A practical guide to solving the stochastic Landau-Lifshitz-Gilbert-Slonczewski equation for macrospin dynamics

In this paper, we discuss the accuracy and complexity of various numerical techniques to solve the stochastic Landau-Lifshitz-Gilbert-Slonczewski (s-LLGS) equation. The s-LLGS equation is widely used by researchers to study the temporal evolution of the macrospin subject to spin torque and thermal noise. The numerical simulation of the s-LLGS equation requires an appropriate choice of stochastic calculus and the numerical integration scheme. In this paper, we focus on implicit midpoint, Heun, and Euler-Heun methods that converge to the Stratonovich solution of the s-LLGS equation. We also demonstrate a new method intended to solve stochastic differential equations (SDEs) with small noise, and test its capability to handle the s-LLGS equation. The choice of specific stochastic calculus while solving SDEs determines which numerical integration scheme to use. In this sense, methods, such as Euler and Gear, which are typically used by SPICE-based circuit simulators do not yield the expected outcome when solving the Stratonovich s-LLGS equation. While the trapezoidal method in SPICE does solve for the Stratonovich solution, its accuracy is limited by the minimum time-step of integration in SPICE. Through several numerical tests, including path-wise error, preservation of the magnetization norm, and 50% magnetization reversal boundary of the macrospin, we clearly illustrate the accuracy of various numerical schemes for solving the s-LLGS equation. The results in this paper will serve as guidelines for researchers to understand the tradeoffs between accuracy and complexity of various numerical methods and the choice of appropriate calculus to handle SDEs.