Efficient numerical schemes for the conserved Allen-Cahn phase-field surfactant system based on high order supplementary variable method

Numerous accurate, efficient, and robust numerical schemes for phase-field surfactant models have been developed, with those ensuring energy stability being particularly attractive. Unconditional energy stability refers to numerical stability without any restrictions on the time step size. Recent research has concentrated on methods such as convex splitting, invariant energy quadratization (IEQ), scalar auxiliary variable (SAV), and Lagrange multiplier methods, which are generally unconditionally energy stable but mostly guarantee modified energy dissipation and are second-order accurate in time. This is often insufficient to meet the high-precision demands of long-term simulations. This paper presents a class of high-order numerical schemes based on the supplementary variable method (SVM) combined with the Runge–Kutta (RK) method. These schemes preserve the original energy dissipation law and can achieve arbitrarily high-order time accuracy. Benchmark numerical examples are provided to illustrate the accuracy and efficiency of these schemes.