Introduction

Coherent control of electronic and lattice degrees of freedom is a novel frontier in materials physics1,2. Experiments have demonstrated ultrafast control over physical properties and interactions3,4, light-induced phase transitions5,6,7, and driven lattice responses8,9, all occurring on femtosecond to few-picosecond timescales. At the same time, a range of longer-lived phenomena—extending to hundreds of picoseconds and beyond—have been observed, such as carrier relaxation in nanostructured semiconductors10, spin dynamics via light-driven phonons11,12, coherent lattice oscillations13,14, and lattice thermalization15.

These dynamics can be probed experimentally using techniques such as transient absorption and reflectivity16,17,18, inelastic X-ray scattering19,20, time- and angle-resolved photoemission spectroscopy21,22 and transient thermal gratings23, among others. In parallel, advances in computing electron and phonon interactions from first principles24,25,26,27 have enabled quantitative simulations of nonequilibrium phenomena such as excited carrier relaxation28,29,30,31,32, high-field transport33, ultrafast spectroscopies34 and more recently simulations of coupled electron and phonon dynamics35,36,37. These methods can unravel microscopic mechanisms of ultrafast dynamics and quantitatively interpret time-domain spectroscopies.

Various first-principles methodologies address distinct regimes of nonequilibrium dynamics, each tailored to specific timescales and interactions. Time-dependent density functional theory can effectively model coherent femtosecond dynamics of electrons following optical excitation38,39. Other approaches such as nonadiabatic molecular dynamics (NAMD)32,40 and nonequilibrium Green’s functions41,42 are also widely used to model femtosecond electronic processes. Recent advances in momentum space NAMD (NAMD_k) have shown promising results for simulating quantum dynamics in graphene and silicon on picosecond timescales43,44, but reaching longer timescales where anharmonic phonon effects become dominant remains challenging.

This work focuses on the real-time Boltzmann transport equation (rt-BTE) method26,30,33,35 with first-principles electron and phonon interactions. The rt-BTE is a set of coupled integro-differential equations for the time-dependent electron and phonon occupations in momentum space, hereon referred to as populations. Both the electron-phonon (e-ph) and phonon-phonon (ph-ph) scattering integrals are computed on dense momentum grids at each time step, with timescales of interest ranging from femtoseconds (fs) for electrons to hundreds of picoseconds (ps) for phonons. Using dense momentum grids ensures precise resolution of scattering processes and energy conservation, which are critical for reliable predictions of the dynamics. The ability to capture anharmonic ph-ph interactions45,46,47,48,49,50 is particularly important for modeling nonequilibrium lattice dynamics at the ps timescale and in materials with strong anharmonicity.

However, due to computational challenges with propagating the rt-BTE in time, only a handful of recent works have shown first-principles simulations of phonon dynamics in the time domain35,36,51. Two key challenges are the different timescales of electron and phonon dynamics and the computational cost of evaluating collision integrals, particularly for ph-ph interactions. Typically, the rt-BTE is evolved with a fixed time step of a few fs to resolve the faster electron dynamics, while the simulation extends to tens of ps for the phonons to reach steady state or thermal equilibrium. At each time step, the ph-ph scattering integral is computed by integrating over a dense momentum grid in the Brillouin zone (BZ) for all phonon modes. In bulk systems, this requires summing over billions of scattering channels, which is several orders of magnitude more expensive than evaluating the e-ph scattering integrals. Therefore, a 10-ps simulation for a 2D material with a few atoms in the unit cell can easily exceed thousands of CPU core hours on modern high-performance computers35. Without a significant reduction in computational cost, simulations of coupled electron and phonon dynamics remain out of reach for bulk materials and all but the simplest 2D materials.

To address these challenges, we explore time integration methods from the SUNDIALS suite52,53, which offers an array of robust and efficient algorithms for integrating differential equations. The ARKODE package54 within SUNDIALS contains adaptive step size Runge-Kutta (RK)55 as well as multirate infinitesimal (MRI) methods56. Adaptive RK methods dynamically adjust the time step to obtain solutions within user-specified solution tolerances while maximizing efficiency, selecting step sizes that reflect the inherent dynamics of the system. MRI methods further improve efficiency by time-evolving different processes or solution components with different step sizes. They are particularly effective when the dynamics are governed by processes with well-separated timescales and the slower processes dominate the computational cost, making them an ideal fit for the rt-BTE. Methods in the MRI family have demonstrated improved efficiency in air pollution models57 and reactive flow simulations58 compared to single-rate integrators and multirate spectral deferred correction methods, respectively. Despite their success in these areas, their application to electron and lattice dynamics simulated from first principles remains unexplored.

In this work, we use adaptive RK and MRI time integration methods to simulate coupled electron and phonon dynamics in the rt-BTE framework. This is done by implementing an interface between the PERTURBO code26 and the SUNDIALS library52,53. We achieve a significant reduction in computational cost while retaining the same accuracy by using different and adaptive time steps for e-ph and ph-ph interactions. We benchmark our approach on graphene, achieving a reduction in computational cost by orders of magnitude for any choice of solution tolerance. Finally, our adaptive and multirate time-stepping scheme allows us to solve the coupled rt-BTE for bulk materials with reasonable computational effort, a goal that has so far been considered out of reach. This is demonstrated by studying nonequilibrium lattice dynamics in silicon and gallium arsenide, and simulating the associated thermal diffuse scattering maps in silicon. Taken together, the methods developed in this work advance accurate simulations of nonequilibrium physics in real materials, opening doors for future studies of ultrafast electronic and lattice dynamics in materials driven by optical or terahertz pulses.

Results

Real-time Boltzmann transport equation

Using the PERTURBO26,35 code, we propagate in time the coupled electron and phonon rt-BTEs in a homogeneously excited material:

$$\begin{array}{lll}\frac{\partial {f}_{n{\bf{k}}}(t)}{\partial t}\,=\,{{\mathcal{I}}}^{e-{\rm{ph}}}[\,{f}_{n{\bf{k}}}(t),{N}_{\nu {\bf{q}}}(t)],\\ \frac{\partial {N}_{\nu {\bf{q}}}(t)}{\partial t}\,=\,{{\mathcal{I}}}^{{\rm{ph}}-e}[\,{f}_{n{\bf{k}}}(t),{N}_{\nu {\bf{q}}}(t)]+{{\mathcal{I}}}^{{\rm{ph}}-{\rm{ph}}}[{N}_{\nu {\bf{q}}}(t)].\end{array}$$
(1)

The electron populations, fnk(t), are labeled by the electronic band index n and crystal momentum k (hole carriers can be simulated with populations 1 − fnk(t)), while the phonon populations, Nνq(t), are labeled by the mode index ν and wave-vector q. In Eq. (1), the e-ph/ph-e scattering integrals, \({{\mathcal{I}}}^{e-{\rm{ph}}}\) and \({{\mathcal{I}}}^{{\rm{ph}}-e}\), and the ph-ph scattering integrals, \({{\mathcal{I}}}^{{\rm{ph}}-{\rm{ph}}}\), are computed from the corresponding interaction matrix elements (see “Methods”).

