A structure-preserving parametric approximation for anisotropic geometric flows via an $\alpha$-surface energy matrix

We propose a structure-preserving parametric approximation for geometric flows with general anisotropic effects. By introducing a hyperparameter $\alpha$, we construct a unified surface energy matrix $\hat{\boldsymbol{G}}_k^\alpha(\theta)$ that encompasses all existing formulations of surface energy matrices, and apply it to anisotropic curvature flow. We prove that $\alpha=-1$ is the unique choice achieving optimal energy stability under the necessary and sufficient condition $3\hat{\gamma}(\theta)\geq\hat{\gamma}(\theta-\pi)$, while all other $\alpha\neq-1$ 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 $\alpha=-1$ and demonstrate the effectiveness and robustness.