A structure-preserving parametric approximation for anisotropic geometric flows via an $α$-surface energy matrix
This work provides a theoretically optimal and robust numerical method for simulating anisotropic geometric flows, which is important for applications in materials science and image processing.
The paper proposes a parametric approximation for anisotropic geometric flows using a unified surface energy matrix with a hyperparameter α, proving that α=-1 uniquely achieves optimal energy stability under a necessary and sufficient condition. Numerical experiments confirm the theoretical optimality.
We propose a structure-preserving parametric approximation for geometric flows with general anisotropic effects. By introducing a hyperparameter $α$, we construct a unified surface energy matrix $\hat{\boldsymbol{G}}_k^α(θ)$ that encompasses all existing formulations of surface energy matrices, and apply it to anisotropic curvature flow. We prove that $α=-1$ is the unique choice achieving optimal energy stability under the necessary and sufficient condition $3\hatγ(θ)\geq\hatγ(θ-π)$, while all other $α\neq-1$ require strictly stronger conditions. The framework extends naturally to general anisotropic geometric flows through a unified velocity discretization that ensures energy stability. Numerical experiments validate the theoretical optimality of $α=-1$ and demonstrate the effectiveness and robustness.