Previous work has simulated coupled carrier and phonon dynamics in graphene by advancing the rt-BTE using the 4th-order Runge-Kutta (RK4) method with a fixed time step size35. While the e-ph and ph-ph scattering matrices can be computed and stored before the real-time simulation, the scattering integrals need to be recomputed at each time step and require integration over the BZ (see “Methods”). Computing the ph-ph scattering integrals takes about two orders of magnitude more runtime than the e-ph integrals due to differences in the scattering processes in momentum space. The computational cost of evaluating all e-ph and ph-ph scattering integrals is determined by the number of possible scattering channels, as constrained by momentum and energy conservation. Once these channels are identified, at each time step, the computational cost depends only on summations over momenta and bands (for electrons) or modes (for phonons) of two particles participating in the collision, while the state of the third particle is determined by conservation laws and does not affect the scaling complexity. Therefore, for e-ph interactions, the cost scales as \({{\mathcal{N}}}_{{\rm{c}}}\,{{\mathcal{N}}}_{{\rm{ph}}}\), where \({{\mathcal{N}}}_{{\rm{c}}}\) is the number of carrier momenta and band indices (labels nk), and \({{\mathcal{N}}}_{{\rm{ph}}}\) is the number of phonon momenta and mode indices (labels νq). In contrast, the computational cost for evaluating all ph-ph scattering integrals scales as \({{\mathcal{N}}}_{{\rm{ph}}}^{2}\). In the low-excitation regime, the carriers can be sampled within a few-eV energy window near the band edge or Fermi energy, thus in practice \({{\mathcal{N}}}_{{\rm{ph}}}\gg {{\mathcal{N}}}_{{\rm{c}}}\). This shows that computing the ph-ph collision integral \({{\mathcal{I}}}^{{\rm{ph}}-{\rm{ph}}}\) and the phonon dynamics is significantly more expensive—in practical calculations by 1–2 orders of magnitude—than computing the dynamics governed by e-ph interactions.

Adaptive and multirate time integration

To improve the computational efficiency, we explore adaptive time-stepping methods, which can adjust the time step size according to the underlying dynamics. Motivated by the different timescales for carrier (fs) and phonon (ps) dynamics, we also apply multirate methods that use different time steps for e-ph and ph-ph scattering integrals.

Adaptive-step explicit Runge-Kutta (ERK) methods target ordinary differential equations (ODEs),

$${y}^{{\prime} }=f(t,y),\quad y({t}_{0})={y}_{0}.$$
(2)

For the coupled rt-BTEs, the solution vector y contains all carrier and phonon populations in the respective momentum grids:

$$y(t)=\left[\begin{array}{c}{f}_{n{\bf{k}}}(t)\\ {N}_{\nu {\bf{q}}}(t)\end{array}\right],$$
(3)

and the right-hand side (RHS) function in Eq. (2) is

$$f(t,y)=\left[\begin{array}{c}{{\mathcal{I}}}^{e-{\rm{ph}}}(y)\\ {{\mathcal{I}}}^{{\rm{ph}}-e}(y)+{{\mathcal{I}}}^{{\rm{ph}}-{\rm{ph}}}(y)\end{array}\right].$$
(4)

With an adaptive integration method, the time step is adjusted to satisfy user-defined error tolerances, the choice of which is critical to adaptive integrator performance. To solve the rt-BTE with adaptive time-stepping, we specify the relative and absolute tolerances for carriers and phonons in SUNDIALS (see “Methods”).

MRI methods target ODEs with the RHS function split into fast and slow parts,

$${y}^{{\prime} }={f}^{f}(t,y)+{f}^{s}(t,y),\quad y({t}_{0})={y}_{0},$$
(5)

where e-ph interactions are considered the fast part and ph-ph interactions the slow component:

$${f}^{f}(t,y)=\left[\begin{array}{c}{{\mathcal{I}}}^{e-{\rm{ph}}}(y)\\ {{\mathcal{I}}}^{{\rm{ph}}-e}(y)\end{array}\right],\quad {f}^{s}(t,y)=\left[\begin{array}{c}0\\ {{\mathcal{I}}}^{{\rm{ph}}-{\rm{ph}}}(y)\end{array}\right].$$
(6)

MRI methods advance the slow dynamics at a fixed time step, hs, while the fast dynamics are evolved by solving an auxiliary ODE with an adaptive time step using a sufficiently accurate method, such as an adaptive ERK method. Both time integration approaches are available in the ARKODE package in SUNDIALS and are accessed through our interface in PERTURBO.

Electron and phonon dynamics in graphene

Leveraging adaptive and multirate time integration methods, we study the coupled dynamics of carriers and phonons in graphene, where evolving the system for long times, up to tens of ps to access lattice dynamics, is computationally expensive. The initial state of excited carriers is set to a hot Fermi-Dirac distribution at 4000 K35 while the phonons are set to a Bose-Einstein distribution at 300 K.

Using a third-order MRI method, we are able to extend the simulations to 80 ps with a reasonable computational cost. Figure 1 shows the carrier and phonon populations along a high-symmetry path at various time snapshots, capturing the main trends of the coupled dynamics. In the first ps, electrons and holes undergo rapid cooling and emit \({{\rm{A}}}_{1}^{{\prime} }\) and E2g optical phonons with momenta near Γ and K, due to the strong e-ph couplings with these phonons22. From 1 to 10 ps, the excess optical phonons decay into acoustic phonons near Γ. The thermalization process for flexural phonons is slow and extends to more than 80 ps.

Fig. 1: Coupled electron and phonon dynamics in graphene.
figure 1

Top row: electron populations, fnk(t) (blue), and hole populations, 1 − fnk(t) (green), mapped onto the band structure at different time snapshots labeled in the panels, with point sizes proportional to the populations. To the left of each panel, we show the carrier populations as a function of energy by averaging over the BZ. Bottom row: change in phonon populations, ΔNνq(t), relative to the initial distribution, shown on the phonon dispersion. The BZ-averaged change in populations is shown to the left of each phonon dispersion plot. At time t = 0, the initial carriers are set to a hot Fermi-Dirac distribution at 4000 K, while the phonons follow a Bose-Einstein distribution at 300 K.

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Figure 2 provides a detailed benchmark of the efficiency improvements obtained with the adaptive and multirate schemes for these graphene simulations. We compare results obtained using RK4 with a fixed time step, an adaptive 4th-order ERK method, and a third-order MRI method. The accuracy and computational cost depend strongly on the choice of the following key parameters: the fixed time step size h in RK4, the fixed slow time step size hs in the MRI method, and the tolerances in the standalone ERK method (and also in the ERK method used for the MRI fast timescale). In Fig. 2, we vary these parameters for each method and plot the carrier and phonon population errors at t = 0.5 ps against the corresponding computational cost (runtime). The error is taken relative to a reference MRI solution with a small step size hs; see “Methods” for details.

