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Boris-type exponential integrators for charged-particle dynamics in strong magnetic fields

Marlis Hochbruck, Sebastian Merk

arXiv:2609.35679v1math.NAcs.NA

TL;DR

For strong-field particle dynamics, standard convergence bounds depend on magnetic-field strength, while existing filtered Boris bounds can require nonresonance conditions and encounter resonant singularities. The method uses filters defined on the eigenvalues 0, ±iτ ∥B∥ of τB, where B represents the map v ↦ v × B. Bounded filters avoid velocity-error blow-up at resonances and allow arbitrary timesteps, at the cost of order reduction near resonances.

  • Problem

    For strong-field particle dynamics, standard convergence bounds depend on magnetic-field strength, while existing filtered Boris bounds can require nonresonance conditions and encounter resonant singularities.

    Source: §1 · pp. 2–3

  • Method

    The method uses filters defined on the eigenvalues 0, ±iτ ∥B∥ of τB, where B represents the map v ↦ v × B.

    Source: §1 · p. 3

  • Results

    Bounded filters avoid velocity-error blow-up at resonances and allow arbitrary timesteps, at the cost of order reduction near resonances.

    Source: §1 · p. 4

  • Takeaways & Limitations

    For constant strong magnetic fields, bounded filters provide a way to avoid resonance blow-up without imposing a nonresonance condition on the timestep.

    Source: §1 · pp. 3–4

    Resonance effects cannot generally be eliminated; they vanish in the stated special case where E′(eqn+1/2) commutes with B for all n.

    Source: §6.3 · p. 20

Abstract

from arXiv · show

We study numerical time integration for the motion of a charged particle in a strong magnetic field, focusing on the regime in which the fast gyration is not resolved by the time step. Standard Boris-type methods are structure preserving and accurate when the gyration is resolved, but their error constants deteriorate with increasing field strength; filtered Boris methods improve this behavior but may suffer from singularities and resonance-induced error blow-up. In this work we consider the case of a constant strong magnetic field and derive a broad family of one-step exponential integrators from the variation-of-constants formula, formulated in terms of matrix-valued filter functions. This framework includes existing filtered Boris variants and permits the construction of schemes with uniformly bounded filters. Using a discrete variation-of-constants representation together with summation-by-parts arguments, we establish error bounds whose constants are independent of the magnetic field strength and characterize the filter conditions required for first- and second-order accuracy. In particular, we show that bounded filters avoid resonance blow-up and allow arbitrary time steps, at the cost of a controlled order reduction near resonant frequencies. Numerical experiments confirm the predicted convergence behavior, the different accuracy of position and velocity components, and the role of the leading error terms across a wide range of step-size-field-strength products.

1. Introduction

Strong-field particle dynamics is difficult to integrate accurately when the timestep does not resolve the fast gyration, because standard error bounds depend on field strength. The paper develops a family of filtered exponential integrators and analyzes how bounded filters avoid resonance blow-up while permitting arbitrary timesteps.

1. Introduction

  • Standard Boris-type methods lose accuracy when the gyration is unresolved, while their error constants grow with the magnetic-field strength.In coupled field simulations, the field may be unknown in advance, making worst-case timestep selection or adaptivity necessary.Source: §1 · p. 2
  • The analysis targets stiff-order bounds independent of ||B||, unlike standard Taylor-based convergence bounds whose constants grow with field strength.Related highly oscillatory integrator analyses also use filter conditions and summation by parts, while prior exponential integrators for this system establish classical rather than field-uniform orders.Source: §1 · pp. 2–3
  • Earlier filtered Boris bounds rely on a filter singular at multiples of 2π, causing velocity-error blow-up and requiring nonresonance conditions for second-order position bounds.When fields are supplied through a Maxwell solver, ensuring nonresonance may require timestep adaptation.Source: §1 · p. 3
  • The analysis is a first step for constant magnetic fields; extension to non-constant fields is left for future work.The motivation includes particle characteristics in Vlasov–Maxwell computations, where electromagnetic fields are computed self-consistently.Source: §1 · pp. 1–3
  • The proposed one-step exponential-integrator family uses matrix-valued filters of τB, includes filtered Boris variants, and also permits more general schemes derived from variation of constants.The analysis uses a discrete variation-of-constants formula and summation by parts to characterize filters and resonance behavior.Source: §1 · p. 3
  • Bounded filters avoid velocity-error blow-up at resonances and permit arbitrary timesteps, with order reduction near resonant frequencies.The analysis is restricted to constant magnetic fields, where the field strength is the only large parameter.Source: §1 · pp. 3–4

2. Review on filtered Boris algorithms

The section reviews filtered Boris algorithms and relates their velocity updates to exponential-integrator formulations. For a constant magnetic field, two filtered formulations are equivalent.

