The design and verification of MuMax3

MuMax3 provides a relax() function that attempts to find the systems’ energy minimum. This function disables the precession term Eq. (2), so that the effective field points towards decreasing energy. Relax first advances in time until the total energy cuts into the numerical noise floor. At that point the state will be close to equilibrium already. We then begin monitoring the magnitude of the torque instead of the energy, since close to equilibrium the torque will decrease monotonically and is less noisy than the energy. So we advance further until the torque cuts into the noise floor as well. Each time that happens, we decrease MaxErr and continue further until MaxErr=10−9. At this point it does not make sense to increase the accuracy anymore (see Fig. 11) and we stop advancing.

This Relax procedure was used in the presented standard problems, where it proved adequate. Typical residual torques after Relax are of the order of 10−4–10−7 γLL T, indicating that the system is indeed very close to equilibrium. Nevertheless, as with any energy minimization technique, there is always a possibility that the system ends up in a saddle point or very flat part of the energy landscape.

Relax internally uses the RK23 solver, which we noticed performs better then RK45 in most relaxation scenarios. Near equilibrium, both solvers tend to take similarly large time steps, but RK23 needs only half as many torque evaluations per step as RK45.