Compact Difference Method Research Articles

Matrix-free TriGlobal adjoint stability analysis of compressible Navier–Stokes equations

A numerical method for TriGlobal (i.e., fully three-dimensional) adjoint stability analysis for compressible flows was developed and is presented in this paper. The developed method solves the adjoint stability problem using a matrix-free method based on Krylov-Schur method and a time-stepping approach. Because of the low memory (RAM) requirement of the matrix-free approach, the developed method can analyze fully three-dimensional flows that are difficult to analyze with conventional matrix-based methods. To perform time-stepping on the adjoint variables, the adjoint equations, including appropriate boundary conditions for the compressible Navier–Stokes equations, were derived. The equations were discretized using a finite compact difference method, and time integration was conducted using the three-step third-order Runge–Kutta method. A flow field around a two-dimensional square cylinder and a cubic cavity flow were analyzed using the developed method, and it was confirmed that the method reproduces the dominant instabilities reported in the literature. In addition, in the square cylinder flow analysis, the receptivity and sensitivity regions of the secondary wake mode, which corresponds to far wake instability, were clarified. Finally, the TriGlobal direct and adjoint stabilities of compressible flows over a finite width cavity were analyzed for the first time, and it was proven that large-scale adjoint stability analysis can be performed with the developed method. The results also show that instability phenomena similar to those obtained with BiGlobal stability analysis appear, but sidewall effects exist.

High-order conservative schemes for the space fractional nonlinear Schrödinger equation

In the paper, the high-order conservative schemes are presented for space fractional nonlinear Schrödinger equation. First, we give two class high-order difference schemes for fractional Risze derivative by compact difference method and extrapolating method, and show the convergence analysis of the two methods. Then, we apply high-order conservative difference schemes in space direction, and Crank-Nicolson, linearly implicit and relaxation schemes in time direction to solve fractional nonlinear Schrödinger equation. Moreover, we show that the arising schemes are uniquely solvable and approximate solutions converge to the exact solution at the rate O(τ2+h4), and preserve the mass and energy conservation laws. Finally, we given numerical experiments to show the efficiency of the conservative finite difference schemes.

A class of compact finite difference schemes for solving the 2D and 3D Burgers’ equations

In this paper, a class of two-level high-order compact finite difference implicit schemes are proposed for solving the Burgers’ equations. Firstly, based on the fourth-order compact finite difference schemes for spatial derivatives and the truncation error remainder correction method for temporal derivative, the high-order compact (HOC) difference method is introduced for solving the one-dimensional (1D) Burgers’ equation. At the same time, the stability of the scheme is analyzed by using the Fourier analysis method. Because only three grid points are involved in each time level. The Thomas algorithm can be directly used to solve the tridiagonal linear system. Then, this method is extended to solve the two-dimensional (2D) and three-dimensional (3D) coupled Burgers’ equations. Finally, numerical experiments are conducted to verify the accuracy and the reliability of the present schemes.

A Linearised Three-Point Combined Compact Difference Method with Weighted Approximation for Nonlinear Time Fractional Klein-Gordon Equations

A Linearised Three-Point Combined Compact Difference Method with Weighted Approximation for Nonlinear Time Fractional Klein-Gordon Equations

High-order compact difference method for two-dimension elliptic and parabolic equations with mixed derivatives

In this article, firstly, based on Taylor series expansion and truncation error correction technology, combined with the fourth-order Padé schemes of the first-order derivatives, a new fourth-order compact difference (CD) scheme is constructed to solve the two-dimensional (2D) linear elliptic equation a mixed derivative. In this new scheme, unknown function and its first-order derivatives are regarded as the unknown variables in calculation. Then, the method is extended to solve the 2D parabolic equation with a mixed derivative. To match the spatial fourth-order accuracy, The backward differentiation formula (BDF) is employed to gain the fourth-order accuracy for the temporal discretization. Truncation error is analyzed to display that the present scheme is fourth-order accuracy in space. In order to solve the resulting large-scale linear equations, a multigrid method is employed to accelerate the convergence speed of the conventional relaxation methods. Finally, numerical results indicate that the present schemes obtain fourth-order convergence and are more accurate than those in the literature.