2. Review on filtered Boris algorithms

  • The reviewed filtered Boris algorithm is presented as a one-step method on a staggered time grid, with its update expressible in the form of the filtered Boris velocity step.The filtered version in uses a different definition of the intermediate velocity from the formulation in.Source: §2 · pp. 4–5
  • Setting ψ = 1 recovers the classical Boris algorithm.Source: §2 · p. 4
  • The velocity update can be interpreted as applying the exponential midpoint rule to the velocity equation.The text establishes this interpretation by rewriting the update.Source: §2 · p. 4
  • For a constant magnetic field, the filtered algorithm in [6] is equivalent to the formulation in [5] when the filter is chosen as specified.The compared algorithms differ in how rotation and full-grid velocity approximations are realized.Source: §2 · p. 5

3. Exponential integrators for the Euler–Lagrange equations

The paper derives exponential-integrator variants by applying variation of constants to the charged-particle equations and approximating the electric-field terms. Different choices of approximation yield methods with different exactness and grid properties.

3. Exponential integrators for the Euler–Lagrange equations

  • The method construction assumes a strong constant magnetic field and a sufficiently regular autonomous electric field, then represents the dynamics as a semilinear system.The electric field is assumed locally Lipschitz continuous.Source: §3 · p. 5
  • Variation of constants supplies the exponential representation used to approximate the position and velocity updates from electric-field values.The construction evaluates the formulas at selected time points and approximates the electric field by suitable constants.Source: §3 · p. 5
  • The implicit scheme is expected to be solvable with a few fixed-point iterations under the assumption that the electric field does not make the equation stiff.Source: §3 · p. 6
  • A symmetric choice of the scheme is exact for constant magnetic and electric fields, while the classical and filtered Boris update approximates sinch by the identity and drops φ2 terms.Source: §3 · p. 6
  • Exactness for constant electric fields depends on the update variant: several are exact, but others fail because they replace the midpoint velocity by an approximation.For variant (3.8a), exactness holds only when E = 0.Source: §3 · p. 6

4. Error analysis: recursion and stability

The section develops a unified error analysis for filtered Boris-type algorithms, using defects and an error recursion to study stability. It gives power bounds for the schemes in Table 2 and distinguishes variants whose stability bounds require nonresonance.

4. Error analysis: recursion and stability

  • The framework targets a unified analysis of filtered Boris variants and filter conditions for order-one error bounds independent of ||B||.The stated O(τ^j) constants are independent of τ and ||B||.Source: §4 · pp. 6–7
  • For method (2.8) and some results, nonresonance is required, so arbitrary time steps are not admissible.The condition separates τ||B|| from nonzero integer multiples of 2π.Source: §4 · pp. 6–7

4.1. Definition of defects and error recursion.

  • The analysis defines defects by inserting the exact solution into the numerical scheme, then obtains an error recursion by subtracting the scheme.The recursion uses the scheme’s block triangular structure.Source: §4.1 · p. 7

4.2. Stability.

  • The schemes in Table 2 satisfy power bounds ||R^n|| ≤ 1 + t_n and ||Υ_n|| ≤ 1 + t_{n+1/2}; variants (3.8) also have bounded S.For filter functions (2.8), the bound on S requires the nonresonance condition.Source: §4.2 · p. 8

5. Error bounds of order one

This section characterizes filters for field-independent order-one error bounds and derives defect estimates that feed into a global error bound. The estimates rely on bounded trajectories and regularity of the electric field.

5. Error bounds of order one

  • The order-one analysis characterizes filter functions that yield error bounds with constants independent of ||B||.The trajectories are assumed to remain in compact sets independent of ||B||, with E in C1 nearby.Source: §5 · p. 9
  • All listed variants have Ψ1(0) = 0, but only variants with Ψ1 = 0 have position defects O(τ^2) with constants independent of ||B||.Other variants have second-order position defects whose constants may depend on ||B||.Source: §5 · p. 11, Table 2 · p. 11
  • With Φ0 = Ψ0 = 0, bounded S, and sufficiently small τ, Theorem 5.4 gives an order-one error bound for the approximation.The smallness condition is 2τ(1+T)CSLE < 1; the bound’s constant is stated as C(LE, K, S, T).Source: §5 · p. 12

6. Error bounds of order two

The section derives second-order bounds by refining the velocity-defect representation and analyzing error components parallel and perpendicular to the magnetic field. It also identifies conditions governing perpendicular velocity error and resonance effects.