Fig. 2: Comparison of error and cost for solving the rt-BTE with different methods.
figure 2

The carrier and phonon population errors at a fixed time (t = 0.5 ps) as a function of CPU core-hour cost, shown for RK4 (orange), ERK (gray), and MRI (green) time integration methods. Errors are computed as \(\parallel {f}_{n{\bf{k}}}(t)-{\tilde{f}}_{n{\bf{k}}}(t)\parallel /\parallel {\tilde{f}}_{n{\bf{k}}}(t)\parallel\) for carriers and \(\parallel {N}_{\nu {\bf{q}}}(t)-{\tilde{N}}_{\nu {\bf{q}}}(t)\parallel /\parallel {\tilde{N}}_{\nu {\bf{q}}}(t)\parallel\) for phonons, with the reference solutions \({\tilde{f}}_{n{\bf{k}}}(t)\) and \({\tilde{N}}_{\nu {\bf{q}}}(t)\) obtained using the MRI with hs = 0.01 fs and tight tolerances (see “Methods”). Each pair of cross and circle with the same color and CPU cost represents carrier and phonon population errors for the same simulation method and parameter settings. These parameter changes are indicated schematically with arrows for each method. For RK4, points from left to right represent results using a progressively smaller time step, h, from 5 fs to 0.01 fs. For MRI, results from left to right correspond to the slow time step, hs, ranging from 20 fs to 0.05 fs. For ERK, the relative tolerance ranges from 10−4 to 10−11 from left to right with a fixed absolute tolerance. See “Methods” for detailed tolerance settings.

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With RK4, the solution error can be reduced systematically by using smaller time steps at the expense of taking longer computation time. When the time step is too long (here h > 5 fs), the populations diverge at long enough simulation times. Error-versus-cost results with the ERK method are obtained by tightening the relative tolerance (rtol) from 10−4 to 10−11 while keeping the absolute tolerance fixed. Results using the MRI method are obtained by varying hs with fixed tolerance values for the adaptive ERK method used for the fast timescale.

Results for the ERK and MRI methods are in the lower-left corner of the error-versus-cost map in Fig. 2, which indicates their superior performance compared to RK4 with a fixed time step. These methods are more efficient over a wide range of tolerances (ERK) and slow time step values (MRI), and are particularly robust in reducing phonon population errors.

Detailed comparisons are helpful to quantify the performance improvement. The carrier population error using the MRI method with hs = 20 fs is comparable to RK4 with a much shorter time step (h = 1 fs, a typical step size chosen for e-ph dynamics35), but the MRI method requires only ~10% of the computational effort of RK4 for a 0.5-ps simulation. In addition, the phonon population error with the MRI method for the same settings is comparable to RK4 with an extremely short time step, h = 0.01 fs, but the computational cost for phonon dynamics is 1000x smaller relative to RK4. For the same computational cost as RK4 with h = 1 fs, the MRI method improves the carrier accuracy by 3 orders of magnitude and the phonon accuracy by 6 orders of magnitude. This dramatic improvement of efficiency and accuracy with the MRI method is game-changing—for example, it enables modeling nonequilibrium lattice dynamics up to long times (>100 ps) and in bulk materials. Compared to RK4, the different error convergence behavior of MRI and ERK arises from the usage of adaptive time-stepping, which takes sufficiently small time steps in the early part of the dynamics, when the system is far from equilibrium, to limit error accumulation (see Supplementary Fig. 1d). In contrast, RK4 uses a fixed time step, leading to a larger integration error during this critical time window. Additional benchmarks for the ERK and MRI methods are shown in Supplementary Figs. 1 and 2.

The carrier and phonon population errors of the coupled dynamics simulated using different time integration methods and time-stepping parameters are shown in Fig. 3. In all simulations, we find that the errors approach constant values after 0.5 ps, showing that the error-cost results in Fig. 2 are representative of the entire dynamics of longer simulation times. With both the RK4 and MRI methods, the respective step sizes h and hs determine how frequently the ph-ph collision integral \({{\mathcal{I}}}^{{\rm{ph}}-{\rm{ph}}}\) is evaluated, which is the main computational cost driver. To make a fair comparison, we set h = hs in both methods. As shown in Fig. 3, with this setup the MRI results are significantly more accurate than RK4, with accuracy greater by over 3 orders of magnitude for h = 0.5 fs and 8 orders of magnitude for h = 5 fs.

Fig. 3: Solution error for the coupled dynamics in graphene.
figure 3

Population errors for a carrier and b phonon dynamics as functions of time using different time integration methods and parameters. The reference solution is the same as in Fig. 2. Orange curves show solution errors for RK4 with fixed time steps h = 5 fs (square) and 0.5 fs (circle), or an operator splitting approach with RK4 using separate time steps for ph-ph (5 fs) and e-ph/ph-e collisions (0.5 fs) (triangle). The green curves represent results for MRI with hs = 5 fs (square) and 0.5 fs (circle). The gray curve shows results for ERK with relative tolerance 10−5. See “Methods” for detailed tolerance settings.

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To highlight the superior performance of the MRI method, we also consider a first-order operator-splitting method where RK4 is employed with different fixed time steps for the fast (e-ph) and slow (ph-ph) components defined in Eq. (5). In this case, each simulation step performs RK4 advances of the fast and slow parts with separate fixed time steps. The results of this method with a 0.5 fs time step for e-ph scattering and 5 fs for ph-ph scattering are plotted with orange triangles in Fig. 3. In terms of both speed and accuracy, this operator-splitting method falls between the RK4 results with h = 0.5 fs and h = 5 fs, without offering a clear improvement over RK4 with a single time step. This time-splitting approach is, therefore, significantly less efficient and accurate than the MRI and ERK methods. The same trends hold for other combinations of fixed fast and slow time steps when using the operator splitting method with RK4.

Our results for graphene demonstrate that the ERK and MRI methods can efficiently solve the coupled rt-BTE by taking advantage of the inherently different timescales of electron and phonon dynamics. More broadly, these findings suggest that adaptive time-stepping can improve the efficiency and accuracy of simulations involving coupled degrees of freedom in materials.

Electron and phonon dynamics in silicon

We address the challenge of simulating lattice dynamics in bulk materials using first-principles e-ph and ph-ph scattering processes on dense momentum grids, choosing silicon (Si) as a case study. This calculation is not feasible with fixed-step time integration methods such as RK4, even with the high-performance rt-BTE parallel implementation in PERTURBO26. As such, we employ a third-order MRI method paired with an adaptive ERK method as the fast timescale integrator for this simulation. At time zero, we initialize excited electrons in the conduction band of Si using a hot Fermi-Dirac distribution at 2000 K (with concentration 4.6 × 1020 cm−3), while the phonons are initially in thermal equilibrium at 300 K.

The coupled electron and phonon dynamics in Si are analyzed in Fig. 4. Figure 4a illustrates the relaxation of excited electron populations in the X valleys of the lowest conduction band. The top two panels show the increase in electron populations fnk(t) in the X valleys during the first picosecond. The bottom panel shows the change in populations from 1 ps to 20 ps, indicating a modest redistribution near the boundaries of the valleys and an even smaller change at their centers. The change in phonon populations for different phonon modes is visualized in Fig. 4b. In the first ps, most phonons are excited in the optical and longitudinal acoustic modes via e-ph interactions, mainly near the edge of the BZ or along the Γ-X high-symmetry line. Between 1 ps and 20 ps, these optically excited phonons decay into acoustic phonons, which progressively thermalize to long-wavelength lattice vibrations with q → 0 (Γ point in Fig. 4b).