A high-order compact difference method on fitted meshes for Neumann problems of time-fractional reaction–diffusion equations with variable coefficients

This paper is concerned with numerical methods for a class of nonhomogeneous Neumann problems of time-fractional reaction–diffusion equations with variable coefficients. The solutions of this kind of problems often have weak singularity at the initial time. This makes the existing numerical methods with uniform time mesh often lose accuracy. In this paper, we propose and analyze a high-order compact finite difference method with nonuniform time mesh. The time-fractional derivative is approximated by Alikhanov’s high-order approximation on a class of fitted time meshes. For the spatial variable coefficient differential operator, a new fourth-order boundary discretization is developed under the nonhomogeneous Neumann boundary condition, and then a new fourth-order compact finite difference approximation on a space uniform mesh is obtained. Under the assumption of the weak initial singularity of solution, we prove that for the general case of the variable coefficients, the proposed method is unconditionally stable and the numerical solution converges to the solution of the problem under consideration. The convergence result also gives an optimal error estimate of the numerical solution in the discrete L2-norm, which shows that the method has the spatial fourth-order convergence, while it attains the temporal optimal second-order convergence provided a proper mesh grading parameter is employed. Numerical results that confirm the sharpness of the error analysis are presented.

Linearized compact difference methods combined with Richardson extrapolation for nonlinear delay Sobolev equations

Delay Sobolev equations (DSEs) are a class of important models in fluid mechanics, thermodynamics and the other related fields. For solving this class of equations, in this paper, linearized compact difference methods (LCDMs) for one- and two-dimensional problems of DSEs are suggested. The solvability and convergence of the methods are analyzed and it is proved under some appropriate conditions that the methods are convergent of order two in time and order four in space. In order to improve the computational accuracy of LCDMs in time, we introduce the Richardson extrapolation technique, which leads to the improved LCDMs can reach the fourth-order accuracy in both time and space. Finally, with several numerical experiments, the theoretical accuracy and computational effectiveness of the proposed methods are further testified.

A compact exponential difference method for multi-term time-fractional convection-reaction-diffusion problems with non-smooth solutions

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 L1 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 L2-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.

A high-order linearized and compact difference method for the time-fractional Benjamin–Bona–Mahony equation

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 H1-norm error estimate is obtained. Numerical results confirm the theoretical convergence result.

A High-Order Compact (HOC) Implicit Difference Scheme and a Multigrid Method for Solving 3D Unsteady Reaction Diffusion Equations

A high-order compact (HOC) implicit difference scheme is proposed for solving three-dimensional (3D) unsteady reaction diffusion equations. To discretize the spatial second-order derivatives, the fourth-order compact difference operators are used, and the third- and fourth-order derivative terms, which appear in the truncation error term, are also discretized by the compact difference method. For the temporal discretization, the multistep backward Euler formula is used to obtain the fourth-order accuracy, which matches the spatial accuracy order. To accelerate the traditional relaxation methods, a multigrid method is employed, and the computational efficiency is greatly improved. Numerical experiments are carried out to validate the accuracy and efficiency of the present method.

A high-order compact difference method for fractional sub-diffusion equations with variable coefficients and nonhomogeneous Neumann boundary conditions

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.

A fast high-order compact difference method for the fractal mobile/immobile transport equation

ABSTRACT In this paper, we present a high-order compact finite difference scheme for fractal mobile/immobile transport model with a Caputo fractional time derivative. The compact finite difference scheme is stable and convergent with convergence order in -norm. Furthermore, because of the non-local property of fractional differential operators, it is necessary to find a fast technique to reduce the computational cost. We develop a fast solution technique which is based on a fast Fourier transform. The computational work will reduce to while the direct method requires the computational cost of , where and τ is the size of time step, and h is the size of space step. Moreover, numerical results are consistent with the theoretical analysis.

