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Adaptively time stepping the stochastic Landau-Lifshitz-Gilbert equation at nonzero temperature: implementation and validation in MuMax3

J. Leliaert, J. Mulkers, J. De Clercq, A. Coene, M. Dvornik, B. Van Waeyenberge

arXiv:1709.01682v1cond-mat.mes-hall

TL;DR

Thermal micromagnetic simulations matter across magnetic-particle and device research, yet fixed time steps had been considered necessary at nonzero temperatures. The paper uses the constrained drift in sLLG to extend higher-order solvers with adaptive stepping, reporting a twenty-fold speedup without loss of accuracy against fixed-step Heun’s method.

  • Problem

    At nonzero temperatures, micromagnetic simulations had been considered to require fixed time steps, despite their growing relevance to magnetic-particle and technological applications.

    Source: §I · pp. 1–2

  • Method

    The authors exploit that the sLLG drift affects only the fixed magnetization length to extend higher-order solvers with adaptive time stepping.

    Source: §I · p. 2, §V · p. 7

  • Results

    Twenty-fold speedup without loss of accuracy compared with RK12 using a small fixed time step.

    Source: §V · p. 7

  • Takeaways & Limitations

    Adaptive time stepping offers inherent error control and makes simulations that previously took too long feasible for the stated micromagnetic research settings.

    Source: §V · p. 7

Abstract

from arXiv · show

Thermal fluctuations play an increasingly important role in micromagnetic research relevant for various biomedical and other technological applications. Until now, it was deemed necessary to use a time stepping algorithm with a fixed time step in order to perform micromagnetic simulations at nonzero temperatures. However, Berkov and Gorn have shown that the drift term which generally appears when solving stochastic differential equations can only influence the length of the magnetization. This quantity is however fixed in the case of the stochastic Landau-Lifshitz-Gilbert equation. In this paper, we exploit this fact to straightforwardly extend existing high order solvers with an adaptive time stepping algorithm. We implemented the presented methods in the freely available GPU-accelerated micromagnetic software package MuMax3 and used it to extensively validate the presented methods. Next to the advantage of having control over the error tolerance, we report a twenty fold speedup without a loss of accuracy, when using the presented methods as compared to the hereto best practice of using Heun's solver with a small fixed time step.

I. INTRODUCTION

Thermal fluctuations make micromagnetic simulations important across magnetic-particle and device research, but analytical approaches can fail for complex systems. The paper motivates adaptive higher-order solvers for stochastic LLG simulations, where fixed-step methods had remained standard.

I. INTRODUCTION

  • Thermal micromagnetic modeling supports nanoparticle characterization and biomedical applications, as well as studies of domain-wall motion and magnetic-storage behavior.For domain-wall creep, motion reflects thermal fluctuations, disorder, and driving forces; full micromagnetic simulations remain prohibitively expensive for all but the smallest systems.Source: §I · p. 1
  • For complex anisotropy or continuous exchange-coupled systems, the Fokker–Planck approach requires approximations or becomes intractable.Analytical solutions are available only in the simplest cases, such as aligned particle anisotropy axes and applied field.Source: §I · p. 2
  • The sLLG approach adds a stochastic thermal field to the effective field, but must be solved repeatedly to estimate averaged quantities.Source: §I · p. 2
  • The sLLG drift affects only magnetization length, which is fixed, enabling higher-order adaptive solvers despite the usual stochastic-integration complications.The paper contrasts this approach with the common practice of using Heun’s RK12 solver with a very small fixed time step.Source: §I · p. 2

II. METHODS

The methods describe magnetization dynamics with the LLG equation and incorporate thermal fluctuations through a stochastic field. The field statistics are specified so that the modeled dynamics are independent of spatial and temporal discretization.

II. METHODS

  • The LLG equation describes nanometer-scale, picosecond magnetization dynamics through precession and damping.Source: §II · p. 2
  • The chosen thermal-noise formulation adds a stochastic field to the effective field in both the precession and damping terms.An alternative is to add thermal fluctuation as an extra torque term.Source: §II · p. 2
  • Brown’s thermal-field theory applies to micromagnetic simulations by treating each finite-difference cell as a single-domain particle.Source: §II · p. 2
  • The thermal field has zero average and is uncorrelated in time and space, with its strength set so averaged dynamics do not depend on spatial or time discretization.The volume used for the fluctuations is that of a single finite-difference cell.Source: §II · p. 2

A. Implementation in MuMax3

MuMax3 is a GPU-accelerated, finite-difference micromagnetics package whose thermal simulations previously used Heun’s method with a very small fixed time step. The implementation also describes solver properties and the trade-offs of higher-order methods in zero-temperature dynamics.

