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A high-order compact finite difference method on nonuniform time meshes is proposed for solving a class of variable coefficient reaction–subdiffusion problems. The solution of such a problem in general has a typical weak singularity at the initial time. Alikhanov’s high-order approximation on a uniform time mesh for the Caputo time fractional derivative is generalised to a class of nonuniform time meshes, and a fourth-order compact finite difference scheme is used for approximating the spatial variable coefficient differential operator. A full theoretical analysis of the stability and convergence of the method is given for the general case of the variable coefficients by developing an analysis technique different from the one for the constant coefficient problem. Taking the weak initial singularity of the solution into account, a sharp error estimate in the discrete $$L^{2}$$ -norm is obtained. It is shown that the proposed method attains the temporal optimal second-order convergence provided a proper mesh parameter is employed. Numerical results demonstrate the sharpness of the theoretical error analysis result.

A high-order Crank-Nicolson-type compact difference method is proposed for a class of time fractional Cattaneo convection-diffusion equations with smooth solutions. The convection coefficient of the equation may be spatially variable. A suitable transformation is adopted to transform the original equation into a reaction-diffusion equation, which is then discretized by a fourth-order compact difference approximation for the spatial derivative and by a second-order Crank-Nicolson-type difference approximation for the time first derivative and the Caputo time fractional derivative. The local truncation error and the solvability of the resulting scheme are discussed in detail. The (almost) unconditional stability of the method and its convergence of second order in time and fourth order in space are rigorously proved using a discrete energy analysis method. A Richardson extrapolation algorithm, including its rigorous convergence analysis, is presented. This extrapolation algorithm improves the temporal accuracy of the computed solution to the third order. An application of the proposed method to the non-smooth solution which has a weak singularity at the initial time is also discussed by introducing a correction term. Numerical results demonstrate the accuracy of the new method and the high efficiency of the Richardson extrapolation algorithm.

A set of high-order compact finite difference methods is proposed for solving a class of Caputo-type fractional sub-diffusion equations in conservative form. The diffusion coefficient of the equation may be spatially variable, and the proposed methods have the global convergence order $$\mathcal{O}(\tau ^{r}+h^{4})$$ , where $$r\ge 2$$ is a positive integer and $$\tau $$ and h are the temporal and spatial steps. Such new high-order compact difference methods greatly improve the known methods in the literature. The local truncation error and the solvability of the methods are discussed in detail. By applying a discrete energy technique to the matrix form of the methods, a rigorous theoretical analysis of the stability and convergence of the methods is carried out for the case of $$2\le r\le 6$$ , and the optimal error estimates in the weighted $$H^{1}$$ , $$L^{2}$$ and $$L^{\infty }$$ norms are obtained for the general case of variable coefficient. Applications are given to two model problems, and some numerical results are presented to illustrate the various convergence orders of the methods.

In a recent paper, Ren and Liu proposed and analyzed a high-order compact finite difference method for a class of fractional sub-diffusion equations with variable coefficients and nonhomogeneous Neumann boundary conditions. In this paper, we point out some deficiencies and errors found in that paper and make the corresponding revisions.

This paper is concerned with high-order numerical methods for a class of fractional mobile/immobile convection–diffusion equations. The convection coefficient of the equation may be spatially variable. In order to overcome the difficulty caused by variable coefficient problems, we first transform the original equation into a special and equivalent form, which is then discretized by a fourth-order compact finite difference approximation for the spatial derivative and a second-order difference approximation for the time first derivative and the Caputo time fractional derivative. The local truncation error and the solvability of the resulting scheme are discussed in detail. The (almost) unconditional stability and convergence of the method are proved using a discrete energy analysis method. A Richardson extrapolation algorithm is presented to enhance the temporal accuracy of the computed solution from the second-order to the third-order. Applications using two model problems give numerical results that demonstrate the accuracy of the new method and the high efficiency of the Richardson extrapolation algorithm.

A modified Crank–Nicolson-type compact alternating direction implicit (ADI) finite difference method is proposed for a class of two-dimensional fractional subdiffusion equations with a time Riemann–Liouville fractional derivative of order $$(1-\alpha )$$ $$(0

This paper is concerned with numerical methods for a class of two-dimensional fractional convection-subdiffusion equations with a time Caputo fractional derivative of order $$\alpha (0

This paper is concerned with numerical methods for a class of time fractional convection-diffusion-wave equations. The convection coefficient in the equation may be spatially variable and the time fractional derivative is in the Caputo sense with the order ? (1 < 2). The class of the equations includes time fractional convection-diffusion-wave/diffusion-wave equations with or without damping as its special cases. In order to overcome the difficulty caused by variable coefficient problems, we first transform the original equation into a special and equivalent form, which is then discretized by a fourth-order compact finite difference method for the spatial derivative and by the L1 approximation coupled with the Crank-Nicolson technique for the time derivative. The local truncation error and the solvability of the method are discussed in detail. A rigorous theoretical analysis of the stability and convergence is carried out using a discrete energy analysis method. The optimal error estimates in the discrete H1, L2 and L? norms are obtained under the mild condition that the time step is smaller than a positive constant, which depends solely upon physical parameters involved (this condition is no longer required for the special case of constant coefficients). Applications using three model problems give numerical results that demonstrate the effectiveness and the accuracy of the proposed method.

A high-order compact finite difference method is proposed for solving a class of time fractional convection-subdiffusion equations. The convection coefficient in the equation may be spatially variable, and the time fractional derivative is in the Caputo’s sense with the order \(\alpha \) (\(0

This paper is concerned with numerical methods for a class of multi-dimensional fractional diffusion-wave equations with a time fractional derivative of order $$\alpha $$ ( $$1

In order to develop the technique of Laser Time Transfer (LTT), Shanghai Astronomical Observatory has built a prototype of space-borne LTT module. The performance of the LTT module and the results of ground tests are discussed in the paper. The average precision of time difference between two rubidium clocks measured by laser pulses is 196 ps, and the uncertainty of measurement for the relative frequency differences is 1.2×10−13/2800 s.

This paper is concerned with finite difference solutions of a system of reaction-diffusion equations with coupled nonlinear boundary conditions and time delays. The reaction functions and the boundary functions are not necessarily quasimonotone, and the time delays may appear in the reaction functions as well as in the boundary functions. The investigation is devoted to the finite difference system for both the time-dependent problem and its corresponding steady-state problem. Some monotone iteration processes for the finite difference systems are given, and the asymptotic behavior of the time-dependent solution in relation to the steady-state solution is discussed. The asymptotic behavior result leads to some local and global attractors of the time-dependent problem, including the convergence of the time-dependent solution to a unique steady-state solution. An application and some numerical results to an enzyme-substrate reaction-diffusion problem are given. All the results are directly applicable to parabolic-ordinary systems and to reaction-diffusion systems without time delays.

Almost monotone approximation is proposed for nonlinear two‐points problem. A general framework is given for studying the existence and uniqueness of numerical solutions. A discrete approximation with high accuracy is constructed. Nonlinear Jacobi iteration and Gauss–Seidel iteration are introduced to save work. The continuous approximation is also considered. The numerical results show the advantages of such an approach.