An extended quadratic auxiliary variable method for the singular Lennard-Jones droplet liquid film model

In this letter, we present an extended quadratic auxiliary variable method for a droplet liquid film model characterized by the singular Lennard-Jones potential. This innovative approach begins with the introduction of three new auxiliary variables, each satisfying a quadratic equation, with the primary objective of reformulating the original model into an equivalent system. The reformulated system exhibits remarkable properties, including the preservation of three quadratic invariants and a quadratic energy dissipation law, where these auxiliary variables satisfy consistent initial conditions. To solve it numerically, we employ the implicit midpoint method in time and the central finite difference scheme in space to produce a second -order fully discrete scheme for the original model. Under consistent initial conditions, our proposed scheme is rigorously proved to preserve the original energy dissipation law at the fully discrete level. Numerical experiments showcase the scheme's accuracy and effectiveness. It is noteworthy that the quadratic auxiliary variable method is extended to gradient flow systems featuring rational energy functionals.

Automated summary

In this article, an extended quadratic auxiliary variable method is introduced for a droplet liquid film model. The method shows good numerical solvability and accuracy.