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Based on quadratic and linear polynomial interpolation approximations and the Crank-Nicolson technique, a new second-order difference approximation method of the Caputo fractional derivative of order α ∈ (1 , 2) is proposed. This method is different from the known second-order methods, which is called the L 2- 1 σ Crank-Nicolson method in this paper. Using the L 2- 1 σ Crank-Nicolson method of the Caputo fractional derivative, a fully discrete L 2- 1 σ Crank-Nicolson difference method for two-dimensional time-fractional wave equations with variable coefficients is developed. The unconditional stability and convergence of the method are rigorously proved. The optimal error estimates in the discrete L 2 -norm and H 1 -norm are obtained under a relatively weak regularity condition. The error estimates show that the method has the second-order convergence in both time and space for all α ∈ (1 , 2). In order to overcome the loss of the temporal convergence order caused by the stronger singularity of the exact solution at the initial time, a nonuniform L 2- 1 σ Crank-Nicolson difference method is also developed on a class of general nonuniform time meshes. Numerical results confirm the theoretical convergence result. The effectiveness of the nonuniform method for non-smooth solutions with the stronger singularity at the initial time is tested on a class of initially graded time meshes. [ABSTRACT FROM AUTHOR]

This paper is concerned with an efficient numerical method for a class of parabolic integro-differential equations with weakly singular kernels. Due to the presence of the weakly singular kernel, the exact solution has singularity near the initial time t = 0. A generalized Crank-Nicolson-type scheme for the time discretization is proposed by designing a product integration rule for the integral term, and a compact difference approximation is used for the space discretization. The proposed method is constructed on the uniform time mesh, but it can still achieve the second-order convergence in time for weakly singular solutions. The unconditional stability and convergence of the method is proved and the optimal error estimate in the discrete L 2 -norm is obtained. The error estimate shows that the method has the second-order convergence in time and the fourth-order convergence in space. The extension of the method to two-dimensional problems is also discussed. A simple comparison is made with several existing methods. Numerical results confirm the theoretical analysis result and show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

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]

Abstract A high-order compact finite difference method is proposed for solving a class of time-fractional sub-diffusion equations. 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). The Caputo time-fractional derivative is discretized by a (3 − α) th-order numerical formula (called the L 2 formula here) which is constructed by piecewise quadratic interpolating polynomials but does not require any sub-stepping scheme for the approximation at the first-time level. The variable coefficient spatial differential operator is approximated by a fourth-order compact finite difference operator. By developing a technique of discrete energy analysis, a full theoretical analysis of the stability and convergence of the method is carried out for the general case of variable coefficient and for all α ∈ (0, 1). The optimal error estimate is obtained in the L 2 norm and shows that the proposed method has the temporal (3 − α) th-order accuracy and the spatial fourth-order accuracy. Further approximations are also considered for enlarging the applicability of the method while preserving its high-order accuracy. Applications are given to three model problems, and numerical results are presented to demonstrate the theoretical analysis results. [ABSTRACT FROM AUTHOR]

A compact alternating direction implicit (ADI) finite difference method is proposed for two-dimensional time fractional sub-diffusion equations with nonhomogeneous Neumann boundary conditions. The unconditional stability and convergence of the method is proved. The error estimates in the weighted L 2 - and L ∞ -norms are obtained. The proposed method has the fourth-order spatial accuracy and the temporal accuracy of order min { 2 − α , 1 + α } , where α ∈ ( 0 , 1 ) is the order of the fractional derivative. In order to further improve the temporal accuracy, two Richardson extrapolation algorithms are presented. Numerical results demonstrate the accuracy of the compact ADI method and the high efficiency of the extrapolation algorithms. [ABSTRACT FROM AUTHOR]

This paper is concerned with numerical methods for a class of time-fractional convection-reaction-diffusion equations. The convection and reaction coefficients of the equation may be spatially variable. Based on the weighted and shifted Grünwald–Letnikov formula for the time-fractional derivative and a compact finite difference approximation for the spatial derivative, we establish an unconditionally stable compact difference method. The local truncation error and the solvability of the resulting scheme are discussed in detail. The stability of the method and its convergence of third-order in time and fourth-order in space are rigorously proved by the discrete energy method. Combining this method with a Richardson extrapolation, we present an extrapolated compact difference method which is fourth-order accurate in both time and space. A rigorous proof for the convergence of the extrapolation method is given. Numerical results confirm our theoretical analysis, and demonstrate the accuracy of the compact difference method and the effectiveness of the extrapolated compact difference method. [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]