A. Implementation in MuMax3

  • MuMax3 numerically solves the LLG equation using a finite-difference discretization on GPU-accelerated micromagnetic software.Source: A. Implementation in MuMax3 · pp. 2–3
  • The thermal field uses a random vector drawn from a standard normal distribution and redetermined after every time step.The random vector is denoted η, and Δt denotes the time step.Source: A. Implementation in MuMax3 · p. 3
  • Thermal simulations in MuMax3 previously used Heun’s RK12 method with a fixed time step of about 5 fs.The paper uses this solver as a comparison when analytical solutions are unavailable.Source: A. Implementation in MuMax3 · p. 3
  • For zero-temperature dynamics, RK45 is the default; higher-order solvers can be unattractive because of stability limits, extra torque evaluations, and memory use.RK45 has fifth-order error convergence and an embedded fourth-order method for estimating error.Source: A. Implementation in MuMax3 · p. 3

B. Sixth order Runge-Kutta-Fehlberg solver

The section introduces RKF56 as a sixth-order solver with an embedded fifth-order error estimate, then validates its implementation against OOMMF and its expected convergence order.

B. Sixth order Runge-Kutta-Fehlberg solver

  • RKF56 provides a sixth-order solution and an embedded fifth-order solution, whose difference supplies the error estimate.The method is introduced because thermal-field-driven simulations need smaller time steps, making higher-order solvers' extra torque evaluations worthwhile.Source: Table I · p. 3, B. Sixth order Runge-Kutta-Fehlberg… · p. 3
  • At nonzero temperatures, RK45 loses its FSAL evaluation savings because the thermal field changes between time steps, so RKF56 can compete without that disadvantage.RKF56 does not have the FSAL property.Source: B. Sixth order Runge-Kutta-Fehlberg… · p. 3
  • MuMax3’s sixth-order Runge-Kutta–Fehlberg solver is tested like its other implemented solvers; its solution to µMag’s standard problem 4 agrees with OOMMF’s, which did not use RK56.Source: B. Sixth order Runge-Kutta-Fehlberg… · p. 3
  • Sixth-order error convergence persists to the single-precision limit for a single-spin precession without damping in a 0.1 T field.The error was measured after one precession.Source: B. Sixth order Runge-Kutta-Fehlberg… · p. 3

C. Time stepping with adaptive time steps

Adaptive stepping must account for the time-step dependence of the thermal field and preserve its statistical properties when a step is rejected. The section also presents validation visuals for solver error and standard-problem dynamics.

C. Time stepping with adaptive time steps

  • A rejected step reduces Δt while increasing the thermal-field magnitude, creating a feedback that makes maintaining large steps important.The thermal-field magnitude varies as 1/√Δt; the passage notes that its size decreases more slowly than solver error as the step is reduced.Source: C. Time stepping with adaptive time… · p. 3
  • Figure 2 plots RKF56 error against time step and displays sixth-order convergence; Figure 1 shows standard-problem-4 solutions for MuMax3 and OOMMF.Figure 1 shows two applied fields and magnetization configurations when ⟨mx⟩ crosses zero.Source: Figure 2 · p. 4, Figure 1 · p. 4
  • Keeping previously drawn random numbers and rescaling the thermal field after a rejected step preserves its statistical properties.Redrawing would apply small fields over longer steps and large fields over shorter ones, changing the distribution.Source: C. Time stepping with adaptive time… · p. 4

III. VALIDATION

Validation compares adaptive simulations with analytical expectations or fixed-step RK12 and tests the expected thermal-field and magnetization spectra for a single spin.

III. VALIDATION

  • The validation uses test cases spanning static and dynamic properties of uncoupled spins and continuous magnets, comparing results with analytical solutions or fixed-step RK12.The section begins with the single-spin spectra.Source: §III · p. 4

A. Spectra of a single spin

B. Equilibrium magnetization

The equilibrium magnetization test checks whether uncoupled spins follow the Langevin function. Adaptive RK45 and RKF56 simulations agree with it across four values of ξ.

