Dongfang Li

LDG method for reaction–diffusion dynamical systems with time delay

Abstract In this paper we introduce a local discontinuous Galerkin method to solve nonlinear reaction–diffusion dynamical systems with time delay. Stability and convergence of the schemes are obtained. Finally, numerical examples on two biologic models are shown to demonstrate the accuracy and stability of the method.

Split Newton iterative algorithm and its application

Abstract Inspired by some implicit–explicit linear multistep schemes and additive Runge–Kutta methods, we develop a novel split Newton iterative algorithm for the numerical solution of nonlinear equations. The proposed method improves computational efficiency by reducing the computational cost of the Jacobian matrix. Consistency and global convergence of the new method are also maintained. To test its effectiveness, we apply the method to nonlinear reaction–diffusion equations, such as Burger’s–Huxley equation and fisher’s equation. Numerical examples suggest that the involved iterative method is much faster than the classical Newton’s method on a given time interval.

Stability and convergence of compact finite difference method for parabolic problems with delay

Abstract The compact finite difference method becomes more acceptable to approximate the diffusion operator than the central finite difference method since it gives a better convergence result in spatial direction without increasing the computational cost. In this paper, we apply the compact finite difference method and the linear θ-method to numerically solve a class of parabolic problems with delay. Stability of the fully discrete numerical scheme is investigated by using the spectral radius condition. When θ ∈ [ 0 , 1 2 ) , a sufficient and necessary condition is presented to show that the fully discrete numerical scheme is stable. When θ ∈ [ 1 2 , 1 ] , the fully discrete numerical method is proved to be unconditionally asymptotically stable. Moreover, convergence of the fully discrete scheme is studied. Finally, several numerical examples are presented to illustrate our theoretical results.

A two-level linearized compact ADI scheme for two-dimensional nonlinear reaction–diffusion equations

Abstract A novel two-level linearized compact alternating direction implicit (ADI) scheme is proposed for solving two-dimensional nonlinear reaction–diffusion equations. The computational cost is reduced by use of the Newton linearized method and the ADI method. The existence and uniqueness of the numerical solutions are proved. Moreover, the error estimates in H 1 and L ∞ norms are presented. Numerical examples are given to illustrate our theoretical results.