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Exact moment models for conservation laws in phase space

Tileuzhan Mukhamet, Katharina Kormann

TL;DR

This work addresses the computational burden of high-dimensional kinetic conservation laws by deriving exact reduced models through Burby’s centered-moment parametrization of the distribution function. It develops exact fluid moment models of degree $k$ closed by a center evolution $\partial_t \bm v = - \bm G(\bm v) \cdot (\nabla \bm v) + \bm F(\bm v)$, with explicit degree-0,1,2 closures, and constructs phase-space particle models around moving centers $\bm Z_p$ that are exact when $\partial_t \bm Z_p = \bm A(\bm Z_p,t)$, including degree-0,1,2 realizations. A hybrid fluid–particle framework combines these ansätze to preserve exactness for the sum, enabling flexible representations of plasmas. The approach is applied to non-relativistic and relativistic Vlasov–Maxwell systems, yielding invariant-preserving, computationally tractable models and addressing practical issues such as current regularization in distributional settings.

Abstract

Moment equations offer a compelling alternative to the kinetic description of plasmas, gases, and liquids. Their simulation requires fewer degrees of freedom than phase space models, yet it can still incorporate kinetic effects to a certain extent. To derive moment equations, we use a parameterization of the distribution function using centered moments, as proposed by Burby. This yields moment equations for which the parameterized distribution function exactly solves the hyperbolic conservation law. Similarly, a particle model is derived based on a parametrization of the distribution function using phase space moments. Finally, we present the application of the method to the non-relativistic and relativistic Vlasov--Maxwell equations.

Exact moment models for conservation laws in phase space

TL;DR

This work addresses the computational burden of high-dimensional kinetic conservation laws by deriving exact reduced models through Burby’s centered-moment parametrization of the distribution function. It develops exact fluid moment models of degree closed by a center evolution , with explicit degree-0,1,2 closures, and constructs phase-space particle models around moving centers that are exact when , including degree-0,1,2 realizations. A hybrid fluid–particle framework combines these ansätze to preserve exactness for the sum, enabling flexible representations of plasmas. The approach is applied to non-relativistic and relativistic Vlasov–Maxwell systems, yielding invariant-preserving, computationally tractable models and addressing practical issues such as current regularization in distributional settings.

Abstract

Moment equations offer a compelling alternative to the kinetic description of plasmas, gases, and liquids. Their simulation requires fewer degrees of freedom than phase space models, yet it can still incorporate kinetic effects to a certain extent. To derive moment equations, we use a parameterization of the distribution function using centered moments, as proposed by Burby. This yields moment equations for which the parameterized distribution function exactly solves the hyperbolic conservation law. Similarly, a particle model is derived based on a parametrization of the distribution function using phase space moments. Finally, we present the application of the method to the non-relativistic and relativistic Vlasov--Maxwell equations.
Paper Structure (11 sections, 7 theorems, 105 equations)

This paper contains 11 sections, 7 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Fluid models for conservation laws
  3. Proof of Theorem \ref{['prop:main_theorem']} in 1u1x
  4. Proof of Theorem \ref{['prop:main_theorem']}
  5. Particle models for conservation laws
  6. Proof of Theorem \ref{['prop:main_theorem_particles']}
  7. Hybrid-model
  8. Application to Plasma
  9. Vlasov--Maxwell
  10. Relativistic Vlasov--Maxwell system
  11. Acknowledgement

Key Result

Lemma 2.1

Let $f(z,t)$ be a smooth real-valued function with compact support. Let $\phi \in C^{\infty}_0(\Omega_u)$ s.t. $\phi(\bm u)=1$ for all $\bm u \in T_{\varepsilon} (\bm v(\bm x, t))$ for some $\varepsilon>0$. Then, the following identity holds

Theorems & Definitions (17)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.1
  • proof
  • Proposition 2.1
  • proof
  • Theorem 2.1
  • Remark 2.1
  • proof : Proof (1u1x)
  • proof
  • ...and 7 more