Xuan Zhao

This paper proposes a novel framework for constructing up to sixth-order, explicit, and unconditional maximum principle preserving extended integrating factor Runge–Kutta (eIFRK) schemes for the Allen-Cahn equation. The proposed framework significantly relaxes the constraints on the exponential coefficients and offers a recursive procedure to establish high-order schemes. Satisfying the requirement of the local truncation error in the numerical integration, derived from Duhamel’s formula, provides an innovative approach to achieve high-order accuracy instead of solving the order conditions directly. Specifically, in order to derive the target order scheme, Gauss quadrature is applied to approximate the implicit integration, where one order lower scheme is employed to compute explicitly at quadrature points. Consequently, maximum bound principle and [Formula: see text] norm analysis are uniformly proved for eIFRK4–6. Numerical experiments on eIFRK schemes validate their accuracy, maximum-norm preservation, and energy stability. In comparison to the classic second- and third-order sIFRK methods [SIAM J. Sci. Comput., Vol 43, 2021, pp. A1780-A1802], the presented high-order eIFRK schemes, particularly eIFRK6, exhibit enhanced efficiency and stability during long-term simulations, especially for large time steps.

The main theoretical obstacle to establishing the original energy dissipation laws of Runge–Kutta methods for phase field equations is verifying the maximum norm boundedness of the stage solutions without assuming global Lipschitz continuity of the nonlinear bulk. We present a unified theoretical framework for the energy stability of three effective classes of Runge–Kutta methods, including the additive implicit-explicit Runge–Kutta, explicit exponential Runge–Kutta, and corrected integrating factor Runge–Kutta methods, for the Swift–Hohenberg and phase field crystal models. By the standard discrete energy argument, it is proven that the three classes of Runge–Kutta methods preserve the original energy dissipation law if the associated differentiation matrices are positive definite. Our main tools include the differential form with the associated differentiation matrix, the discrete orthogonal convolution kernel, and the principle of mathematical induction. Many existing Runge–Kutta methods in the literature are revisited by evaluating the lower bound on the minimum eigenvalues of the associated differentiation matrices. Our theoretical approach paves a new way toward the internal nonlinear stability of Runge–Kutta methods for dissipative semilinear parabolic problems.

This article considers the stability of nabla linear time invariant (LTI) fractional order systems with the order $\alpha \in (0,+\infty)$ . First, the stable criterion is developed, by using the nabla Laplace transform. Compared with the existing case of $\alpha \in (0,1)$ , our work introduces a wide range of dynamic behaviors for future applications. Second, many essential properties are discussed for the developed criterion, including the changing trend of the stable/unstable region regarding the order, the containment relationship between the imaginary axis, the negative semi-axis and the stable region, the evolution of the modulus with the absolute value of argument for the point lying in the critical stable region. Third, the linear matrix inequality (LMI) condition is tentatively derived to evaluate the stability. Finally, the elaborated results are supported by three illustrative numerical examples.