Energy quadratization Runge–Kutta scheme for the conservative Allen–Cahn equation with a nonlocal Lagrange multiplier

In this study, we present a high-order energy stable scheme for the conservative Allen–Cahn equation with a nonlocal Lagrange multiplier by combining the concept of energy quadratization and the Runge–Kutta method. Under the stability condition for the Runge–Kutta coefficients, we analytically demonstrate that the scheme is unconditionally stable with respect to the reformulated energy. Additionally, we develop a Newton-type fixed point iteration method to implement the scheme, enabling the achievement of a fast iterative convergence. Numerical experiments are presented to demonstrate the accuracy and energy stability of the proposed scheme.