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

TL;DR

The paper develops a projection-free, linearly implicit $(\theta,\mu)$-method for flows of harmonic maps into spheres with a unit-length constraint, achieving unconditional energy stability and quadratic accuracy in the constraint violation under a discrete regularity condition. It extends to variable time stepping, preserving energy decay and constraint control while enabling acceleration toward stationary states and better handling of singularities. A detailed comparison with implicit Euler and linearly implicit BDF2 methods shows competitive or superior constraint-violation behavior, with the midpoint variant often offering improved accuracy. Numerical experiments in $H^1$ and $L^2$ gradient flows, including singular scenarios and adaptive stepping, demonstrate robustness, stability, and practical efficiency for constrained geometric PDEs representative of micromagnetics and related applications.

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.

Figures (4)

• Figure 1: Evolution of the energy (left) and the constraint violation $L^1$-error (right) for the approximations generated by the BDF2 and the midpoint (MP) methods ($H^1$-gradient flow, $\tau = 2^{-5}$).
• Figure 2: Pictures of the initial value (left) and the final value (right). The picture of the final value refers to the results obtained for $\tau = 2^{-14}$. The color scale refers to the third component of the field, which attains values between -1 (blue) and 1 (red).
• Figure 3: Evolution in $[0,1/5]$ of the energy (left), the $L^2$-norm of the update (middle), and the constraint violation $L^1$-error (right) for the approximations generated by the midpoint method ($\tau = 2^{-14}$).
• Figure 4: Evolution in $[0,1/5]$ of the energy (top-left), the $L^2$-norm of the update (top-right), the constraint violation $L^1$-error (bottom-left), and the step size (bottom-right) for the approximations generated by the midpoint method with adaptively adjusted step size ($\tau_1 = 2^{-14}$).

Theorems & Definitions (19)

• Remark 2.1: relevant choices for $\theta$ and $\mu$
• Remark 2.2
• Proposition 2.1: energy identity
• proof
• Lemma 2.1: discrete chain rule
• proof
• Proposition 2.2: constraint violation error
• proof
• Proposition 2.3: bound of the constraint violation error
• proof