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

¹ Department of Computer Science and Engineering, University of Ioannina, 451 10 Ioannina, Greece
² Institute of Applied and Computational Mathematics, FORTH, 700 13 Heraklion, Greece
³ Department of Applied Mathematics, University of Freiburg, 79104 Freiburg i. Br., Germany
⁴ Department of Mathematics, University of Bologna, 40126 Bologna, Italy
⁵ School of Science, Harbin Institute of Technology, Shenzhen 518055, P.R. China

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 11 September 2025
Accepted: 9 February 2026

Abstract

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.