Fig. 4: Coupled dynamics of electrons and phonons in silicon.
figure 4

a Electron populations, fnk(t) in the lowest conduction band at 0 ps and 1 ps, and the change in the electron populations from 1 ps to 20 ps, Δfnk, shown in a 2D plane of the first BZ at kz = 0. b Change in phonon populations, ΔNνq(t), in the first BZ at times 1 ps and 20 ps, plotted separately for transverse acoustic (TA1 + TA2), longitudinal acoustic (LA), transverse optical (TO1 + TO2), and longitudinal optical (LO) modes. c Fractional change in diffuse scattering intensity, ΔI(q, t), for the (100) plane at 1 ps and 20 ps simulation times.

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Experiments have employed optical pump pulses to excite electrons in silicon and track their ultrafast dynamics18,59,60,61. In the absence of first-principles simulations, the two-temperature model62 is commonly used to describe the energy transfer from excited carriers to the lattice during the relaxation. A three-temperature model (3TM) has also been used18, which separately tracks the effective temperatures of electrons, acoustic phonons, and optical phonons. However, the 3TM fails to capture key aspects of the dynamics, such as the distinct relaxation times of different phonon modes and the long-time dynamics of acoustic phonons. This underscores the advantage of the rt-BTE as a first-principles, mode-resolved framework for simulating coupled carrier-lattice dynamics beyond the limitations of empirical models. Supplementary Fig. 5 shows the time evolution of effective phonon temperatures, highlighting the nonthermal behavior in the early stage of the dynamics and the different effective temperatures and dynamics of individual phonon modes.

Using the time-dependent phonon populations, we can connect our results with widely used experimental probes of time-domain lattice dynamics, particularly ultrafast diffraction techniques. Momentum-resolved thermal diffuse X-ray scattering (TDS) has been used extensively to determine phonon dispersions and study ultrafast phonon dynamics20,63,64. The transient TDS at scattering vector q at time t, I(q, t), is obtained from the time-dependent phonon populations, phonon frequencies, and structure factors Fν (q, t), as described in ref. 65 and “Methods”.

Figure 4c shows the relative intensity change \(\Delta I({\bf{q}},t)=\frac{I({\bf{q}},t)-I({\bf{q}},t=0)}{I({\bf{q}},t=0)}\) at 1 ps and 20 ps on the (100) plane, where Γ000 is at the center of the plot, and Γ220 and Γ121 are labeled in the plot. At 1 ps, the TDS primarily shows an increase in optical phonons due to electron cooling. At 20 ps, the main contribution to the TDS is from acoustic phonons, with the LA modes yielding a signal close to Γ and the TA modes the “double-bar” patterns.

Electron and phonon dynamics in GaAs

To demonstrate the general applicability of our time-integration techniques beyond nonpolar materials, we simulate gallium arsenide (GaAs), a prototypical polar semiconductor where long-range e-ph interactions dominate the carrier cooling process66,67,68. In our simulation, the electrons are excited by a pump pulse centered at 1.88 eV, corresponding to an excitation of 0.45 eV above the band gap. The pulse amplitude is chosen to generate a carrier density of 1.1 × 1019 cm−3 at the end of the 50 fs pulse duration. We simulate the electron populations in the lowest conduction band and the excess phonon populations following the pulse and show them in Fig. 5. (Holes are omitted from the simulation due to their minor contribution to phonon dynamics.) The electrons exhibit slow relaxation in the time window from 0 to 20 ps, driven primarily by the accumulation of longitudinal optical phonons (LO) generated through the Fröhlich interaction69. The slow decay of these LO phonons creates a bottleneck that slows the electron cooling process70. This result is shown in Supplementary Fig. 6, where we analyze the contribution of nonequilibrium phonons to carrier cooling by simulating the same dynamics but including only a subset of interactions in the rt-BTE, and then comparing the corresponding time-dependent electronic temperatures.

Fig. 5: Coupled electron and phonon dynamics in GaAs.
figure 5

Top row: electron populations, fnk(t), mapped onto the band structure at different time snapshots labeled in the panels, with point sizes proportional to the populations. Bottom row: excess phonon populations, ΔNνq(t), shown on the phonon dispersion. To the left of each panel: populations (electrons) or change of populations (phonons) as a function of energy averaged over the BZ. At time t = 0, the electrons are excited via a pump pulse 0.45 eV above the band gap, leading to a carrier concentration of 1.1 × 1019 cm−3 at the end of the 50 fs pulse. The phonons are initially set at a Bose-Einstein distribution at 300 K.

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These results show that adaptive and multirate time-stepping of the rt-BTEs enables first-principles simulations of phonon dynamics up to long timescales in bulk crystals, as well as modeling TDS experiments that are widely used to probe ultrafast lattice dynamics.

Discussion

Time step adaptivity is a key factor in improving the accuracy and efficiency of rt-BTE simulations. Therefore, it is instructive to analyze the time step sizes selected by the algorithm over the course of the simulation as the dynamics evolve. At early stages of the carrier dynamics in the graphene simulation, the adaptive step size in the ERK method is close to zero, and then it increases to about 5 fs after 400-fs simulation time (see Supplementary Fig. 1d). For both carrier and phonon populations, ERK and MRI methods exhibit distinctly different error convergence characteristics compared to RK4, when parameters are set to match the computational cost. This difference underscores the need to resolve early-time dynamics, when the system is far from equilibrium, to minimize error propagation to longer times. With reasonable tolerances and slow time-step size choices, the average adaptive step with the ERK method and the fast time step in the MRI method both reach a steady-state value of 5 fs, suggesting that adaptive methods can find the characteristic timescales of physical interactions in the system.

In addition to improving efficiency, using adaptive methods eliminates the need to converge the solution with respect to the chosen time step. In adaptive ERK methods, the only external parameter is the desired level of accuracy for the solution, while in the MRI method the slow time step is estimated using the ph-ph coupling strength. Both methods will produce greater efficiency than fixed time step methods without any fine-tuning of simulation parameters. (The only point to note is that if the chosen slow time-step size is unnecessarily small, the solution will be more accurate than required, as the adaptive fast time step is constrained to be smaller than the slow time step, leading to a higher computation cost.) As SUNDIALS continues to expand the capabilities of multirate methods, in future work it will become possible to further improve the efficiency by having the algorithm select both fast and slow time steps based on user-defined tolerances.

The range of nonequilibrium dynamics that can be addressed with multirate methods is not limited to the coupled electron and phonon system studied here. For example, the nonequilibrium dynamics of excitons and other elementary excitations can also be studied from first principles, including their couplings with the lattice71,72,73. Several frameworks beyond the rt-BTE, including time-dependent DFT with (non)adiabatic lattice dynamics, master-equation methods such as the Liouville or Lindblad equations74, and nonequilibrium Green’s functions75 could all greatly benefit from the multirate and adaptive time-stepping shown in this work. This advance provides a new tool for addressing a potentially wide range of problems in nonequilibrium materials physics, including modeling and interpreting time-domain spectroscopy and diffraction experiments.

In summary, we applied adaptive and multirate time-stepping methods to accelerate simulations of coupled electron and phonon dynamics using the rt-BTE. We achieve a 10- to 100-fold speedup relative to fixed time step methods with the same accuracy, or orders of magnitude greater accuracy when setting simulation parameters to enforce equal computational cost. Beyond computational efficiency, our approach can automatically adapt the time integration to the intrinsic timescales of physical interactions in the system. This advance sets the stage for studying nonequilibrium dynamics at longer timescales and for a wider range of coupled degrees of freedom in materials, including electron, spin, phonon, and other excitations, opening the door to exploring new physical regimes. Future directions include adding explicit treatments of light pulses, coherent electron and phonon dynamics, and higher-order phonon interactions to explore novel quantum states and dynamical control of materials, such as phonon-driven Floquet engineering and time-domain tuning of physical properties, order parameters, spin texture, and crystal structure.