[Display omitted] • Russel-Saunders coupling scheme might be more feasible for U 5 f electron. • U j = 5/2, j = 7/2 manifolds are in metallic and insulating states, respectively. • Occupation numbers of U 5 f electrons n f ∼ 1.8585 in UF 6. • U 5 f electrons exhibit inter-configuration fluctuation due to 5 f 1/5 f 2/5 f 3 mixing. • Charge transfer, hybridization and delocalization promote bonding of U 5 f orbitals. A quantum mechanics calculation is performed on uranium hexafluoride (UF 6) by using density functional theory (DFT) combing with dynamical mean-field theory (DMFT) scheme, taking into account the spin–orbit coupling (SOC) interaction and the on-site 5 f -5 f Coulomb repulsion. Results demonstrate that SOC-splitting j = 5/2 and j = 7/2 manifolds are in the metallic and insulating regimes, respectively, albeit both are in weakly correlated states. Ratio of the quasi-particle weights R = Z j =7/2 / Z j =5/2 is about 1, suggesting that UF 6 is not in the so-called orbital-selective localized state. Russel-Saunders (LS) coupling scheme is feasible for U 5 f electrons. U 5 f -conduction electrons hybridization, 5 f -5 f correlation, SOC interaction and final state effects yield a complicated spectrum function. Occupation probability analysis shows that 5 f electrons exhibit the so-called inter-configuration fluctuation (ICF) due to admixing of 5 f 1 and 5 f 3 atomic configurations into 5 f 2 configuration with an average occupation number of n 5 f ∼ 1.8585, mainly arising from the complicated quantum mechanics processes including the dual nature of localized and itinerant 5 f electrons, a quantum mechanical mixture of the energetically degenerate 5 fn configurations, as well as the flexible outer electronic configuration of U ions. Finally, the momentum-resolved electronic spectrum function (the so-called quasiparticle band structure) is also discussed. [ABSTRACT FROM AUTHOR]

A Crank–Nicolson-type compact locally one-dimensional (LOD) finite difference method is proposed for a class of two-dimensional modified anomalous fractional sub-diffusion equations with two time Riemann–Liouville fractional derivatives of orders ( 1 − α ) and ( 1 − β ) ( 0 < α , β < 1 ) . The resulting scheme consists of simple tridiagonal systems and all computations are carried out completely in one spatial direction as for one-dimensional problems. This property evidently enhances the simplicity of programming and makes the computations more easy. The unconditional stability and convergence of the scheme are rigorously proved. The error estimates in the standard H 1 - and L 2 -norms and the weighted L ∞ -norm are obtained and show that the proposed compact LOD method has the accuracy of the order 2 min { α , β } in time and 4 in space. A Richardson extrapolation algorithm is presented to increase the temporal accuracy to the order min { α + β , 4 min { α , β } } if α ≠ β and min { 1 + α , 4 α } if α = β . A comparison study of the compact LOD method with the other existing methods is given to show its superiority. Numerical results confirm our theoretical analysis, and demonstrate the accuracy and the effectiveness of the compact LOD method and the extrapolation algorithm. [ABSTRACT FROM AUTHOR]

This paper is concerned with a new numerical method for time-fractional mobile/immobile diffusion equations with weakly singular solutions. We propose an averaged L 1 -type compact difference method, which improves the temporal convergence order of the traditional L 1 -type method and is also superior to the WSGD-type and L 2 − 1 σ -type methods in terms of regularity requirements. Taking into account the weak singularity of the solution at the initial time, we prove that the proposed method is unconditionally convergent with the convergence order O (τ 2 | ln τ | + h 4) , where τ and h are the sizes of the time and spatial steps, respectively. Numerical results confirm the theoretical convergence result. [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 higher-order compact locally one-dimensional (LOD) finite difference method for two-dimensional nonhomogeneous parabolic differential equations is proposed. The resulting scheme consists of two one-dimensional tridiagonal systems, and all computations are implemented completely in one spatial direction as for one-dimensional problems. The solvability and the stability of the scheme are proved almost unconditionally. The error estimates are obtained in the discrete and norms, and show that the proposed compact LOD method has the accuracy of the second-order in time and the fourth-order in space. Two Richardson extrapolation algorithms are presented to increase the accuracy to the fourth-order and the sixth-order in both time and space when the time step is proportional to the spatial mesh size. Numerical results demonstrate the accuracy of the compact LOD method and the high efficiency of its extrapolation algorithms. [Copyright &y& Elsevier]

