Fourth-order Compact Difference Scheme Research Articles

Pointwise error estimate and stability analysis of fourth-order compact difference scheme for time-fractional Burgers’ equation

A novel implicit high-order compact difference scheme is established for the time-fractional Burgers’ equation based on a newly developed nonlinear compact operator and the reduction order technique under the uniform mesh and graded mesh. Uniqueness, boundedness, convergence in the pointwise sense and stability in L2-norm of the discrete solutions are proved by the Browder fixed point theorem and energy argument. Numerical examples with the smooth solution and nonsmooth solution are provided to confirm numerical theoretical results and demonstrate the efficiency of the high-order compact difference scheme.

A fourth-order block-centered compact difference scheme for nonlinear contaminant transport equations with adsorption

Nonlinear contaminant transports through porous media are important in many scientific and engineering applications. In this paper, we develop and analyze fourth-order block-centered compact difference scheme (BCCDS) for the nonlinear contaminant transport equations with adsorption process in porous media. Based on block-centered mesh, a fourth order compact difference scheme of solution and its flux is derived. We prove the mass conservation of the proposed scheme and its unconditional stability. We analyze the convergence and obtain the fourth-order error estimate under the smooth regularity of exact solution. Numerical experiments are presented to show the performance of the schemes.

Fourth-order compact difference schemes for the two-dimensional nonlinear fractional mobile/immobile transport models

In this article, we develop fourth-order compact difference schemes based on the linearized generalized BDF2-θ to solve the two-dimensional nonlinear fractional mobile/immobile (M/I) transport equations. We derive theoretical results, including unconditional stability and error estimates associated with solution regularity. Finally, we provide extensive numerical examples with smooth solutions to demonstrate the effectiveness and accuracy of the schemes and develop the corrected compact difference scheme to recovery convergence results with nonsmooth solutions.

Numerical investigation of double-diffusive convection in rectangular cavities with different aspect ratio I: High-accuracy numerical method

In this paper, being based on the idea of dispersion-relation-preserving optimization, a class of high-order high-resolution upwind compact schemes with a free parameter and their consistent boundary scheme are proposed for the solution of the governing equations of the 2D double-diffusive convection. The first derivative and the second derivative, in the vorticity equation, the temperature equation and the concentration equation, are discretized by using the optimal upwind compact scheme on a uniform mesh and the fourth-order symmetrical Padé compact scheme, respectively. The pressure Poisson equation is approximated using a fourth-order compact difference scheme constructed currently on the nine-point two dimensional stencil. The fourth-order Runge-Kutta method is utilized for the temporal discretization. To assess numerical capability of the proposed algorithm, particularly its spatial behavior, the problems about the convection diffusion equation, the nonlinear Burgers equation and the nature convection flows in the square cavity with adiabatic horizontal walls and differentially heated vertical walls are computed by using the newly proposed algorithm. The numerical results are in excellent agreement with the benchmark solutions and some of the accurate results available in the literature, and show that effectiveness, accuracy and the advantage of better resolution of the present method. After that, steady and unsteady solutions for the double-diffusive convection in a rectangular cavity are also used to assess the efficiency of the present algorithm. The period of oscillation and flow field profiles are in great agreement with the data in the literature for buoyancy ratio, λ=1. The typical separation and secondary vortices at the bottom corner of the cavity as well as the top corner can be captured well for various buoyancy ratio. These indicate that the present method is suitable for simulating effectively the unsteady double-diffusive convection.

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.

Study on the dynamics of a nonlinear dispersion model in both 1D and 2D based on the fourth-order compact conservative difference scheme

In this paper, the nonlinear dispersion wave model in both 1D and 2D is studied by the compact finite difference method, which is called the generalized Rosenau–RLW equation. A fourth-order compact three-level and linearized difference scheme that maintains the original conservative properties of equation is proposed. The discrete mass conservation and discrete energy conservation of compact difference scheme are obtained. The solvability of numerical scheme is obtained. By using the discrete energy method, convergence and unconditional stability can also be obtained without relying on the grid ratio, and the optimal error estimates in the L∞ norm are fourth-order and second-order accuracy for the spatial and temporal step sizes, respectively. The scheme is conservative so can be used for long time computation. Numerical experiment results show that the theory is accurate and the method is efficient and reliable. Finally, the new numerical scheme is used to study the nonlinear dynamic of 1D generalized Rosenau–RLW equation and the wave interference of 2D generalized Rosenau–RLW equation, focusing on the influence of nonlinear convection term on the wave and the conservation of wave propagation.

Three-dimensional forward modelling of gravity field vector and its gradient tensor using the compact difference schemes

SUMMARY The traditional gravity forward modelling methods for solving partial differential equations (PDEs) only can yield second-order accuracy. When computing the gravity field vector and gradient tensor from the obtained potential, those numerical differentiation approaches will inevitably lose accuracy. To mitigate this issue, we propose an efficient and accurate 3-D forward modelling algorithm based on a fourth-order compact difference scheme. First, a 19-point fourth-order compact difference scheme with general meshsizes in x-, y- and z-directions is adopted to discretize the governing 3-D Poisson’s equation. The resulting symmetric positive-definite linear systems are solved by the pre-conditioned conjugate gradient algorithm. To obtain the first-order (i.e. the gravity field vector) and second-order derivatives (i.e. the gravity gradient tensor) with fourth-order accuracy, we seek to solve a sequence of tridiagonal linear systems resulting from the above mentioned finite difference discretization by using fast Thomas algorithm. Finally, two synthetic models and a real topography relief are used to verify the accuracy of our method. Numerical results show that our method can yield a nearly fourth-order accurate approximation not only to the gravitational potential, but also to the gravity field vector and its gradient tensor, which clearly demonstrates its superiority over the traditional PDE-based methods.

