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This paper deals with the discrete maximum principle and energy stability of a new difference scheme for solving Allen-Cahn equations. By combining the second-order central difference approximation in space and the Crank-Nicolson method with Newton linearized technique in time, a two-level linearized difference scheme for Allen-Cahn equations is derived, which can yield accuracy of order two both in time and space. Under appropriate conditions, the scheme is proved to be uniquely solvable and able to preserve the maximum principle and energy stability of the equations in the discrete sense. With some numerical experiments, the theoretical results and computational effectiveness of the scheme are further illustrated.

Efficient long-time integration of nonlinear fractional differential equations is significantly challenging due to the integro-differential nature of the fractional operators. In addition, the inherent non-smoothness introduced by the inverse power-law kernels deteriorates the accuracy and efficiency of many existing numerical methods. We develop two efficient first- and second-order implicit-explicit (IMEX) methods for accurate time-integration of stiff/nonlinear fractional differential equations with fractional order α ∈ ( 0 , 1 ] and prove their convergence and linear stability properties. The developed methods are based on a linear multi-step fractional Adams-Moulton method (FAMM), followed by the extrapolation of the nonlinear force terms. In order to handle the singularities nearby the initial time, we employ Lubich-like corrections to the resulting fractional operators. The obtained linear stability regions of the developed IMEX methods are larger than existing IMEX methods in the literature. Furthermore, the size of the stability regions increase with the decrease of fractional order values, which is suitable for stiff problems. We also rewrite the resulting IMEX methods in the language of nonlinear Toeplitz systems, where we employ a fast inversion scheme to achieve a computational complexity of O ( N log ⁡ N ) , where N denotes the number of time-steps. Our computational results demonstrate that the developed schemes can achieve global first- and second-order accuracy for highly-oscillatory stiff/nonlinear problems with singularities.

This paper is concerned with numerical analysis and computation for neutral equations with piecewise constant argument (NEPCA). The block boundary value methods (BBVMs) are extended to solve NEPCA. Under the conditions of preconsistency and p-order consistency, it is proved that the convergence order of the extended BBVMs can arrive at p. Moreover, a sufficient and necessary condition of the numerical asymptotical stability is derived for linear NEPCA. The presented numerical examples further testify the computational effectiveness of the methods and the obtained theoretical results.

In this paper, by combining the p-order block boundary value methods with the m-th Lagrange interpolation, a class of new numerical methods for solving nonlinear fractional differential equations with the γ-order ( 0 γ 1 ) Caputo derivatives are obtained. It is proved under some appropriate conditions that the induced methods are convergent of order min ⁡ { p , m − γ + 1 } and globally stable. Several numerical examples are given to illustrate the theoretical results and the computational effectiveness and accuracy of the methods.

This paper deals with nonlinear stochastic functional differential equations with piecewise continuous arguments (SFDEPCAs). Based on an adaptation to the underlying one-leg θ-methods for ODEs, a class of new one-parameter methods for nonlinear SFDEPCAs are introduced. The mean-square exponential stability criteria of analytical and numerical solutions are derived. Under the suitable conditions, it is proved that the one-parameter methods are convergent with strong order 1/2. Some numerical experiments are given to illustrate the theoretical results and computational advantages of the induced methods.

In this paper, a class of new compact difference schemes is presented for solving the fourth-order time fractional sub-diffusion equation of the distributed order. By using an effective numerical quadrature rule based on boundary value method to discretize the integral term in the distributed-order derivative, the original distributed order differential equation is approximated by a multi-term time fractional sub-diffusion equation, which is then solved by a compact difference scheme. It is shown that the suggested compact difference scheme is stable and convergent in L ∞ norm with the convergence order O ( τ 2 + h 4 + ( Δ γ ) p ) when a boundary value method of order p is used, where τ , h and Δγ are the step sizes in time, space and distributed-order variables, respectively. Numerical results are reported to verify the high order accuracy and efficiency of the suggested scheme. Moreover, in the example, comparisons between some existing methods and the suggested scheme is also provided, showing that our method doesn't compromise in computational time.