Abstract: This paper is concerned with an existing compact finite difference ADI method, published in the paper by Liao et al. (2002) , for solving systems of two-dimensional reaction–diffusion equations with nonlinear reaction terms. This method has an accuracy of fourth-order in space and second-order in time. The existence and uniqueness of its solution are investigated by the method of upper and lower solutions, without any monotone requirement on the nonlinear reaction terms. The convergence of the finite difference solution to the continuous solution is proved. An efficient monotone iterative algorithm is presented for solving the resulting discrete system, and some techniques for the construction of upper and lower solutions are discussed. An application using a model problem gives numerical results that demonstrate the high efficiency and advantages of the method. [Copyright &y& Elsevier]

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 time-delayed Lotka–Volterra competition reaction–diffusion system with homogeneous Neumann boundary conditions. Some explicit and easily verifiable conditions are obtained for the global asymptotic stability of all forms of nonnegative semitrivial constant steady-state solutions. These conditions involve only the competing rate constants and are independent of the diffusion–convection and time delays. The result of global asymptotic stability implies the nonexistence of positive steady-state solutions, and gives some extinction results of the competing species in the ecological sense. The instability of the trivial steady-state solution is also shown. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of this paper is to give a numerical treatment for a class of strongly nonlinear two-point boundary value problems. The problems are discretized by fourth-order Numerov''s method, and a linear monotone iterative algorithm is presented to compute the solutions of the resulting discrete problems. All processes avoid constructing explicitly an inverse function as is often needed in the known treatments. Consequently, the full potential of Numerov''s method for strongly nonlinear two-point boundary value problems is realized. Some applications and numerical results are given to demonstrate the high efficiency of the approach. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to investigate the existence of iterative solutions for a class of 2nth-order nonlinear multi-point boundary value problems. The multi-point boundary condition under consideration includes various commonly discussed boundary conditions, such as three- or four-point boundary condition, (n +2)-point boundary condition and 2(n − m)-point boundary condition. The existence problem is based on the method of upper and lower solutions and its associated monotone iterative technique. A monotone iteration is developed so that the iterative sequence converges monotonically to a maximal solution or a minimal solution, depending on whether the initial iteration is an upper solution or a lower solution. Two examples are presented to illustrate the results. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the existence and uniqueness of a solution for a class of th-order nonlinear multi-point boundary value problems. The existence of a solution is proven by the method of upper and lower solutions without any monotone condition on the nonlinear function. A sufficient condition for the uniqueness of a solution is given. It is also shown that there exist two sequences which converge monotonically from above and below, respectively, to the unique solution. Two examples are presented to illustrate the results. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the computational algorithms for finite difference discretizations of a class of nonlinear reaction–diffusion–convection equations with nonlinear boundary conditions. A higher-order monotone iterative method is presented for solving the finite difference discretizations of both the time-dependent problem and the corresponding steady-state problem. This method leads to an efficient linear iterative algorithm which yields two sequences of iterations that converge monotonically to a unique solution of the system. The monotone property of the iterations gives concurrently improved upper and lower bounds of the solution in each iteration. It is shown that the rate of convergence for the sum of the two produced sequences is of order , where is a positive integer depending on the construction of the method, and under an additional requirement, the higher-order rate of convergence is attained for one of these two sequences. An application is given to an enzyme–substrate reaction–diffusion problem, and some numerical results are presented to illustrate the effectiveness of the proposed method. [Copyright &y& Elsevier]

Abstract: This paper is to investigate the asymptotic behavior of solutions for a time-delayed Lotka–Volterra -species mutualism reaction–diffusion system with homogeneous Neumann boundary condition. It is shown, under a simple condition on the reaction rates, that the system has a unique bounded time-dependent solution and a unique constant positive steady-state solution, and for any nontrivial nonnegative initial function the corresponding time-dependent solution converges to the constant positive steady-state solution as time tends to infinity. This convergence result implies that the trivial steady-state solution and all forms of semitrivial steady-state solutions are unstable, and moreover, the system has no nonconstant positive steady-state solution. A condition ensuring the convergence of the time-dependent solution to one of nonnegative semitrivial steady-state solutions is also given. [Copyright &y& Elsevier]