Methods

Electron-phonon and phonon-phonon scattering from first principles

We carry out first-principles calculations of e-ph and ph-ph scattering in graphene, Si, and GaAs. We obtain the electronic ground state and band structure from density functional theory (DFT)76 with the QUANTUM ESPRESSO77 package, using the local density approximation and norm-conserving pseudopotentials.

The e-ph matrix elements are defined as26:

$${g}_{mn\nu }\,({\bf{k}},{\bf{q}})=\sqrt{\frac{\hslash }{2{\omega }_{\nu {\bf{q}}}}}\sum\limits_{\kappa \alpha }\frac{{e}_{\nu {\bf{q}}}^{\kappa \alpha }}{\sqrt{{\mu }_{\kappa }}}\langle {\psi }_{m{\bf{k}}+{\bf{q}}}| {\partial }_{{\bf{q}},\kappa \alpha }V| {\psi }_{n{\bf{k}}}\rangle$$
(7)

and represent the probability amplitude to scatter from an initial electronic state \(\left\vert {\psi }_{n{\bf{k}}}\right\rangle\), with band index n, crystal momentum k, and energy εnk, to a final state \(\left\vert {\psi }_{m{\bf{k}}+{\bf{q}}}\right\rangle\) by emitting or absorbing a phonon with mode index ν, wave-vector q, frequency ωνq, and displacement eigenvector eνq due to the e-ph perturbation potential ∂q,καV, where μκ is the mass of atom κ in the unit cell, and α labels Cartesian components. The matrix elements are first computed on a coarse momentum grid using density functional perturbation theory (DFPT)24. In the PERTURBO code, the e-ph matrix elements are then transformed to a maximally-localized Wannier basis (generated using WANNIER9078) and interpolated to finer electron and phonon momentum grids.

For the rt-BTE dynamics, the e-ph and ph-e scattering integrals at time t read35,79

$$\begin{array}{lll}{{\mathcal{I}}}^{e-{\rm{ph}}}[\,{f}_{n{\bf{k}}},{N}_{\nu {\bf{q}}}]\,=\,-\displaystyle\frac{2\pi }{\hslash }\frac{1}{{{\mathcal{N}}}_{{\bf{q}}}}\sum\limits_{m\nu {\bf{q}}}| {g}_{mn\nu }({\bf{k}},{\bf{q}}){| }^{2}\\\qquad\qquad\qquad\qquad\;\;\times \left[{F}_{{\rm{em}}}\times \delta ({\varepsilon }_{n{\bf{k}}}-\hslash {\omega }_{\nu {\bf{q}}}-{\varepsilon }_{m{\bf{k}}+{\bf{q}}})\right.\\\qquad\qquad\qquad\qquad\;\; \left.+\,{F}_{{\rm{abs}}}\times \delta ({\varepsilon }_{n{\bf{k}}}+\hslash {\omega }_{\nu {\bf{q}}}-{\varepsilon }_{m{\bf{k}}+{\bf{q}}})\right],\\ {{\mathcal{I}}}^{{\rm{ph}}-e}[{f}_{n{\bf{k}}},{N}_{\nu {\bf{q}}}]\,=\;\;-\displaystyle\frac{4\pi }{\hslash }\frac{1}{{{\mathcal{N}}}_{{\bf{k}}}}\sum\limits_{mn{\bf{k}}}| {g}_{mn\nu }({\bf{k}},{\bf{q}}){| }^{2}\\\qquad\qquad\qquad\qquad\;\;\, \times\, {F}_{{\rm{abs}}}\times \delta ({\varepsilon }_{n{\bf{k}}}+\hslash {\omega }_{\nu {\bf{q}}}-{\varepsilon }_{m{\bf{k}}+{\bf{q}}}),\end{array}$$
(8)

where \({{\mathcal{N}}}_{{\bf{k}}}\) and \({{\mathcal{N}}}_{{\bf{q}}}\) are the total number of momentum grid points for electrons and phonons, respectively. In practice, a subset of the \({{\mathcal{N}}}_{{\bf{k}}}\) electron momenta within a given energy range of the band edge or Fermi energy is included in the dynamics (and in the e-ph scattering integral summations) to reduce computational cost. The emission and absorption factors used above are defined as35

$$\begin{array}{rcl}{F}_{{\rm{em}}}&=&{f}_{n{\bf{k}}}(1-{f}_{m{\bf{k}}+{\bf{q}}})(1+{N}_{\nu {\bf{q}}})\\ &&-(1-{f}_{n{\bf{k}}}){f}_{m{\bf{k}}+{\bf{q}}}{N}_{\nu {\bf{q}}},\end{array}$$
(9)

and

$$\begin{array}{lll}{F}_{{\rm{abs}}}\,=\,{f}_{n{\bf{k}}}(1-{f}_{m{\bf{k}}+{\bf{q}}}){N}_{\nu {\bf{q}}}\\\qquad\quad\;-(1-{f}_{n{\bf{k}}}){f}_{m{\bf{k}}+{\bf{q}}}(1+{N}_{\nu {\bf{q}}}).\end{array}$$
(10)

The delta functions enforcing energy conservation in Eq. (8) are approximated by the Gaussian function \(\delta (E)\approx \frac{\pi }{\sigma }\exp [-{(E/\sigma )}^{2}]\), where σ is a broadening parameter typically set to a few meV.

The second- and third-order inter-atomic force constants (IFCs), \({\Phi }_{\alpha \beta }(l\kappa ;{l}^{{\prime} }{\kappa }^{{\prime} })\) and \({\Psi }_{\alpha \beta \gamma }(l\kappa ;{l}^{{\prime} }{\kappa }^{{\prime} };{l}^{{\prime}\prime}{\kappa }^{{\prime}\prime})\), are computed by expanding the potential energy of the system with respect to small atomic displacements u:

$$\begin{array}{lll}U\,=\,{U}_{0}+\frac{1}{2!}\sum\limits_{l{l}^{{\prime} }}\sum\limits_{\kappa {\kappa }^{{\prime} }}\sum\limits_{\alpha \beta }{\Phi }_{\alpha \beta }(l\kappa ;{l}^{{\prime} }{\kappa }^{{\prime} }){u}_{l\kappa }^{\alpha }{u}_{{l}^{{\prime} }{\kappa }^{{\prime} }}^{\beta }\\\quad +\frac{1}{3!}\sum\limits_{l{l}^{{\prime} }{l}^{{\prime}\prime}}\sum\limits_{\kappa {\kappa }^{{\prime} }{\kappa }^{{\prime}\prime}}\sum\limits_{\alpha \beta \gamma }{\Psi }_{\alpha \beta \gamma }(l\kappa ;{l}^{{\prime} }{\kappa }^{{\prime} };{l}^{{\prime}\prime}{\kappa }^{{\prime}\prime}){u}_{l\kappa }^{\alpha }{u}_{{l}^{{\prime} }{\kappa }^{{\prime} }}^{\beta }{u}_{{l}^{{\prime}\prime}{\kappa }^{{\prime}\prime}}^{\gamma }.\end{array}$$
(11)

Here, U0 is the total potential energy for atoms in equilibrium, and the vector ulκ is the displacement from equilibrium of atom κ in the l-th unit cell, while α, β, and γ label Cartesian coordinates. We compute the IFCs using the temperature-dependent effective potential (TDEP) method80, which computes the atomic forces on structures with random thermal displacements distributed according to a canonical ensemble. These calculations are carried out at 300 K employing supercells with dimensions 12 × 12 × 1 for graphene, 6 × 6 × 6 for Si, and 8 × 8 × 8 for GaAs. The second-order IFCs are used to compute the phonon frequencies and eigenvectors81, and the third-order IFCs to compute the ph-ph matrix elements.

