The Landau-Lifshitz-Gilbert equation governs magnetization dynamics — how magnetic moments in a material precess and relax toward equilibrium. The equation has a geometric property: the magnetization vector maintains unit length throughout its evolution. The physics demands it. Each magnetic moment represents a spin direction, and spin directions are points on a sphere, not points in unbounded space.
Standard numerical methods don't automatically preserve this. A naive time-stepping scheme will let the magnetization vector drift off the unit sphere — slightly at first, then catastrophically as errors accumulate. The traditional fix is renormalization: after each time step, project back onto the sphere. This works but is inelegant. It corrects the symptom (wrong length) without respecting the cause (the equation's geometric structure).
A new fractional step method splits the LLG equation into a precession part and a damping part, each solved in a way that independently preserves unit length. The precession step is a rotation (which automatically preserves length). The damping step is handled as a constrained evolution on the sphere. Neither part requires renormalization. The splitting preserves the structure, and the structure preserves the physics.
The deeper point: the Landau-Lifshitz-Gilbert equation lives on a manifold (the unit sphere), not in Euclidean space. Numerical methods that respect the manifold don't just produce better numbers — they produce the right kind of numbers. The constraint isn't a correction; it's the domain.