Stabilization of linearly implicit schemes for dissipative systems with quartic potentials

This paper introduces a novel framework for constructing stable, structure-preserving, linearly implicit schemes to dissipative systems with quartic polynomial potentials. Conventional approaches, such as those relying on the discrete gradient method, often depend on ad-hoc linearization techniques and tend to yield unstable numerical schemes. In contrast, the proposed method offers a systematic approach to linearization through the application of polar forms of the potentials and the Ito-Abe discrete gradient method. By embedding the original potential into a higher-dimensional space via its polar form, the approach enables regularized linearizations that ensure the stability of the resulting schemes. The paper explores the connection between the stability of the obtained schemes and the properties of the transformed potentials, such as their real Waring rank and coercivity. Moreover, explicit conditions on the regularization parameter required to maintain stability are derived.