The ph-ph matrix elements describe the interaction of three phonons with momenta \(\bf{q}\), \({{\bf{q}}}^{{\prime} }\), and \(\bf{q}^{\prime\prime}\) and mode indices ν, \({\nu }^{{\prime} }\), and ν″. They are computed as Fourier transforms of the third-order IFCs using

$$\begin{array}{lll}{\Psi }_{\nu {\nu }^{{\prime} }{\nu }^{{\prime}\prime}}\,({\bf{q}},{{\bf{q}}}^{{\prime} },{{\bf{q}}}^{{\prime}\prime})\,=\,\frac{1}{6}\sqrt{\frac{{\hslash }^{3}}{8{\omega }_{\nu {\bf{q}}}{\omega }_{{\nu }^{{\prime} }{{\bf{q}}}^{{\prime} }}{\omega }_{{\nu }^{{\prime}\prime}{{\bf{q}}}^{{\prime}\prime}}}}\Delta ({\bf{q}}+{{\bf{q}}}^{{\prime} }+{{\bf{q}}}^{{\prime}\prime})\\\qquad\qquad\qquad\qquad \times \sum\limits_{{l}^{{\prime} }{l}^{{\prime}\prime}}\sum\limits_{\kappa {\kappa }^{{\prime} }{\kappa }^{{\prime}\prime}}\sum\limits_{\alpha \beta \gamma }\exp \left[i({{\bf{q}}}^{{\prime} }\cdot {{\bf{R}}}_{{l}^{{\prime} }}+{{\bf{q}}}^{{\prime}\prime}\cdot {{\bf{R}}}_{{l}^{{\prime}\prime}})\right]\\\qquad\qquad\qquad \frac{{e}_{\nu {\bf{q}}}^{\kappa \alpha }{e}_{{\nu }^{{\prime} }{{\bf{q}}}^{{\prime} }}^{{\kappa }^{{\prime} }\beta }{e}_{{\nu }^{{\prime}\prime}{{\bf{q}}}^{{\prime}\prime}}^{{\kappa }^{{\prime}\prime}\gamma }}{\sqrt{{\mu }_{\kappa }{\mu }_{{\kappa }^{{\prime} }}{\mu }_{{\kappa }^{{\prime}\prime}}}}{\Psi }_{\alpha \beta \gamma }(0\kappa ;{l}^{{\prime} }{\kappa }^{{\prime} };{l}^{{\prime}\prime}{\kappa }^{{\prime}\prime})\end{array}$$
(12)

where Rl is the lattice vector of the unit cell l, and the delta function \(\Delta ({\bf{q}}+{{\bf{q}}}^{{\prime} }+{{\bf{q}}}^{{\prime}\prime})\) conserves crystal momentum, ensuring \({\bf{q}}+{{\bf{q}}}^{{\prime} }+{{\bf{q}}}^{{\prime}\prime}\) is equal to zero or a reciprocal lattice vector.

The ph-ph scattering integral can be written as

$$\begin{array}{lll}{{\mathcal{I}}}^{{\rm{ph}}-{\rm{ph}}}[{N}_{\nu {\bf{q}}}]\,=\,-\frac{36\pi }{\hslash }\frac{1}{{{\mathcal{N}}}_{{\bf{q}}}}\sum\limits_{{\nu }^{{\prime} }{\nu }^{{\prime}\prime}}\sum\limits_{{{\bf{q}}}^{{\prime} }{{\bf{q}}}^{{\prime}\prime}}| {\Psi }_{\nu {\nu }^{{\prime} }{\nu }^{{\prime}\prime}}({\bf{q}},{{\bf{q}}}^{{\prime} },{{\bf{q}}}^{{\prime}\prime}){| }^{2}\\\qquad\qquad\qquad \times \left[{G}_{{\rm{em}}}\times \delta (-\hslash {\omega }_{\nu {\bf{q}}}+\hslash {\omega }_{{\nu }^{{\prime} }{{\bf{q}}}^{{\prime} }}+\hslash {\omega }_{{\nu }^{{\prime}\prime}{{\bf{q}}}^{{\prime}\prime}})\right.\\\qquad\qquad\qquad \left.+2\,{G}_{{\rm{abs}}}\times \delta (\hslash {\omega }_{\nu {\bf{q}}}-\hslash {\omega }_{{\nu }^{{\prime} }{{\bf{q}}}^{{\prime} }}+\hslash {\omega }_{{\nu }^{{\prime}\prime}{{\bf{q}}}^{{\prime}\prime}})\right],\end{array}$$
(13)

where

$$\begin{array}{lll}{G}_{{\rm{em}}}\,=\,{N}_{\nu {\bf{q}}}({N}_{{\nu }^{{\prime} }{{\bf{q}}}^{{\prime} }}+1)({N}_{{\nu }^{{\prime}\prime}{{\bf{q}}}^{{\prime}\prime}}+1)\\\qquad -({N}_{\nu {\bf{q}}}+1){N}_{{\nu }^{{\prime} }{{\bf{q}}}^{{\prime} }}{N}_{{\nu }^{{\prime}\prime}{{\bf{q}}}^{{\prime}\prime}},\end{array}$$
(14)

and

$$\begin{array}{lll}{G}_{{\rm{abs}}}\,=\,{N}_{\nu {\bf{q}}}({N}_{{\nu }^{{\prime} }{{\bf{q}}}^{{\prime} }}+1){N}_{{\nu }^{{\prime}\prime}{{\bf{q}}}^{{\prime}\prime}}\\\qquad -({N}_{\nu {\bf{q}}}+1){N}_{{\nu }^{{\prime} }{{\bf{q}}}^{{\prime} }}({N}_{{\nu }^{{\prime}\prime}{{\bf{q}}}^{{\prime}\prime}}+1).\end{array}$$
(15)

The Wannier interpolation, e-ph and ph-ph matrix computation, and ultrafast dynamics simulation described above are implemented in the PERTURBO package26. Both MPI and OpenMP parallelization are employed to sum over scattering channels when computing e-ph and ph-ph matrices, and to sum over k- and q-points when computing the scattering integrals in Eqs. (8) and (13).