A high-order compact difference method for time fractional Fokker–Planck equations with variable coefficients

A high-order compact finite difference method is proposed for time fractional Fokker–Planck equations with variable convection coefficients. This method leads to a very simple and yet efficient compact finite difference scheme with high-order accuracy. It is also very convenient for us to give the corresponding analysis of stability and convergence using a discrete energy method. The proposed method is unconditionally stable and convergent with the convergence order $$\mathcal{O}(\tau ^{2}+h^{4})$$, where $$\tau $$ and h are the step sizes in time and space, respectively. Thus, it improves the convergence order of some recently developed methods. Numerical results confirm the theoretical analysis and demonstrate the high efficiency of this novel method.

A Blended Compact Difference (BCD) Method for Solving 3D Convection–Diffusion Problems with Variable Coefficients

In this study, we present a fourth-order and a sixth-order blended compact difference (BCD) schemes for approximating the three-dimensional (3D) convection–diffusion equation with variable convective coefficients. The proposed schemes, where transport variable, its first and second derivatives are carried as the unknowns, combine virtues of compact discretization, fourth-order Padé scheme and sixth-order combined compact difference (CCD) scheme for spatial derivatives and can efficiently capture numerical solutions of linear and nonlinear convection–diffusion equations with Dirichlet boundary conditions. The fourth-order scheme requires only 7 grid points and the sixth-order scheme requires 19 grid points. The distinguishing feature of the present method is that methodologies of explicit compact difference and implicit compact difference are blended together. The truncation errors of the two difference schemes are analyzed for the interior grid points, respectively. Simultaneously, a sixth-order accuracy scheme is proposed to compute the first and second derivatives of the grid points on boundaries. Finally, the presented methods are applied to several test problems from the literature including linear and nonlinear problems. It is found that the presented schemes exhibit good performance.

A class of high-order compact difference schemes for solving the Burgers’ equations

In this paper, a class of high-order compact difference method is introduced for solving the Burgers’ equations. Firstly, a linear high-order compact difference scheme is proposed to solve the one-dimensional Burgers’ equation. The scheme is fourth-order accurate in space and second-order accurate in time. Linear stability analysis is conducted to show the scheme is conditionally stable. Because only three grid points are involved in each time level, Thomas algorithm can be directly used to solve the tridiagonal linear system. Then, this method is extended to solve the two-dimensional and three-dimensional coupled Burgers’ equations. Numerical experiments are carried out to demonstrate the accuracy and dependability of the present method.

A higher-order blended compact difference (BCD) method for solving the general 2D linear second-order partial differential equation

A higher-order blended compact difference (BCD) scheme is proposed to solve the general two-dimensional (2D) linear second-order partial differential equation. The distinguishing feature of the present method is that methodologies of explicit compact difference and implicit compact difference are blended together. Sixth-order accuracy approximations for the first- and second-order derivatives are employed, and the original equation is also discretized based on a 9-point stencil, which is different from the work of Lee et al. (J. Comput. Appl. Math. 264:23–37, 2014). A truncation error analysis is performed to show that the scheme is of sixth-order accuracy for the interior grid points. Simultaneously, sixth-order accuracy schemes are proposed to compute the grid points on the boundaries for the first- and second-order derivatives. Numerical experiments are conducted to demonstrate the accuracy and efficiency of the present method.