Abstract: This paper is concerned with finite difference solutions of a Michaelis–Menten-type ratio-dependent predator–prey system with diffusion. The system is discretized by the finite difference method, and the investigation is devoted to the finite difference system for the time-dependent solution and its asymptotic behavior in relation to the various steady-state solutions. Three monotone iterative schemes for the computation of the time-dependent solution are presented, and the sequences of iterations are shown to converge monotonically to a unique positive solution. A simple and easily verifiable condition on the rate constants is obtained, which ensures that for every nontrivial nonnegative initial function the corresponding time-dependent solution converges either to a unique positive steady-state solution or to a semitrivial steady-state solution. The above results lead to computational algorithms for the solution as well as the global asymptotic stability of the system. Some numerical results are given. All the conclusions are directly applicable to the finite difference solution of the corresponding ordinary differential system. [Copyright &y& Elsevier]

Abstract: This paper is concerned with finite difference solutions of a Lotka–Volterra reaction–diffusion system with three-competing species. The reaction–diffusion system is discretized by the finite difference method, and the investigation is devoted to the finite difference system for the time-dependent solution and its asymptotic behavior in relation to the corresponding steady-state problem. Three monotone iterative schemes for the computation of the time-dependent solution are presented, and the sequences of iterations are shown to converge monotonically to a unique positive solution. Also discussed is the asymptotic behavior of the time-dependent solution in relation to various steady-state solutions. A simple condition on the competing rate constants is obtained, which ensures that for every nontrivial nonnegative initial function the corresponding time-dependent solution converges either to a unique positive steady-state solution or to one of the semitrivial steady-state solutions. The above results lead to the coexistence and permanence of the competing system as well as computational algorithms for numerical solutions. Some numerical results from these computational algorithms are given. All the conclusions for the reaction–diffusion equations are directly applicable to the finite difference solution of the corresponding ordinary differential system. [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]

Abstract: This paper is concerned with three 3-species time-delayed Lotka–Volterra reaction–diffusion models with homogeneous Neumann boundary condition. Some simple conditions are obtained for the global asymptotic stability of the nonnegative semitrivial constant steady-state solutions. These conditions are explicit and easily verifiable, and they involve only the reaction rate constants and are independent of the diffusion and time delays. The result of global asymptotic stability not only implies the nonexistence of positive steady-state solution but also gives some extinction results of the models in the ecological sense. The instability of some nonnegative semitrivial constant steady-state solutions is also shown. The conclusions for the reaction–diffusion systems are directly applicable to the corresponding ordinary differential systems. [Copyright &y& Elsevier]

Abstract: This paper is concerned with finite difference solutions of a class of fourth-order nonlinear elliptic boundary value problems. The nonlinear function is not necessarily monotone. A new monotone iterative technique is developed, and three basic monotone iterative processes for the finite difference system are constructed. Several theoretical comparison results among the various monotone sequences are given. A simple and easily verified condition is obtained to guarantee a geometric convergence of the iterations. Numerical results for a model problem with known analytical solution are given. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the extrapolation algorithm of Numerov''s scheme for semilinear and strongly nonlinear two-point boundary value problems. The asymptotic error expansion of the solution of Numerov''s scheme is obtained. Based on the asymptotic error expansion, Richardson''s extrapolation is constructed, and so the accuracy of the numerical solution is greatly increased. Numerical results are presented to demonstrate the efficiency of the extrapolation algorithm. [Copyright &y& Elsevier]

Abstract: This paper is concerned with a Lotka-Volterra cooperation-diffusion model with a saturating interaction term for one species. The goal of the paper is to investigate the asymptotic behavior of the time- dependent solution in relation to the corresponding steady-state solutions under homogeneous Neumann boundary condition. Some simple and easily verifiable conditions are given to the rate constants so that for every nontrivial nonnegative initial function the corresponding time-dependent solution converges to one of the nonnegative constant steady-state solutions as time tends to infinity. This convergence result leads to the existence and uniqueness of a positive (or nonnegative) steady-state solution and the global asymptotic stability of a given nonnegative constant steady-state solution. In terms of ecological dynamics, it also gives some coexistence, permanence and extinction results for the model. [Copyright &y& Elsevier]

