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This paper is concerned with the construction and analysis of a high-order compact finite difference method for a class of time-fractional sub-diffusion equations under the Robin boundary condition. The diffusion coefficient of the equation may be spatially variable and the time-fractional derivative is in the Caputo sense with the order α ∈ (0 , 1). A (3 − α) th-order numerical formula (called the L 2 formula here) without any sub-stepping scheme for the approximation at the first-time level is applied to the discretization of the Caputo time-fractional derivative. A new fourth-order compact finite difference operator is constructed to approximate the variable coefficient spatial differential operator under the Robin boundary condition. By developing a technique of discrete energy analysis, the unconditional stability of the proposed method and its convergence of (3 − α) th-order in time and fourth-order in space are rigorously proved for the general case of variable coefficient and for all α ∈ (0 , 1). Further approximations are considered for enlarging the applicability of the method while preserving its high-order accuracy. Numerical results are provided to demonstrate the theoretical analysis results. [ABSTRACT FROM AUTHOR]

This paper is devoted to the construction and analysis of compact finite difference methods for a class of time-fractional convection-reaction-diffusion equations with variable coefficients. Based on some new techniques coupled with the - approximation formula of the time-fractional derivative and a fourth-order compact finite difference approximation to the spatial derivative, a compact finite difference method is proposed for the equations with spatially variable convection and reaction coefficients. The local truncation error and the solvability of the method are discussed in detail. The unconditional stability of the resulting scheme and also its convergence of second order in time and fourth order in space are rigorously proved using a discrete energy analysis method. The proposed method is further extended to the more general case when the convection and reaction coefficients are variable both spatially and temporally. A high-order combined compact finite difference method is also proposed. Numerical results demonstrate the effectiveness of the methods. [ABSTRACT FROM AUTHOR]

This paper is concerned with a compact finite difference method with non-isotropic mesh sizes for a two-dimensional fourth-order nonlinear elliptic boundary value problem. By the discrete energy analysis, the optimal error estimates in the discrete L 2 , H 1 and L ∞ norms are obtained without any constraint on the mesh sizes. The error estimates show that the compact finite difference method converges with the convergence rate of fourth-order. Based on a high-order approximation of the solution, a Richardson extrapolation algorithm is developed to make the final computed solution sixth-order accurate. Numerical results demonstrate the high-order accuracy of the compact finite difference method and its extrapolation algorithm in the discrete L 2 , H 1 and L ∞ norms. [ABSTRACT FROM AUTHOR]

In this paper, we study numerical methods for a class of two-dimensional semilinear elliptic boundary value problems with variable coefficients in a union of rectangular domains. A compact finite difference method with an anisotropic mesh is proposed for the problems. The existence of a maximal and a minimal compact difference solution is proved by the method of upper and lower solutions, and two sufficient conditions for the uniqueness of the solution are also given. The optimal error estimate in the discrete L ∞ norm is obtained under certain conditions. The error estimate shows the fourth-order accuracy of the proposed method when two spatial mesh sizes are proportional. By using an upper solution or a lower solution as the initial iteration, an “almost optimal” Picard type of monotone iterative algorithm is developed for solving the resulting nonlinear discrete system efficiently. Numerical results are presented to confirm our theoretical analysis. [ABSTRACT FROM AUTHOR]

Abstract: A fourth-order compact finite difference method is proposed for a class of nonlinear th-order multi-point boundary value problems. The multi-point boundary condition under consideration includes various commonly discussed boundary conditions, such as the three- or four-point boundary condition, -point boundary condition and -point boundary condition. The existence and uniqueness of the finite difference solution are investigated by the method of upper and lower solutions, without any monotone requirement on the nonlinear term. The convergence and the fourth-order accuracy of the method are proved. An efficient monotone iterative algorithm is developed for solving the resulting nonlinear finite difference systems. Various sufficient conditions for the construction of upper and lower solutions are obtained. Some applications and numerical results are given to demonstrate the high efficiency and advantages of this new approach. [Copyright &y& Elsevier]

Abstract: This paper is concerned with a compact finite difference method for solving systems of two-dimensional reaction–diffusion equations. This method has the accuracy of fourth-order in both space and time. The existence and uniqueness of the finite difference solution are investigated by the method of upper and lower solutions, without any monotone requirement on the nonlinear term. Three monotone iterative algorithms are provided for solving the resulting discrete system efficiently, and the sequences of iterations converge monotonically to a unique solution of the system. A theoretical comparison result for the various monotone sequences is given. The convergence of the finite difference solution to the continuous solution is proved, and Richardson extrapolation is used to achieve fourth-order accuracy in time. An application is given to an enzyme–substrate reaction–diffusion problem, and some numerical results are presented to demonstrate the high efficiency and advantages of this new approach. [Copyright &y& Elsevier]

Abstract: A compact finite difference method with non-isotropic mesh is proposed for a two-dimensional fourth-order nonlinear elliptic boundary value problem. The existence and uniqueness of its solutions are investigated by the method of upper and lower solutions, without any requirement of the monotonicity of the nonlinear term. Three monotone and convergent iterations are provided for resolving the resulting discrete systems efficiently. The convergence and the fourth-order accuracy of the proposed method are proved. Numerical results demonstrate the high efficiency and advantages of this new approach. [Copyright &y& Elsevier]

Abstract: A compact finite difference method is proposed for a general class of th-order Lidstone boundary value problems. The existence and uniqueness of the finite difference solution is investigated by the method of upper and lower solutions, without any monotone requirement on the nonlinear term. A monotone iteration process is provided for solving the resulting discrete system efficiently, and a simple and easily verified condition is obtained to guarantee a geometric convergence of the iterations. The convergence of the finite difference solution and the fourth-order accuracy of the proposed method are proved. Numerical results demonstrate the high efficiency and advantages of this new approach. [Copyright &y& Elsevier]