For graphene, the phonon scattering rates are converged for ph-ph matrix elements computed on a q-grid with size 200 × 200 × 1, while full convergence of e-ph scattering requires a 400 × 400 × 1 q-grid. The nonequilibrium dynamics calculation in Fig. 1 employs this dense 400 × 400 × 1 grid, with ph-ph matrix elements interpolated on the fly. The efficiency analyses in Figs. 2, 3 and Supplementary Figs. 1–3 are obtained using 200 × 200 × 1 k- and q-grids. A small phonon frequency cutoff of 1 meV is employed. We use a Gaussian broadening of 20 meV for the collision integrals \({{\mathcal{I}}}^{e-{\rm{ph}}}\) and \({{\mathcal{I}}}^{{\rm{ph}}-e}\), and 2 meV for \({{\mathcal{I}}}^{{\rm{ph}}-{\rm{ph}}}\).

For silicon, the ph-ph matrix elements are computed on a 40 × 40 × 40 q-grid with a phonon frequency cutoff of 0.2 meV. The e-ph matrix elements and the dynamics are computed on a finer 80 × 80 × 80 q-grid. During the dynamics calculations, the ph-ph matrix elements are interpolated onto this finer grid. The Gaussian broadening is 5 meV for \({{\mathcal{I}}}^{e-{\rm{ph}}}\) and \({{\mathcal{I}}}^{{\rm{ph}}-e}\) and 1.1 meV for \({{\mathcal{I}}}^{{\rm{ph}}-{\rm{ph}}}\). Both the scattering rates and the dynamics have been tested for convergence with respect to broadening and grid size parameters. To find optimal grids for the nonequilibrium lattice dynamics, we compute mode-resolved three-phonon scattering rates at 300 K with the same ph-ph matrix elements employed in the time-domain simulations (see Supplementary Fig. 4). The coupled dynamics employ the MRI method in ARKODE with hs = 50 fs. The fast timescale is integrated with the adaptive ERK method in ARKODE using a relative tolerance rtol = 10−5 and absolute tolerances atolc = 10−9 for carriers and atolph = 10−11 for phonons. Convergence at these tolerance values is confirmed by comparing with tests using tighter tolerances and smaller hs values.

For GaAs, ph-ph matrix elements are computed on a 40 × 40 × 40 q-grid with a phonon frequency cutoff of 0.2 meV. The e-ph matrix elements and the dynamics are computed on a finer 120 × 120 × 120 q-grid. Gaussian broadening values of 8 meV and 0.56 meV are used for computing the \({{\mathcal{I}}}^{e-{\rm{ph}}}\) and \({{\mathcal{I}}}^{{\rm{ph}}-{\rm{ph}}}\) collision integrals, respectively. The MRI method is used with hs = 50 fs, rtol = 10−7 and atolc = 10−10 for carriers and atolph = 10−12 for phonons.

Auger and phonon-induced recombination processes, which typically occur on longer timescales than those considered in this work82, are not expected to significantly impact the carrier dynamics and therefore omitted from the simulations. However, incorporating these processes into the rt-BTE remains a promising direction for future work.

The transient TDS at scattering vector q and time t, I(q, t), is given by

$$I({\bf{q}},t)\propto \sum\limits_{j}\frac{1}{{\omega }_{\nu {{\bf{q}}}^{{\prime} }}}\left[{N}_{\nu {{\bf{q}}}^{{\prime} }}(t)+\frac{1}{2}\right]| {F}_{\nu }\,({\bf{q}},t){| }^{2},$$
(16)

where \({{\bf{q}}}^{{\prime} }={\bf{q}}-{{\bf{K}}}_{{\bf{q}}}\) is the momentum q folded to the first BZ, and Kq is the nearest reciprocal lattice vector to q. The one-phonon structure factor Fν (q, t) is defined as

$${F}_{\nu }\,({\bf{q}},t)=\sum\limits_{\kappa }\frac{{f}_{\kappa }({\bf{q}})}{\sqrt{{\mu }_{\kappa }}}\exp [-{M}_{\kappa }({\bf{q}})]({\bf{q}}\cdot {{\bf{e}}}_{\nu {{\bf{q}}}^{{\prime} }}^{\kappa })\,\exp (-i{{\bf{K}}}_{{\bf{q}}}\cdot {{\bf{r}}}_{\kappa })$$
(17)

where fκ(q) is the atomic scattering factor83, rκ is the equilibrium position of atom κ in the unit cell, and Mκ(q) is the Debye-Waller factor.

Adaptive integrators and implementation

ERK method

We describe how a single step is carried out with an adaptive ERK method. For Eq. (2), to advance from tk−1 to tk = tk−1 + hk with a method of order p, we use

$${z}_{i}={y}_{k-1}+{h}_{k}\mathop{\sum }\limits_{j=1}^{i-1}{A}_{i,j}\,f({t}_{k,j},{z}_{j}),\,i=1,\ldots ,S,$$
(18a)
$${y}_{k}={y}_{k-1}+{h}_{k}\mathop{\sum }\limits_{i=1}^{S}{b}_{i}\,f({t}_{k,i},{z}_{i}),$$
(18b)
$${\tilde{y}}_{k}={y}_{k-1}+{h}_{k}\mathop{\sum }\limits_{i=1}^{S}{\tilde{b}}_{i}\,f({t}_{k,i},{z}_{i}),$$
(18c)

where there are S stages, zi, at times tk,j = tk−1 + cjhk, and the coefficients that define the method for obtaining the new solution, yk, are given in the corresponding Butcher tableau with \(A\in {{\mathbb{R}}}^{S\times S}\), \(b\in {{\mathbb{R}}}^{S}\), and \(c\in {{\mathbb{R}}}^{S}\). The coefficients \(\tilde{b}\in {{\mathbb{R}}}^{S}\) are used to obtain an embedded solution, \({\tilde{y}}_{k}\), (typically of order p − 1) and the difference between the solution and embedding, \({y}_{k}-{\tilde{y}}_{k}\), provides an estimate of the local truncation error (LTE) for adapting the step size55. The ERK results above utilize the default adaptive ERK method in ARKODE, which is a five-stage, fourth-order method from Zonneveld84.

An attempted time step is accepted if it satisfies ∥LTE∥WRMS ≤ 1 in the weighted root-mean-square (WRMS) norm,

$$\parallel\!\! v{\parallel }_{{\rm{WRMS}}}={\left(\frac{1}{{{\mathcal{N}}}_{y}}\mathop{\sum }\limits_{i = 1}^{{{\mathcal{N}}}_{y}}{\left({v}_{i}{w}_{i}\right)}^{2}\right)}^{1/2},$$
(19)

where \({{\mathcal{N}}}_{y}={{\mathcal{N}}}_{{\rm{c}}}+{{\mathcal{N}}}_{{\rm{ph}}}\) is the length of the vector v, and the weights wi are defined by the most recent solution yn−1, the relative tolerance (rtol) and, the vector of absolute tolerances (atol) as

$${w}_{i}={\left({\rm{rtol}}| {y}_{n-1,i}| +{{\rm{atol}}}_{{\rm{i}}}\right)}^{-1}.$$
(20)

The relative tolerance controls the number of digits of accuracy in the solution, while the absolute tolerance sets the level below which differences in small solution components are ignored. If a step attempt is rejected, a new step size is computed based on the LTE, and the step is repeated. After successfully completing a step, the LTE is similarly used to determine the step size for the next step attempt.

