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A Hybridizable Discontinuous Galerkin Method for Time-Modulated Drude--Lorentz Media

Jeonghun J. Lee

arXiv:2610.21991v1math.NAcs.NA

TL;DR

Time-modulated dispersive media need a dynamical constitutive law because mechanisms with the same frozen permittivity can produce different temporal scattering. This paper develops continuous-state models and an event-aligned Maxwell–material HDG method, and its analyses and experiments show that switch-time displacement can change convergence and stability predictions.

  • Problem

    Time-modulated dispersive media require dynamical constitutive laws because mechanisms with the same frozen Drude–Lorentz permittivity can produce different temporal scattering.

    Source: §1 · p. 1

  • Method

    The paper formulates four continuous-state Drude–Lorentz models with distinct transmission rules and analyzes an HDG method using electric displacement.

    Source: §1 · p. 2, §2.1 · p. 3

  • Results

    A displaced switch can reduce midpoint accuracy to first order, produce a model-2 electric-field pulse, and reverse a fixed-mode growth/decay prediction.

    Source: §8 · p. 29

  • Takeaways & Limitations

    Event alignment matters: midpoint's second-order rate requires physical switch times to be grid nodes, while model-2 field convergence may fail uniformly in time on mismatched histories.

    Source: §8 · p. 29

    The spatial convergence theorem applies to compatible switching histories; lower regularity calls for reduced approximation estimates, and constants are not uniform in zero-coupling or zero-resonance limits.

    Source: §4.3 · p. 13

Abstract

from arXiv · show

We formulate and analyze a hybridizable discontinuous Galerkin (HDG) method for Maxwell's equations coupled to four time-modulated Drude--Lorentz models: modulated in-coupling, modulated polarization density, modulated current density, and modulated effective mass. The use of electric displacement and model-dependent continuous material states gives a common treatment of temporal interfaces while preserving the distinct physical transmission rules. We derive energy and stability estimates, establish spatial and event-aligned time-error bounds, and separate time-integration error from error caused by misplaced switching times. Numerical experiments in three dimensions and in reduced modal systems confirm the predicted spatial, temporal, and switch-timing behavior, including the field-jump effect specific to the polarization-density model.

1 Introduction

Time modulation can produce different temporal scattering even when mechanisms share the same frozen Drude–Lorentz permittivity, because their material evolution and interface continuity rules differ. The paper formulates four continuous-state models and analyzes an event-aligned Maxwell–material HDG scheme, including errors from displaced switches.

1 Introduction

  • Four modulation mechanisms with the same frozen Drude–Lorentz permittivity can produce different temporal scattering coefficients because material evolution and temporal-interface continuity differ.The constitutive law must be dynamical rather than a frequency-domain permittivity with parameters replaced by functions of time.Source: §1 · p. 1
  • Existing work includes time-domain Maxwell HDG and DG analyses, while related smooth-coefficient estimates differ from this paper's treatment of coefficient jumps and switch-time displacement.The present estimates concern auxiliary dispersive states with constant background permittivity and permeability.Source: §1 · pp. 1–2
  • Local polarization equations add no spatial derivatives, so their unknowns need no additional skeleton variables; computational advantages over other solvers remain to be demonstrated at equal response accuracy.The method is presented as a reference solver for time-modulated wave propagation and eventual inverse design.Source: §1 · p. 2
  • Switch displacement creates error distinct from midpoint integration error, can reduce midpoint accuracy to first order, and in model 2 requires separating state convergence from field convergence.The model-2 electric constitutive reconstruction can be discontinuous.Source: §1 · p. 2
  • The paper formulates four continuous-state models with distinct transmission rules and derives model-specific energy balances and spatial and event-aligned time-error estimates for an HDG scheme.The work addresses a different issue from prior DG analyses of dispersive Maxwell systems: the four modulation mechanisms' distinct temporal transmission rules.Source: §1 · pp. 1–2

2 Models, transmission conditions, and continuous energy

The section formulates four discontinuous-coefficient Drude–Lorentz models using states continuous at temporal events, with model-specific constitutive coordinates and transmission rules. It also derives model-specific energy balances and an energy amplification bound.