B. Equilibrium magnetization

  • The test assesses whether uncoupled spins in an applied field reach the equilibrium magnetization described by the Langevin function L(ξ).Source: B. Equilibrium magnetization · p. 5
  • Adaptive RK45 and RKF56 simulations agree with the Langevin function across four ξ values.The values were realized using two cell sizes and two temperatures, over a large range of applied fields.Source: B. Equilibrium magnetization · p. 5
  • The four ξ values were varied through two cell sizes and two temperatures.Source: B. Equilibrium magnetization · p. 5

C. Thermal switching

The paper tests thermal switching of uncoupled spins against the predicted switching rate. Simulated rates agree quantitatively with theory for the tested cell and anisotropy settings.

C. Thermal switching

  • The switching-rate prediction applies to a uniaxial macrospin in the high-energy-barrier limit.Source: C. Thermal switching · p. 5
  • For an ensemble initialized with aligned spins, switching produces exponentially decaying magnetization with decay constant 1/2ν, used to estimate the numerical rate.Source: C. Thermal switching · p. 5
  • MuMax3 switching-rate simulations agree quantitatively with theoretical predictions for 10 nm cells at two anisotropy values.The values tested were K=1×10^4 or 2×10^4 J/m^3.Source: C. Thermal switching · p. 5, Figure 6 · p. 5

D. Thermally excited magnetization spectrum

The paper validates adaptive solvers on a thermally excited disk spectrum and thermal domain-wall diffusion. The spectrum matches the fine-step reference, while the diffusion results support a relatively loose tolerance for that problem.

D. Thermally excited magnetization spectrum

E. Thermal diffusion of a domain wall

  • The domain-wall test simulates thermally driven diffusion of a transverse wall in a permalloy nanowire.The wall is simulated for 100 ns and the run is repeated with many random seeds.Source: E. Thermal diffusion of a domain wall · p. 6
  • At ϵ=10^-3, domain-wall diffusion results are adequate for practical purposes because simulation-to-simulation variance exceeds the solver error.The diffusion constant is compared with the model prediction of approximately 310 nm^2/ns; MuMax3 retains a default tolerance of 10^-5 because suitable tolerances depend on the system.Source: E. Thermal diffusion of a domain wall · p. 6, Table II · p. 6

IV. PERFORMANCE

Adaptive RK45 and RKF56 use larger time steps than fixed-step RK12 at comparable estimated error. The benchmark reports a twenty-fold speedup, with performance depending on tolerance, temperature, and hardware.

IV. PERFORMANCE

  • A twenty-fold speedup is reported for adaptive time stepping on the disk-spectrum and domain-wall benchmarks, without loss of accuracy at comparable estimated error.The comparison uses fixed-step RK12 at 5 fs as the reference; runtimes are for 10 ns of dynamics.Source: §IV · pp. 6–7
  • For uncoupled-spin switching, reported speedups range from 20 to 10,000 depending on hardware and accuracy.Time steps can exceed those required by RK12 by more than 1,000 times at the same accuracy, since random-number generation becomes limiting when demagnetizing-field calculations are absent.Source: §IV · p. 7
  • RK45 is faster at higher ϵ and temperatures, whereas RKF56 is faster at low ϵ and temperatures.RKF56 generally pays off only at very high temperatures or very small ϵ.Source: §IV · p. 7

V. CONCLUSIONS

The fixed-length magnetization makes it possible to extend high-order solvers with adaptive time stepping at nonzero temperatures. The methods provide inherent error control and report a twenty-fold speedup without loss of accuracy versus small-fixed-step RK12.

V. CONCLUSIONS

  • The drift term acts only along magnetization length, which is fixed, allowing existing high-order solvers to be extended straightforwardly for adaptive stepping at nonzero temperatures.The authors also implemented a sixth-order Runge-Kutta-Fehlberg solver.Source: §V · p. 7
  • The authors extensively validated the new solver and adaptive time stepping at nonzero temperature, and included the methods in open-source MuMax3.Figure 9 benchmarks runtime for 10 ns of magnetization dynamics and average time step on an NVIDIA Titan Xp GPU.Source: §V · p. 7, Figure 9 · p. 7
  • Twenty-fold speedup without loss of accuracy versus the commonly best practice of small-fixed-step RK12, with inherent error control unavailable to fixed-step methods.The authors say this makes previously infeasible simulations feasible and useful for continuous and uncoupled magnetic systems.Source: §V · p. 7
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