Abstract: This work is concerned with the convergence of a monotone method for fourth-order semilinear elliptic boundary value problems. A comparison result for the rate of convergence is given. The global error is analyzed, and some sufficient conditions are formulated for guaranteeing a geometric rate of convergence. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the computational algorithms for finite difference solutions of a class of semilinear elliptic boundary value problems. An accelerated monotone iterative scheme is presented by using the method of upper and lower solutions. The rate of convergence of the iterations is estimated by the infinity norm, and the rate of convergence is quadratic for a larger class of nonlinear functions, including monotone nonincreasing functions. An application is given to a logistic model problem in ecology. [Copyright &y& Elsevier]

A parallel monotone iterative relaxation method for a class of two-dimensional discrete boundary value problems is established, and the sequence of iterations is shown to converge monotonically either from above or below to a solution of the problem. This monotone convergence results yields a parallel computational algorithm as well as an existence-comparison result for the solutions. To compute the sequence of iterations, the Thomas algorithm can be used in the same fashion as for one-dimensional problem. The existence and comparison results of the upper and lower solutions are given. The local as well as global existence-uniqueness of the solution are obtained. The global convergence of the iterations is investigated, and the influence of the parameters on the rate of convergence of the iterations is analyzed. Numerical results are given to corroborate the analytical results. [Copyright &y& Elsevier]

By a proper transformation, a class of strongly nonlinear two-point boundary value problems are transformed into semilinear problems so that the well-known Numerov''s method can be applied. Some applications and numerical results are presented to demonstrate the efficiency of the approach. [Copyright &y& Elsevier]

• Multi-term time-fractional convection-reaction-diffusion problems with non-smooth solutions are considered. • A compact exponential finite difference method is proposed • Taking into account the initial weak singularity of the solution, the stability and convergence of the method is rigorously proved. • The spatial fourth-order convergence and the temporal optimal convergence are obtained. • Numerical results confirm the theoretical convergence result. This paper is concerned with a numerical method for a class of one-dimensional multi-term time-fractional convection-reaction-diffusion problems, where the differential equation contains a sum of the Caputo time-fractional derivatives of different orders between 0 and 1. In general the solutions of such problems typically exhibit a weak singularity at the initial time. A compact exponential finite difference method, using the well-known L 1 formula for each time-fractional derivative and a fourth-order compact exponential difference approximation for the spatial discretization, is proposed on a mesh that is generally nonuniform in time and uniform in space. Taking into account the initial weak singularity of the solution, the stability and convergence of the method is proved and the optimal error estimate in the discrete L 2-norm is obtained by developing a discrete energy analysis technique which enables us to overcome the difficulties caused by the nonsymmetric discretization matrices. The error estimate shows that the method has the spatial fourth-order convergence, and reveals how to select an appropriate mesh parameter to obtain the temporal optimal convergence. The extension of the method to two-dimensional problems is also discussed. Numerical results confirm the theoretical convergence result, and show the applicability of the method to convection dominated problems. [ABSTRACT FROM AUTHOR]

This paper is concerned with a numerical method for the time-fractional Benjamin–Bona–Mahony (BBM) equation whose solution typically exhibits a weak singularity at the initial time. Lyu and Vong (2019) presented a linearized difference method of second-order in space and third-order in time. We improve their result by proposing a linearized and compact difference method which is fourth-order in space while keeping third-order in time. By using discrete energy analysis, the unconditional convergence of the proposed method is rigorously proved and the optimal H 1 -norm error estimate is obtained. Numerical results confirm the theoretical convergence result. [ABSTRACT FROM AUTHOR]

Beidou is the regional satellite navigation system in China, consisting of three kinds of orbiting satellites, MEO, GEO and IGSO, with the orbital altitudes of 21500-36000km. For improving the accuracy of satellites orbit determination, calibrating microwave measuring techniques and providing better navigation service, all Beidou satellites are equipped with laser retro-reflector arrays (LRAs) to implement high precision laser ranging. The paper presents the design of LRAs for Beidou navigation satellites and the method of inclined installation of LRAs for GEO satellites to increase the effective reflective areas for the regional ground stations. By using the SLR system, the observations for Beidou satellites demonstrated a precision of centimeters. The performances of these LRAs on Beidou satellites are very excellent. [ABSTRACT FROM AUTHOR]