2 Models, transmission conditions, and continuous energy

  • The source-free Maxwell system uses electric displacement and one electric Lorentz pole, on a PEC-bounded polyhedron or a periodic cell.The mesh resolves fixed spatial material interfaces; coefficients are piecewise smooth between global event times.Source: §2 · pp. 2–3

2.1 Continuous-state formulations

  • All four formulations use material states that are absolutely continuous within event-free intervals and continuous across temporal events.The equations define the discontinuous-coefficient model without expanding derivatives of step coefficients into distributions.Source: §2.1 · p. 3

2.1.1 Model 1: modulated in-coupling

  • Model 1 represents modulated in-coupling, or oscillator-strength modulation, with state Y1 = (D, H, P, J).Source: §2.1.1 · p. 3

2.1.2 Model 2: modulated polarization density

  • Model 2 uses Q = P/s as its continuous material coordinate; a switch in s can therefore produce a physical jump in P.The transmission rules keep D, H, Q, and V continuous rather than imposing continuity of P.Source: §2.1.2 · pp. 3–4

2.1.3 Model 3: modulated current density

  • Model 3 localizes its integral constitutive law with an independent memory state, so in general P ≠ sQ.Its state components remain continuous at events, but the physical energy does not control the independent P state, which is estimated separately.Source: §2.1.3 · p. 4

2.1.4 Model 4: modulated effective mass

  • Model 4 represents effective-mass modulation through s multiplying the polarization current and the reciprocal factor b/s in the restoring term.At b = 0, models 3 and 4 have coincident field and polarization dynamics, although the coercivity assumption used below fails.Source: §2.1.4 · p. 4
  • The analysis gives model-specific continuous energy balances and accounts separately for modulation work and energy changes at temporal events.The resulting amplification bound is an energy estimate, not a sharp Floquet growth prediction.Source: §2.1.4 · pp. 5–6

3 HDG formulation and semidiscrete stability

The first-order Maxwell–material system is discretized with two HDG variants, and its material states are transferred continuously across temporal events while fields and traces are reconstructed. The semidiscrete energy balance supports stability analysis, while variant B’s common spatial rate is optimal for the magnetic degree, not an extra-order electric-field result.

3 HDG formulation and semidiscrete stability

  • The method differs from stationary superconvergent Maxwell HDG constructions by discretizing a first-order time-dependent Maxwell–material system without a divergence multiplier.It uses the full tangential electric trace and a semidiscrete energy identity rather than a stationary curl–curl duality argument.Source: §3 · p. 7
  • Variant B uses degree k + 1 for electric/material fields, degree k for magnetic fields, and stabilization τK = θK/hK.Its proven common rate is optimal for the magnetic degree, not an extra-order electric-field result.Source: §3 · p. 7

3.1 Temporal interfaces and initialization

  • At temporal events, the method copies continuous material states, then recomputes the electric field and algebraic trace using the new coefficients.The hybrid trace is not an independent differential state and need not be copied.Source: §3.1 · pp. 7–8
  • Initialization uses elementwise L2 projections, with degree m for displacement and material states and degree k for the magnetic field.Model 3 initialization includes its independent memory state.Source: §3.1 · p. 8

3.2 Solvability and local elimination

  • For fixed element fields, the trace is unique, and eliminating it yields a finite-dimensional ODE for the continuous states with a unique solution on each interval.The solution continues across events by the prescribed transmission map.Source: §3.2 · p. 8
  • In an implicit time step, element-local material variables and electromagnetic fields can be eliminated, leaving a global system only for the electric trace.Implicit Euler and midpoint solvability follow from an energy inequality.Source: §3.2 · p. 8

3.3 Semidiscrete balance and stability

  • The semidiscrete energy balance holds on event-free intervals, and the event identities lead to an interval-wise stability bound.The bound uses nonnegative loss and dissipation terms.Source: §3.3 · p. 8
  • The semidiscrete energy controls D_h = ϵ0E_h + P_h, while the model-3 estimate does not directly control P_h alone.The paper distinguishes control of the physical displacement from coercivity over all five states.Source: §3.3 · p. 9

4 Spatial error analysis

The spatial analysis derives projection-based error identities and integrates their energy estimates across temporal events. Under its regularity and transmission hypotheses, enriched variant B attains a common-state rate of order h^(k+1), while equal-degree variant A has a lower rate.

