7.2.3 Heun scheme

The improved Euler or Heun method [39] is an example of a predictor-corrector method. The predictor is given by a simple Euler type integration. If we consider the Langevin equation (), the predictor is

$$\bar M_i = M_i(t) + A_i(\mathbf{M}, t)\Delta t + B_{ik}(\mathbf{M}, t)\Delta W_k \quad.$$ (7.10)

$\Delta t$ is the discretization time step and

$$\Delta W_k=\int_t^{t+\Delta t}H_{\mathrm{th},k}(t') \,d{t'}\,$$

are Gaussian random numbers, whose first two moments are given by

$$\langle {\Delta W_k} \rangle =0 \, , \quad \langle {\Delta W_k \Delta W_l} \rangle = 2 D \delta_{kl} \Delta t$$

$2 D$ is the variance of the stochastic thermal field (), which is given by ().

The Heun scheme is then given by

$$M_i(t+\Delta t) = M_i(t)+ \frac{1}{2} \left[ A_i(\mathbf{\bar M}, t+\Delta t)+A_i(\mathbf{M}, t) \right] \Delta t +$$

$$\frac{1}{2} \left[ B_{ik}(\mathbf{\bar M}, t+\Delta t)+B_{ik}(\mathbf{M}, t) \right] \Delta W_k \quad.$$ (7.11)

The stochastic Heun scheme converges in quadratic mean to the solution of the general system of Langevin equations () when interpreted in the sense of Stratonovich.

To conclude, there are two main reasons for the choice of the Heun scheme for the numerical integration of the stochastic Landau-Lifshitz equation: First, the Heun scheme yields Stratonovich solutions of the stochastic differential equations without alterations to the deterministic drift term. Secondly, the deterministic part of the differential equations is integrated with a second order accuracy in $\Delta t$, which renders the Heun scheme numerically more stable than Euler type schemes.