6. Error bounds of order two

  • The second-order analysis needs E in C2 because the second time derivative of E(q(t)) cannot be bounded independently of ||B|| under the weaker regularity.This motivates examining dominant velocity-error terms and their propagation before proving second-order position estimates.Source: §6 · p. 12

6.1. Refined representation of the velocity defect.

  • The velocity defect is represented as two parts that can be handled differently in the error recursion.The representation is derived under the filter condition Φ0 = 0.Source: §6.1 · p. 12

6.2. Errors in parallel and perpendicular direction.

  • For the parallel and perpendicular components, summation by parts supports the componentwise bounds; the theorem’s constants are independent of ||B||.The theorem also requires the stated filter conditions and a smallness condition involving τ, T, LE, and CS.Source: §6.2 · pp. 14, 16
  • Theorem 6.5 gives field-independent bounds: position and parallel velocity errors are O(τ^2), while perpendicular velocity error is O(τ).The result assumes the stated filter conditions and a sufficiently small τ satisfying 2τ(1+T)LECS < 1.Source: §6.2 · p. 16

6.3. Detailed investigation of the velocity error.

  • The filters in (2.8) uniquely make Φ⊥ = 0 and yield second-order total error under the nonresonance assumption.For these filters, the second-order bound requires nonresonance.Source: §6.3 · p. 20
  • Resonance effects cannot generally be eliminated, but vanish when E′(q_{n+1/2}) commutes with B for every n.Under that commutation condition, Theorem 6.5 and Lemma 6.7 hold without the nonresonance condition.Source: §6.3 · p. 20

7. Numerical examples

The experiments vary θ to examine resonance robustness, convergence, componentwise errors, and leading-order predictions across filter choices and electric fields.

7. Numerical examples

  • Away from resonances, both filter choices show second-order position convergence; bounded-filter position peaks remain below the O(τ) envelope.The position-error curves follow the O(τ^2) reference away from resonances, while the predicted first-order loss is not visible at resonances for (3.7b), (3.8b).Source: Figure 1 · p. 22, §7 · pp. 22–23
  • At 2kπ, singular filters produce velocity-error peaks growing like reciprocal distance, whereas (3.7b), (3.8b) avoid blow-up and show no visible first-order behavior.Away from resonances, the bounded-filter velocity error remains below the first-order reference line.Source: Figure 1 · p. 22, §7 · p. 23
  • With (3.7b), (3.8b), parallel position error is essentially θ-independent, perpendicular position error decays like 1/(τ∥B∥), and resonances appear mainly in velocity components.The linear electric field reproduces the nonlinear field's qualitative behavior; the panels differ in error-constant size.Source: Figure 2 · p. 23, §7 · pp. 23–24
  • For large stepsizes, the non-resonant contribution grows like τ^2 and eventually dominates the resonance spikes.Source: §7 · p. 23
  • For perpendicular velocity error, Lemma 6.8's leading-order curves closely match measurements across θ, including the non-monotone maximum near π and slow decay beyond it.Deviations appear near 2kπ; odd-π resonances from off-diagonal terms vanish for the ϕ± filters (3.7a).Source: Figure 3 · p. 24, §7 · p. 24
  • For filter choice (3.7b), (3.8a), Ψ1 ≠ 0, so the perpendicular position error is generally first order; its propagated term leaves remaining components second-order only while negligible, then first-order.Source: §7 · pp. 24–25
  • When E′ commutes with B, parallel errors lack resonance effects and the perpendicular velocity spike at π disappears, while the leading-order Φ⊥ behavior remains.The experiment compares commuting and noncommuting field pairs.Source: §7 · pp. 25–26, Figure 5 · p. 26

Appendix A. Useful identities

Appendix A collects identities and lemmas used in the proofs.

Appendix A. Useful identities

  • Appendix A gathers identities and lemmas used in the preceding proofs.Source: Appendix A · p. 27
  • One proof begins by invoking identity (2.4), though the passage does not specify the accompanying identity.Source: Appendix A · p. 27
  • Another proof uses identities (A.1h) and (A.1g) to establish a lemma.Source: Appendix A · p. 27
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