4.1 Projection identities and the exact error equations

  • The projected material equations are exact for both variants, and the projected electromagnetic error admits an energy identity.The material equations have elementwise constant scalar coefficients, so the electric projection applies exactly without a common projection of E and H.Source: §4.1 · p. 10
  • Variant A’s residual rates differ from variant B’s balanced residual powers; increasing τ alone in the equal-degree method leaves an electric contribution of order h^(2k).For variant B, the magnetic contribution gains a factor hK while the electric contribution starts with a higher-degree trace estimate.Source: §4.1 · p. 11

4.2 Events and integrated energy estimate

  • The integrated energy estimate includes all events, where model 2’s electric error can jump even though the error in (D, H, Q, V) remains continuous.Model 3 uses a reduced continuous error state.Source: §4.2 · p. 11

4.3 Spatial convergence theorem

  • Variant B achieves a common-state spatial rate of order h^(k+1) for models 1, 2, and 4, with the same rate extending to model 3’s full state.For model 3, the initial estimate is for (E, H, Q, V); adding P and D preserves the rate under the stated assumptions.Source: §4.3 · p. 12
  • Variant A’s common-state estimate is order h^(k+1/2), and the analysis does not upgrade it to order h^(k+1).Source: §4.3 · p. 13
  • The spatial theorem requires solution regularity and compatible switching traces; a heterogeneous model-2 event can create an electric jump incompatible with tangential continuity.Incompatible switching data may fail the stated regularity hypothesis, and constants are not claimed uniform in the zero-coupling or zero-resonance limit.Source: §4.3 · p. 13

5 Event-aligned time discretization: stability and fully discrete error

The time analysis aligns steps with coefficient events and establishes stability and fully discrete error estimates for backward Euler and implicit midpoint. The bounds separate spatial and time-integration errors, while event misplacement requires separate analysis.

5 Event-aligned time discretization: stability and fully discrete error

  • Both backward Euler and implicit midpoint are unconditionally energy stable for the stated models, with unique solvability and no CFL condition.For model 3, the assertions hold in the reduced energy, with its independent polarization controlled by summing its update.Source: §5 · p. 14

5.1 Backward Euler: fully discrete error

  • Variant B’s backward Euler estimate gives common space-time convergence on every positive-step event-aligned partition, without a maximum-step restriction.The theorem assumes the stated spatial hypotheses and piecewise time regularity.Source: §5.1 · p. 15
  • For model 3, the backward Euler common space-time rate applies first to the reduced state and extends to P and D under corresponding regularity.Source: §5.1 · p. 16

5.2 Implicit midpoint: fully discrete error

  • The midpoint estimate is second order in time on a fixed mesh; a mesh-uniform bound requires semidiscrete regularity quantities bounded independently of h, which is not established.The same rate holds for model 3’s reduced state and extends to P and D with an additional term.Source: §5.2 · pp. 17–18
  • The estimates for forced systems assume exact Galerkin loads; numerically approximated loads introduce an additional residual not included in those bounds.Source: §5.2 · p. 18

5 Event-aligned time discretization: stability and fully discrete error

  • Event alignment ensures smooth-step time defects; a step crossing a physical jump requires separate timing-error analysis.Event-aligned grids contain every prescribed event, so no open time step contains one.Source: §5 · p. 13, §5.2 · p. 18

6 Switch-time perturbations and convergence reduction

Section 6 separates error from time integration from error due to displaced switching times. It shows that switch displacement can limit midpoint accuracy and, in model 2, produce a larger transient electric-field error than the continuous-state error suggests.

6.1 State error caused by switch displacement

  • The continuous-state error from displaced events is bounded by contributions supported on the intervals where the physical and represented schedules differ; for HDG, uniformity in h requires uniform constants.The comparison separates discretization error against the shifted solution from shift error against the physical solution.Source: §6.1 · p. 19, §6 · p. 19
  • When |δ| is comparable to ∆t and K(θ)Y0 is nonzero, the first-order timing term dominates midpoint’s second-order integration error.The timing error vanishes when the event is fitted exactly; midpoint state bounds also include the usual spatial and time-integration terms.Source: §6.1 · p. 20

6.2 Sharpness for a spatially exact Lorentz mode

  • For a spatially constant Lorentz mode, both HDG variants are exact in space, yet displaced-schedule errors are bounded above and below by positive multiples of ∆t; fitting the switch gives O(∆t²) midpoint error.Even exact integration on the displaced schedule has first-order error relative to the fixed physical problem.Source: §6.2 · p. 21