In this work, we employ the default error controller in ARKODE, the PID controller85,86,87, for step size selection,

$${h}^{{\prime} }={h}_{k}{\varepsilon }_{k}^{-{a}_{1}/{p}_{* }}\,{\varepsilon }_{k-1}^{-{a}_{2}/{p}_{* }}\,{\varepsilon }_{k-2}^{-{a}_{3}/{p}_{* }},$$
(21)

where \({h}^{{\prime} }\) is the new step size and εn, εn−1, and εn−2 are the WRMS norms of the error estimates from the current and prior two time steps, respectively. The values of εn−1 and εn−2 are initialized to 1 and updated as steps are accepted. We use the default PID parameter values (a1 = 0.58, a2 = 0.21, and a3 = 0.1) except for p* where we use the method order, p, rather than the embedding order. Additionally, we disable the step size adjustment thresholds, allowing any step size change between each step rather than retaining the current step size if the step growth factor is small. These adjustments to the default values are shown to be more efficient and lead to fewer failed steps, larger step sizes, more frequent changes in step size, and smoother step size profiles compared to the current default settings.

MRI method

Explicit MRI methods for Eq. (5) with \(\hat{S}\) stages advance the solution from tk−1 to tk = tk−1 + hs with the following algorithm:

  1. (1)

    Set z1 = yk−1 and \({\hat{t}}_{k,1}={t}_{k-1}\)

  2. (2)

    For \(i=2,\ldots ,\hat{S}\)

    • (a) Let \({\hat{t}}_{k,i}={t}_{k-1}+{\hat{c}}_{i}{h}_{s}\) and \(v({\hat{t}}_{k,i-1})={z}_{i-1}\)

    • (b) Solve \({v}^{{\prime} }(t)={f}^{f}(t,v)+{r}_{i}(t)\) on \(t\in [{\hat{t}}_{k,i-1},{\hat{t}}_{k,i}]\) where:

      $$\begin{array}{lll}{r}_{i}(t)\,=\,\frac{1}{\Delta {\hat{c}}_{i}}\mathop{\sum }\limits_{j=1}^{i-1}{\gamma }_{i,j}\left(\tau \right){f}^{s}({\hat{t}}_{k,j},{z}_{j}),\\\qquad \tau \,=\,(t-{\hat{t}}_{k,i-1})/(\Delta {\hat{c}}_{i}{h}_{s}),\\\qquad \Delta {\hat{c}}_{i}\,=\,{\hat{c}}_{i}-{\hat{c}}_{i-1}\end{array}$$

    • (c) Set \({z}_{i}=v({\hat{t}}_{k,i})\)

  3. (3)

    Set \({y}_{k}={z}_{\hat{S}}\)

The abscissae are sorted, \(0={\hat{c}}_{1}\le \cdots \le {\hat{c}}_{\hat{S}}=1\), and they define the intervals over which the fast auxiliary ODE in step 2(b) is solved to compute the stage values zi. If \(\Delta {\hat{c}}_{i}=0\), solving the fast auxiliary ODE in step 2(b) reduces to a standard ERK stage update as in Eq. (18). The coefficient function,

$${\gamma }_{i,j}(\tau )=\mathop{\sum }\limits_{k=1}^{K}{\Omega }_{i,j,k}\,{\tau }^{k-1},$$
(22)

is a polynomial in time with coefficients \({\Omega }_{i,j,k}\in {{\mathbb{R}}}^{\hat{S}\times \hat{S}\times K}\) that define the coupling between slow and fast timescales.

The results above utilize the default MRI algorithm in ARKODE, a third-order multirate infinitesimal step (MIS) method57,88,89 based on the explicit method in ref. 90. MIS methods are a subset of MRI methods where the coupling coefficients are uniquely defined from an ERK method with \(S=\hat{S}-1\) stages and sorted abscissae. For third-order accuracy, the ERK method must also satisfy an additional order condition90. In the MIS case, the forcing function ri(t) reduces to a constant value (K = 1) that depends on the stage index, i. The resulting MIS method has abscissae \(\hat{c}={[{c}_{1}\cdots {c}_{S}1]}^{T}\) and coupling coefficients:

$${\Omega }_{i,j,1}=\left\{\begin{array}{ll}0,\quad &\,{\text{if}}\,\ i=1,\\ {A}_{i,j}-{A}_{i-1,j},\quad &\,{\text{if}}\,\ 2\le i\le S,\\{b}_{j}-{A}_{S-1,j},\quad &\,\text{if}\,\ i=S+1=\hat{S}\end{array}\right. .$$
(23)

In theory, the auxiliary ODE in step 2(b) is solved exactly with an infinitesimally small step size; however, in practice, it is solved using any sufficiently accurate method. In this work, we use the same ERK method applied in the single-rate adaptive integration tests as the fast timescale integrator.

Choices of tolerances and reference solutions

Population errors in Figs. 2 and 3 are computed as \(\parallel\!\! {f}_{n{\bf{k}}}(t)-{\tilde{f}}_{n{\bf{k}}}(t)\parallel\!\! /\!\!\parallel {\tilde{f}}_{n{\bf{k}}}(t)\!\!\parallel\) for carriers and \(\parallel\!\! {N}_{\nu {\bf{q}}}(t)-{\tilde{N}}_{\nu {\bf{q}}}(t)\parallel\!\! /\!\!\parallel {\tilde{N}}_{\nu {\bf{q}}}(t)\!\!\parallel\) for phonons. In both figures, the reference solutions \({\tilde{f}}_{n{\bf{k}}}(t)\) and \({\tilde{N}}_{\nu {\bf{q}}}(t)\) are obtained using the MRI method with a very small slow time step, hs = 0.01 fs, and tight tolerances in the fast timescale integration, rtol = 10−10 and atolc = atolph = 10−15 for both carriers and phonons. Analyzing solution errors with different reference solutions (Supplementary Fig. S3) confirms that the MRI method converges significantly faster than RK4 across time step sizes. In Fig. 2, tolerance values for the MRI simulations (excluding the reference solution) range from 10−4 to 10−11 for the relative tolerance, while the absolute tolerances are fixed as atolc = 10−9 for carriers and atolph = 10−12 for phonons. For the ERK results, the relative tolerances also range from 10−4 to 10−11, and the absolute tolerances are fixed at atolc = atolph = 10−15. In Fig. 3, tolerances for both the ERK results and the fast timescale integration in the MRI method are set to rtol = 10−5, atolc = 10−9, and atolph = 10−12.

Perturbo interface with SUNDIALS

This work required the development of a Fortran interface between PERTURBO and SUNDIALS. While SUNDIALS is primarily written in C, the library provides Fortran interface modules generated by SWIG-Fortran91 to facilitate interoperability with Fortran codes such as PERTURBO. For explicit methods, such as the ERK and MRI schemes considered in this work, interfacing with SUNDIALS requires providing the time integrator with a vector object to operate on state data, together with a single right-hand side function for an ERK method or fast and slow right-hand side functions for an MRI method. To enable SUNDIALS to operate on PERTURBO data, the 2D arrays for carrier and phonon populations are copied into a contiguous array under the native SUNDIALS serial vector. The right-hand side functions provided to SUNDIALS are thin wrappers that unpack the input vector data into 2D Fortran arrays, compute the scattering integrals in parallel with the same subroutines utilized by the native RK4 method, and then pack the results into the output SUNDIALS vector. Overall, the implemented interface is versatile and efficient. For all calculations shown in this work, the CPU time spent on interfacing between PERTURBO and SUNDIALS is negligible compared to the cost of the dynamics.