A multi-physical structure-preserving method and its analysis for the conservative Allen-Cahn equation with nonlocal constraint (Q6624873)

The paper is a significant contribution to the numerical analysis field, addressing the challenges of preserving critical physical properties in the numerical solution of the conservative Allen-Cahn equation. The authors investigate the conservative Allen-Cahn equation, which is characterised by its adherence to three pivotal physical principles: the mass conservation law, the energy dissipation law, and the maximum bound principle (MBP). These properties are essential for accurately modelling various scientific phenomena but are notoriously difficult to preserve simultaneously in numerical computations. To tackle this challenge, the authors propose a novel multi-physical structure-preserving method. Their approach combines the averaged vector field method for temporal discretisation with the central finite difference scheme for spatial discretisation, enabling the simultaneous preservation of all three physical principles at the fully discrete level.\N\NTo solve the resulting nonlinear scheme, the authors design an innovative linear iteration algorithm. This algorithm is rigorously analysed and shown to satisfy the MBP while exhibiting a contraction mapping property in the discrete \( L^\infty \) norm. Furthermore, concise error estimates are established for non-uniform time meshes, ensuring the method's reliability and accuracy. The theoretical claims are substantiated through extensive numerical experiments, including benchmark problems that validate the scheme's second-order accuracy in both space and time, as well as its efficiency when coupled with an adaptive time-stepping strategy.\N\NThe authors' findings are notable for their theoretical and practical implications. They demonstrate that the proposed method not only maintains the critical physical properties of the model but also offers robust convergence guarantees and computational efficiency. This work significantly advances the field of numerical solutions for gradient flow models, providing a powerful tool for researchers and practitioners dealing with the conservative Allen-Cahn equation and similar systems.\N\NIn conclusion, this manuscript presents a robust and innovative numerical method that addresses a critical gap in preserving multiple physical structures in the conservative Allen-Cahn equation. The proposed scheme is expected to have far-reaching implications, offering a framework for developing similarly structured-preserving methods in other complex systems. The work is a valuable addition to the numerical algorithms literature and will likely inspire further advancements in this domain.