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Robust and conservative dynamical low-rank methods for the Vlasov equation via a novel macro-micro decomposition

Jack Coughlin, Jingwei Hu, Uri Shumlak

TL;DR

The paper tackles the conservation shortcomings of dynamical low-rank methods for the Vlasov equation by introducing a macro-micro decomposition that splits f into a fixed-rank macro part N and a micro part g. The micro component is evolved with a dynamical low-rank approach while the macro part is advanced using standard conservative discretizations, enabling exact local conservation of charge and, with appropriate field coupling, energy or current. The authors develop first- and second-order time integrators that preserve these conservation properties and prove the corresponding local laws, all while remaining compatible with both projector-splitting and BUG-style DLR schemes. They demonstrate the framework with two velocity-space discretizations (asymmetrically-weighted Hermite and Legendre-based finite differences) and benchmark it on classic plasma problems, including weak and strong Landau damping and the two-stream instability. The results show that the method achieves expected temporal accuracy, preserves invariants, and provides substantial computational efficiency advantages, with flexibility to adapt velocity discretizations to problem specifics.

Abstract

Dynamical low-rank (DLR) approximation has gained interest in recent years as a viable solution to the curse of dimensionality in the numerical solution of kinetic equations including the Boltzmann and Vlasov equations. These methods include the projector-splitting and Basis-update & Galerkin (BUG) DLR integrators, and have shown promise at greatly improving the computational efficiency of kinetic solutions. However, this often comes at the cost of conservation of charge, current and energy. In this work we show how a novel macro-micro decomposition may be used to separate the distribution function into two components, one of which carries the conserved quantities, and the other of which is orthogonal to them. We apply DLR approximation to the latter, and thereby achieve a clean and extensible approach to a conservative DLR scheme which retains the computational advantages of the base scheme. Moreover, our approach requires no change to the mechanics of the DLR approximation, so it is compatible with both the BUG family of integrators and the projector-splitting integrator which we use here. We describe a first-order integrator which can exactly conserve charge and either current or energy, as well as an integrator which exactly conserves charge and energy and exhibits second-order accuracy on our test problems. To highlight the flexibility of the proposed macro-micro decomposition, we implement a pair of velocity space discretizations, and verify the claimed accuracy and conservation properties on a suite of plasma benchmark problems.

Robust and conservative dynamical low-rank methods for the Vlasov equation via a novel macro-micro decomposition

TL;DR

The paper tackles the conservation shortcomings of dynamical low-rank methods for the Vlasov equation by introducing a macro-micro decomposition that splits f into a fixed-rank macro part N and a micro part g. The micro component is evolved with a dynamical low-rank approach while the macro part is advanced using standard conservative discretizations, enabling exact local conservation of charge and, with appropriate field coupling, energy or current. The authors develop first- and second-order time integrators that preserve these conservation properties and prove the corresponding local laws, all while remaining compatible with both projector-splitting and BUG-style DLR schemes. They demonstrate the framework with two velocity-space discretizations (asymmetrically-weighted Hermite and Legendre-based finite differences) and benchmark it on classic plasma problems, including weak and strong Landau damping and the two-stream instability. The results show that the method achieves expected temporal accuracy, preserves invariants, and provides substantial computational efficiency advantages, with flexibility to adapt velocity discretizations to problem specifics.

Abstract

Dynamical low-rank (DLR) approximation has gained interest in recent years as a viable solution to the curse of dimensionality in the numerical solution of kinetic equations including the Boltzmann and Vlasov equations. These methods include the projector-splitting and Basis-update & Galerkin (BUG) DLR integrators, and have shown promise at greatly improving the computational efficiency of kinetic solutions. However, this often comes at the cost of conservation of charge, current and energy. In this work we show how a novel macro-micro decomposition may be used to separate the distribution function into two components, one of which carries the conserved quantities, and the other of which is orthogonal to them. We apply DLR approximation to the latter, and thereby achieve a clean and extensible approach to a conservative DLR scheme which retains the computational advantages of the base scheme. Moreover, our approach requires no change to the mechanics of the DLR approximation, so it is compatible with both the BUG family of integrators and the projector-splitting integrator which we use here. We describe a first-order integrator which can exactly conserve charge and either current or energy, as well as an integrator which exactly conserves charge and energy and exhibits second-order accuracy on our test problems. To highlight the flexibility of the proposed macro-micro decomposition, we implement a pair of velocity space discretizations, and verify the claimed accuracy and conservation properties on a suite of plasma benchmark problems.
Paper Structure (22 sections, 7 theorems, 111 equations, 6 figures)

This paper contains 22 sections, 7 theorems, 111 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Properties of the Vlasov equation with Dougherty collisions
  3. A novel macro-micro-decomposition of the Vlasov equation
  4. Time discretization
  5. Dynamical low-rank approximation of $g$
  6. First-order integrator
  7. Proof of conservation
  8. Second-order time integrator
  9. Proof of charge and energy conservation
  10. Substeps for low-rank factors
  11. Spatial discretization
  12. Finite difference discretization of hyperbolic terms
  13. Two velocity space discretizations
  14. Asymmetrically-weighted Hermite spectral method
  15. Spectral filtering of Hermite modes
  16. ...and 7 more sections

Key Result

Theorem 3.1

Define the conserved quantities of charge, current, and kinetic energy density at time level $t^n$ as follows: The first-order integration algorithm of Section sec:first-order-integrator satisfies three local source-balance laws

Figures (6)

  • Figure 1: Convergence plot demonstrating first and second-order accuracy in time of the respective time integrators. The first-order integrator is unstable for $\Delta t = 8.0e-3$.
  • Figure 2: Weak Landau damping example, demonstrating exact conservation of charge and energy with the second-order time integrator.
  • Figure 3: Collisional Landau damping example.
  • Figure 4: Strong Landau damping example.
  • Figure 5: Conservation plots in the strong Landau damping example. We observe the benefit of overall second-order accuracy in the improved conservation of current in (b), compared to the conservation error of energy in (a).
  • ...and 1 more figures

Theorems & Definitions (15)

  • Theorem 3.1
  • proof
  • Corollary 3.1
  • proof
  • Corollary 3.2
  • proof
  • Remark 1
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • ...and 5 more