theory. This approach allows for the application of Noether's theorem to analyse the conservation properties of the system, both at the continuous and the discrete level. In numerical examples, the conservation properties of the derived schemes are analysed. In case of guiding centre dynamics, momentum in the toroidal direction of a tokamak is preserved exactly. The particle energy exhibits an error, but the absolute value of this error stays constant during the entire simulation. Therefore numerical dissipation is absent. In case of the kinetic theory, the total number of particles, total linear momentum and total energy are preserved exactly, i.e., up to machine accuracy. In case of magnetohydrodynamics, the total energy, cross helicity and the divergence of the magnetic field are preserved up to machine precision.

Motivation & Objective

• To develop structure-preserving numerical schemes for plasma physics models that conserve key physical invariants such as energy, momentum, and magnetic flux.
• To extend variational integrator theory beyond systems with standard Lagrangians by incorporating Ibragimov’s theory of integrating factors and adjoint equations.
• To ensure long-term numerical stability and accuracy by preserving geometric structures like symplecticity and momentum conservation in particle, kinetic, and fluid plasma models.
• To demonstrate that discrete variational integrators yield unconditionally stable simulations with no artificial dissipation, even over long integration times.

Proposed method

• Discretize the continuous Lagrangian and the action principle first, then derive discrete equations of motion via the discrete variational principle, ensuring intrinsic conservation properties.
• Apply Noether’s theorem at both continuous and discrete levels to identify conserved quantities such as energy, momentum, and cross helicity.
• Use Ibragimov’s method to construct a Lagrangian for systems like Vlasov-Poisson and MHD that do not naturally admit a standard Lagrangian formulation.
• Implement a grid-based discrete Nambu bracket formulation for the Vlasov-Poisson system, using functional derivatives and symmetrized Poisson brackets to preserve antisymmetry and conservation laws.
• Construct the discrete action functional $ \mathcal{A}_d[\varphi] = \sum_{\square} \mathcal{L}_d \circ j^1\varphi(\square) $ over spacetime cells to derive consistent time-stepping schemes.
• Symmetrize the discrete Nambu bracket using all even and odd permutations to maintain antisymmetry and geometric consistency on the discrete level.

Experimental results

Research questions

• RQ1 Can variational integrators be constructed for plasma models that lack a standard Lagrangian formulation, such as the Vlasov-Poisson and MHD systems?
• RQ2 How can the discrete variational principle be applied to preserve conservation laws like energy, momentum, and magnetic divergence in plasma simulations?
• RQ3 To what extent do variational integrators prevent numerical dissipation and spurious energy loss compared to standard time integrators?
• RQ4 Can the extended Lagrangian framework via Ibragimov’s theory be systematically applied to derive structure-preserving schemes for general plasma systems?

Key findings

• For guiding centre dynamics, toroidal momentum is conserved exactly, and particle energy error remains bounded with no numerical dissipation over long simulations.
• In the Vlasov-Poisson system, total particle count, linear momentum, and energy are conserved exactly up to machine precision, with no artificial loss or gain.
• For magnetohydrodynamics, total energy, cross helicity, and the divergence of the magnetic field are preserved to machine precision, ensuring stable and physically consistent simulations.
• The discrete Nambu bracket formulation reproduces the Arakawa discretization, showing consistency between variational and Hamiltonian approaches in structure preservation.
• The symmetrized discrete Nambu bracket ensures antisymmetry and geometric consistency, which is essential for preserving the underlying Poisson structure in kinetic systems.