Projection-free approximation of flows of harmonic maps with quadratic constraint accuracy and variable step sizes

We construct and analyze a projection-free linearly implicit method for the approximation of flows of harmonic maps into spheres. The proposed method is unconditionally energy stable and, under a sharp discrete regularity condition, achieves second-order accuracy with respect to the constraint violation. Furthermore, the method accommodates variable step sizes to speed up the convergence to stationary points and to improve the accuracy of the numerical solutions near singularities, without affecting the unconditional energy stability and the constraint violation property. We illustrate the accuracy in approximating the unit-length constraint and the performance of the method through a series of numerical experiments, and compare it with the linearly implicit Euler and two-step BDF methods.