A Local Macroscopic Conservative (LoMaC) Low Rank Tensor Method for the Vlasov-Maxwell System

Jul 16, 2026

The work addresses the computational difficulty of solving the Vlasov-Maxwell system arising from high-dimensional phase space, nonlinearity, and the need to respect conservation properties. It introduces a Local Macroscopic Conservative (LoMaC) low-rank tensor method that leverages the tensor-friendly structure of the Vlasov equation and uses low-rank hierarchical Tucker decomposition to approximate high-dimensional solutions. The algorithm concurrently evolves mass, momentum, and energy conservation laws with a high-order conservative scheme and ensures matching macroscopic observables via conservative orthogonal projection, validated by numerical tests.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages28

Reading Time1 h 11 min

Introduction

Vlasov-Maxwell system and conservation laws

Deterministic solvers and dimensionality challenge

Dimension reduction via low-rank tensor decomposition

Related work and low-rank dynamical approaches

arXiv :2607 .01381v1 [math .NA] 1 Jul 2026 A Local Macroscopic Conservative (LoMaC) low rank tensor method for the Vlasov-Maxwell system Shadi Heenatigala1 and Wei Guo2 Abstract. The main computational challenges of solving the Vlasov-Maxwell (VM) system include the high dimensionality of the phase space, nonlinearity, inherent conservation properties, among others. In this paper, we develop a novel Local Macroscopic Conservative (LoMaC) low rank tensor method for the VM system, as a continuation of our previous work (arXiv:2207.00518) . The method takes advantage of the tensor friendly structure of the Vlasov equation and employs the low rank hierarchical Tucker decomposition to approximate the Vlasov solution in high dimensions. Hence, the curse of dimensionality can be mitigated. Furthermore, to realize the LoMaC property, the algorithm simultaneously evolves the conservation laws of mass, momentum and energy alongside the Vlasov equation using a high order conservative method with the kinetic ﬂux vector splitting. By a conservative orthogonal projection, the low rank solution is guaranteed to have the same macroscopic observables updated from the conservation laws. A collection of numerical tests on the VM system are presented to demonstrate the eﬃciency and eﬃcacy of the proposed algorithm. Key Words: Low rank; hierarchical Tucker decomposition; Vlasov-Maxwell system; conservative truncation; LoMaC. 1 Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX, 70409 . E-mail: shadiheenati[gala92@gmail.com](gala92@gmail.com). 2 Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX, 70409 . E-mail: [weimath.guo@ttu.edu. Research](weimath.guo@ttu.edu. Research) is supported by NSF grant NSF-DMS-2111383 and Air Force Oﬃce of Scientiﬁc Research FA9550-22-1-0390 . 1 Introduction In this paper, we are concerned with the Vlasov-Maxwell (VM) system, known as one of the most fundamental models in plasma physics. The VM system describes the dynamics of charged particles from the statistical mechanics viewpoint. In particular, the dimensionless single species nonrelativistic VM system in 3D3V (three-dimensional physical space and three-dimensional velocity space) reads ∂tf + v · ∇xf + (E + v × B) · ∇vf = 0 (1.1) ∂tE = ∇x × B − J, ∂tB = −∇x × E (1.2) ∇x · E = ρ − ρi , ∇x · B = 0 , (1.3) where the unknown function f (x, v, t) of the Vlasov equation (1.1) is the probability distribution function of electrons on the phase space Ωx × Ωv , x = (x1 , x2 , x3 ) and v = (v1 , v2 , v3 ) . The electromagnetic ﬁelds E = (E1 , E2 , E3 ) and B = (B1 , B2 , B3 ) are determined from Maxwell’s equations (1.2)-(1.3) . The density ρ and current density J are given by ρ (x, t) = ZΩv f (x, v, t)dv, J (x, t) = ZΩv vf(x, v, t)dv. The ion is assumed to be ﬁxed as a uniform neutralizing background with density ρi. Furthermore, by taking the ﬁrst few moments of the VM system, a set of macroscopic equations can be derived ∂tρ + ∇x · J = 0 (1.4) ∂tP + ∇x · σ = ∇x · ÅE ⊗ E + B ⊗ B − 12 (|E|2 + |B|2 )Iã + Eρi (1.5) ∂te + ∇x · Q = ∇x · (E × B) . (1.6) Here P (x, t) = J (x, t) + E × B denotes the momentum density. e (x, t) = κ (x, t) + ~~1~~2 |E|2 + ~~1~~2 |B|2 denotes the energy density as the sum of the kinetic energy density κ (x, t) =RΩv ~~1~~2 |v|2 f (x, v, t)dv, electric energy density ~~1~~2 |E|2 , and the magnetic energy density ~~1~~2 |B|2 . σ (x, t) =RΩv v⊗vf(x, v, t)dvand Q (x, t) = ~~1~~2 RΩv v|v|2 f (x, v, t)dv are the ﬂux terms, and I is the identity matrix. Equations (1.4)-(1.6) correspond to the conservation laws of mass, momentum and energy, respectively. Note that the total momentum of the single species VM system is conserved only if RΩx E dx = 0, which can be seen from (1.5) . The numerical challenges include the high dimensionality of phase space, inherent conservation properties with respect to the macroscopic equations (1.4)-(1.6), nonlinearity, multi-scale featu

FAQs

What key computational challenges does the paper target for the Vlasov-Maxwell system?

The method is designed to address high phase-space dimensionality, nonlinearity, and the requirement to preserve inherent conservation properties, especially at the discrete level for macroscopic invariants.

How does the proposed LoMaC low-rank tensor method reduce the curse of dimensionality?

It approximates the Vlasov solution using a low-rank hierarchical Tucker decomposition, exploiting the tensor-friendly structure of the Vlasov equation to mitigate dimensional growth.

How is the LoMaC (local macroscopic conservation) property enforced?

The algorithm evolves conservation laws of mass, momentum, and energy alongside the Vlasov equation using a high-order conservative method with kinetic flux vector splitting, then applies a conservative orthogonal projection to keep macroscopic observables consistent.

Jul 16, 2026

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