Exponential high-order compact finite difference method for convection-dominated diffusion problems on nonuniform grids

In this article, an exponential high-order compact (EHOC) difference method is introduced for solving the convection-dominated convection diffusion problems on nonuniform grid without any coordinate transformation from the physical space to the computational space. The derived EHOC schemes on nonuniform grid can not only preserve nonoscillation property and yield high accuracy approximation solutions, but also efficiently handle the convection diffusion problems with boundary layers by employing a flexible discretized grid that can be adapted to the singularity in the domain. To demonstrate the performances in computational accuracy, efficiency, and stability of the proposed EHOC schemes, several problems with boundary or interior layers where sharp gradients may appear are solved numerically. The results obtained by the present EHOC schemes on nonuniform grid are compared with analytical solutions and those using the EHOC on uniform grid and other methods. The present EHOC schemes on nonuniform grid provide excellent results for all test problems. It is shown that the proposed EHOC schemes on nonuniform grid are accurate, efficient, and stable, as well as have the advantage of better-scale resolution for convection-dominated diffusion problems.

A second-order accurate numerical method with graded meshes for an evolution equation with a weakly singular kernel

A second-order accurate numerical method with graded meshes is proposed and analyzed for an evolution equation with a weakly singular kernel. The graded meshes are employed to compensate for the singular behavior of the exact solution at t = 0 . For the time discretization, the product integration rule is used to approximate the Riemann–Liouville fractional integral, a generalized Crank–Nicolson time-stepping is considered and shown that the error is of order k 2 , where k denotes the maximum time step. A fully discrete difference scheme is constructed with space discretization by compact difference method. Numerical experiment is carried out to support the theoretical results. The comparison between the method on uniform grids and graded grids shows the efficiency of our method.

A compact streamfunction-velocity scheme for the 2-D unsteady incompressible Navier-Stokes equations in arbitrary curvilinear coordinates

A streamfunction-velocity formulation-based compact difference method is suggested for solving the unsteady incompressible Navier-Stokes equations in the arbitrary curvilinear coordinates, in which the streamfunction and its first derivatives as the unknown variables are utilized. Numerical examples, involving the boundary layer problem, a constricted channel flow, driven polar cavity flow and trapezoidal cavity flow problem, are solved by the present method. Numerical results demonstrate the accuracy of the proposed scheme and exhibit the numerical capability to simulate the flow problems on geometries beyond rectangular. For driven polar cavity flow problem, the results show that the flow for Re = 5 000 is not steady but time-periodic, and the critical Reynold number (Rec) for the occurrence of a Hopf bifurcation is given.

An efficient, high-order method for solving Poisson equation for immersed boundaries: Combination of compact difference and multiscale multigrid methods

A new efficient and high-order accurate sharp-interface method for solving the Poisson equation on irregular domains and non-uniform meshes is presented. The approach is based on a combination of a fourth-order compact finite difference scheme and a multiscale multigrid (MSMG) method. The key aspect of the new method is that the regular compact finite difference stencil is modified at the irregular grid points near the immersed boundary to obtain a sharp interface solution while maintaining the formal fourth-order accuracy. The MSMG method is designed based on the standard multigrid V-cycle technique to solve the system of equations derived from the fourth-order compact discretization, while the corresponding multigrid relaxation, restriction and prolongation operators are properly constructed for non-uniform grids with immersed boundaries. The contribution of the present work is the design of a fourth-order-accurate Poisson solver whose accuracy, efficiency and computational cost are independent of the complexity of the geometry and the presence or not of an immersed boundary. The new method is demonstrated and validated for a number of problems including smooth and jagged boundaries. The test cases confirm that the new method is fourth-order accurate in the maximum norm whether an immersed boundary is present or not and on uniform or non-uniform meshes. Furthermore, the computational efficiency of the new method is demonstrated with regard to convergence rate and run time, which shows that the MSMG method is equally efficient for domains with immersed boundaries as for simple domains. The new compact difference method is evaluated by comparison with the standard fourth-order (non-compact) finite difference approximation in terms of both accuracy and computational efficiency. The new compact difference scheme yields indeed more accurate numerical solutions. The striking difference between the two schemes is the much higher computational efficiency: The number of V-cycles needed to reach the discretization error is significantly lower for the new compact method compared to the standard difference scheme. As a result, the new compact scheme requires only a fraction of the computer time for convergence in comparison to the standard fourth-order difference scheme.