In this paper, the generalized polynomial chaos (gPC) method is extended to solve nonlinear random delay differential equations (NRDDEs). The error estimation of the method is derived, which arises mainly from a finite-dimensional noise assumption, projection error and discretization error. When the error from the finite-dimensional noise assumption can not be omitted, the error of the method converges to a limit inferior which is just the error from the finite-dimensional noise assumption. With some numerical experiments, the obtained theoretical results are further illustrated.

This paper deals with a class of functional differential equations with piecewise continuous arguments. Block boundary value methods (BBVMs) are extended to solve this class of equations. It is shown under the Lipschitz condition that the order of convergence of an extended block boundary value method coincides with its order of consistency. Moreover, we study the linear stability of the extended methods and give the corresponding asymptotical stability criterion. In the end, with several numerical examples, the theoretical results and the computational effectiveness of the methods are further illustrated.

In this paper, we investigate the convergence behavior of discontinuous Galerkin methods for solving a class of delay differential equations. Although discontinuities may occur in various orders of the derivative of the solutions, we show that the m-degree DG solutions have ( m + 1 ) th order accuracy in L ∞ norm. Numerical experiments confirm the theoretical results of the methods.

This paper deals with the error analysis of generalized polynomial chaos (gPC) for nonlinear random ordinary differential equations. The analysis shows that the global error mainly relies on the projection error and the numerical error. For the deterministic systems obtained from the gPC method, a kind of numerical approach with error analysis is given. At last, a numerical experiment is carried out to support the theoretical results.

In this paper, a new three-level compact alternating direction implicit (ADI) difference scheme is derived for solving a kind of nonlinear wave equations. Basing on a fourth-order approximation to the exact solution at the first time level, it is shown by the energy method that the numerical solution is conditionally convergent with an order of O(@Dt^2+h"x^4+h"y^4) in H^1- and L^~-norms. A new Richardson extrapolation formula based on three time-grid parameters is given to get numerical solution of fourth-order accuracy in both time and space. The performance of the new algorithm is illustrated by numerical experiments.

In this paper, we construct a class of extended block boundary value methods (B"2VMs) for Volterra delay integro-differential equations and analyze the convergence and stability of the methods. It is proven under the classical Lipschitz condition that an extended B"2VM is convergent of order p if the underlying boundary value methods (BVM) has consistent order p. The analysis shows that a B"2VM extended by an A-stable BVM can preserve the delay-independent stability of the underlying linear systems. Moreover, under some suitable conditions, the extended B"2VMs can also keep the delay-dependent stability of the underlying linear systems. In the end, we test the computational effectiveness by applying the introduced methods to the Volterra delay dynamical model of two interacting species, where the theoretical precision of the methods is further verified.

In this paper, a class of block boundary value methods (BBVMs) for the initial value problems of delay differential equations are suggested. It is proven under the classical Lipschitz condition that a BBVM is convergent of order p if it is consistent of order p. Several linear stability criteria for the BBVMs are derived. Numerical experiments further confirm the convergence and the effectiveness of the methods.

In this paper, a class of extended one-leg methods for nonlinear neutral delay-integro-differential equations are suggested. The nonlinear stability of the methods are investigated. It is proven that the extended one-leg methods are globally and asymptotically stable under the suitable conditions. In the end, numerical experiments illustrate the computational effectiveness of the methods.

This paper deals with convergence and stability analysis of Runge–Kutta methods with the Lagrangian interpolation (RKLMs) for a class of delay systems. We show that DA-, DAS- and ASI-stability of the Runge–Kutta methods (RKMs) for ODEs imply GDN-stability of the corresponding RKLMs for DDEs, and that a DA-, DAS- and ASI-stable RKM with stage order p , together with a Lagrangian interpolation of order q , results in a D-convergent RKLM of order min {p,q+1} .