6.3 Electric-field error in model 2

  • In model 2, an O(|δ|) continuous-state error can coexist with an electric-field error of order |δ|^1/2 in L2 over time, while the final-time field error is O(|δ|) after both switches.The transient pulse arises from the discontinuous constitutive reconstruction and persists even when both histories are integrated exactly; event fitting removes it.Source: §6.3 · p. 22

7 Numerical experiments

The experiments test spatial and temporal convergence, switch-timing errors, and their effects on electric-field convergence and stability classification. Results include the expected fixed-mesh time orders and show that misplaced switches can change both convergence behavior and a modal stability prediction.

7.1 Spatial convergence

  • Variant B’s last-grid rates are 1.11–1.12 for k = 0 and 2.15–2.17 for k = 1, consistent with its proved hk+1 common-state rate; variant A approaches orders 0.9 and 1.9.The measurements do not establish an additional electric-field order; Tables 1–4 report maximum sampled field-plus-state errors and adjacent-grid rates.Source: Table 3 · p. 23, Table 1 · p. 23, Table 2 · p. 23

7.3 Switch-time displacement reduces midpoint accuracy

  • Fitted midpoint error is second order, whereas exact flow and midpoint on a displaced schedule have first-order error; event alignment is necessary to retain midpoint’s order relative to the physical history.A constant mode is represented exactly by the periodic HDG space, and a propagating fixed-mesh mode shows the same distinction across all four models.Source: §7.3 · p. 24, Table 7 · p. 26

7.2 Temporal convergence with smooth and switching coefficients

  • Across all four models and both smooth and switching histories, finest backward Euler rates are 0.981–0.987 and midpoint rates round to 2.000 at fixed mesh.This verifies the expected orders, including with a correctly fitted event, but does not establish a mesh-uniform midpoint constant.Source: §7.2 · p. 24, Table 5 · p. 25, Table 6 · p. 26

7.5 A displaced switch can reverse the stability prediction

  • At γ = 1.521573577390525, the physical mode grows while the displaced-switch calculation predicts decay; after 5,000 periods, their norms are 18.0186 and 0.0555291, respectively.The fitted calculation reaches 18.5088; this is a fixed-mode example, not a general Floquet result for the HDG discretization.Source: Figure 2 · p. 28, §7.5 · p. 28

8 Discussion and conclusions

The paper analyzes two HDG variants for four time-modulated Drude–Lorentz models, establishing energy estimates and spatial and temporal error results. Experiments confirm the predicted convergence behavior while showing that displaced switching times can reduce midpoint accuracy and disrupt field convergence in model 2.

8 Discussion and conclusions

  • The paper proposes two HDG variants for four Drude–Lorentz models with time-dependent coefficients and analyzes their error convergence.The models’ coefficient discontinuities are included in the energy estimates.Source: §8 · p. 29
  • Spatial common-state rates are O(hk+1/2) for variant A and O(hk+1) for variant B, with event-aligned backward Euler bounds of O(hk+1 + ∆t) for variant B.Both methods are energy stable on event-aligned grids without a CFL restriction; midpoint is second order in ∆t on each fixed mesh under the stated regularity assumptions.Source: §8 · p. 29
  • Fixed-mesh experiments show the expected first- and second-order temporal convergence, while spatial experiments reach nearly hk+1 convergence even for variant A.The variant A spatial result exceeds its proved rate.Source: §8 · p. 29
  • Displacing physical switch times can reduce midpoint accuracy to first order, produce a model-2 electric-field pulse, and reverse a fixed mode’s growth/decay prediction.The pulse prevents uniform-in-time field convergence on mismatched histories; a mesh-uniform second-order midpoint bound and sharper electric-field estimates remain open.Source: §8 · p. 29
  • The conclusions concern finite-time estimates; proving infinite-time exponential decay would require additional conditions balancing dissipation against modulation and analysis of stationary components.The latter includes the independent memory state in model 3.Source: §8 · p. 29

Declaration of AI-assisted writing

The manuscript discloses that ChatGPT assisted with drafting and editing, and that the author reviewed and revised the text and takes responsibility for its final content.

Declaration of AI-assisted writing

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