L-stability and Error Analysis of a Fourth-Order Exponential Time Differencing Runge–Kutta RDP Scheme for the Allen–Cahn Equation

This paper presents a theoretical analysis of a fourth-order exponential time differencing Runge–Kutta rational approximation scheme for the Allen–Cahn equation based on real and distinct poles, aiming to establish a theoretical framework for the method combining exponential time differencing with rational approximation. Starting from the scalar linearized equation, we rigorously prove that the scheme is L-stable and that its matrix norm is stable with three types of boundary conditions, revealing that the L-acceptable property of the RDP rational approximation is the key to ensuring stability. Regarding the error analysis, we establish that the fully discrete scheme attains fourth-order temporal convergence, which theoretically confirms that the scheme achieves fourth-order accuracy. Numerical examples in both 2D and 3D verify the accuracy and effectiveness of the method.