An explicit fourth-order compact difference scheme for solving the 2D wave equation

In this paper, an explicit fourth-order compact (EFOC) difference scheme is proposed for solving the two-dimensional(2D) wave equation. The truncation error of the EFOC scheme is O({tau ^{4}} + {tau ^{2}}{h^{2}} + {h^{4}}), i.e., the scheme has an overall fourth-order accuracy in both time and space. Because the scheme is explicit, it does not need any iterative processes. Afterwards, the stability condition of the scheme is obtained by using the Fourier analysis method, which has a wider stability range than other explicit or alternation direction implicit (ADI) schemes. Finally, some numerical experiments are carried out to verify the accuracy and stability of the present scheme.

Spectral graph wavelet regularization and adaptive wavelet for the backward heat conduction problem

This paper proposes a new regularization technique using spectral graph wavelet for backward heat conduction problem (BHCP) on the graph. The method uses the fourth-order compact difference for an approximation of differential operators. Meanwhile, the error estimate between the exact and wavelet regularized solution is derived. Adaptive node arrangement is obtained using spectral graph wavelet. The essence of the method is that the same operator can be used for the construction of the spectral graph wavelet and the approximation of the differential operator. The proposed method is applied to solve BHCP on graph arising from regular one-dimensional and two-dimensional mesh. Numerical results show that we can obtain a stable solution using spectral graph wavelet regularization. Moreover, spectral graph wavelet adaptivity and fourth-order compact difference scheme lead to a very efficient solution.

Conservative and fourth-order compact difference schemes for the generalized Rosenau–Kawahara–RLW equation

In this article, we present two conservative and fourth-order compact finite-difference schemes for solving the generalized Rosenau–Kawahara–RLW equation. The proposed schemes are energy-conserved, convergent, and unconditionally stable, and the numerical convergence orders in both $$l_{2}$$ -norm and $$l_{\infty }$$ -norm are of $$O(\tau ^{2}+h^{4})$$ . Numerical experiments demonstrate that the present schemes are efficient and reliable.

Compact filtering as a regularization technique for a backward heat conduction problem

In this paper, the backward heat conduction problem is solved numerically using a fourth-order compact difference scheme. For the regularization of this ill-posed problem, fourth-order compact filtering which is a spatial filtering technique is proposed. The second derivative approximation in compact difference scheme has been compared with the fourth-order standard finite difference scheme via Fourier analysis. The comparison shows that the resolution ability of the compact difference scheme is better than the standard finite difference scheme. Meanwhile, an error estimate between the exact and regularized solution is derived. Numerical results are provided using the proposed method. The comparison of CPU time shows that the fourth-order compact difference scheme takes lesser CPU time than the fourth-order standard finite difference scheme.

Fourth-Order Compact Difference Scheme for the Backward Heat Conduction Problem

In this paper, the backward heat conduction problem is solved numerically using the fourth-order compact difference scheme. For regularization of this ill-posed problem, the quasi-reversibility technique is used. Compact finite difference approximation of the second derivative has been compared with the fourth-order standard finite difference approximation via Fourier analysis. The comparison shows that the resolution ability of the compact difference approximation is better than the standard finite difference approximation. The discrete dispersion relation for the compact difference scheme and finite difference scheme is obtained. Two test problems are considered for the numerical experiments. The CPU time taken by a fourth-order compact difference scheme is obtained and compared with the fourth-order standard finite difference scheme which shows that the fourth-order compact difference scheme takes lesser CPU time.

A reduced fourth-order compact difference scheme based on a proper orthogonal decomposition technique for parabolic equations

In this paper, we present a reduced high-order compact finite difference scheme for numerical solution of the parabolic equations. CFDS4 is applied to attain high accuracy for numerical solution of parabolic equations, but its computational efficiency still needs to be improved. Our approach combines CFDS4 with proper orthogonal decomposition (POD) technique to improve the computational efficiency of the CFDS4. The validation of the proposed method is demonstrated by four test problems. The numerical solutions are compared with the exact solutions and the solutions obtained by the CFDS4. Compared with CFDS4, it is shown that our method has greatly improved the computational efficiency without a significant loss in accuracy for solving parabolic equations.

A geometric full multigrid method and fourth-order compact scheme for the 3D Helmholtz equation on nonuniform grid discretization

A full multigrid method and fourth-order compact difference scheme is designed to solve the 3D Helmholtz equation on unequal mesh size. Three dimensional restriction and prolongation operators of the multigrid method on unequal grids could be constructed based on volume law. Two numerical experiments are implemented, and the results show the computational efficiency and accuracy of the full multigrid method. The study also illustrates that the full multigrid method with fourth-order compact difference scheme has great advantages in computation, which has been time consuming, and in iterative convergence efficiency.

A Fourth-Order Compact and Conservative Difference Scheme for the Generalized Rosenau-Korteweg De Vries Equation in Two Dimensions

A Fourth-Order Compact and Conservative Difference Scheme for the Generalized Rosenau-Korteweg De Vries Equation in Two Dimensions

High-accuracy numerical methods for a parabolic system in air pollution modeling

We present two approaches for enhancing the accuracy of the second-order finite difference approximations of two-dimensional semilinear parabolic systems. These are the fourth-order compact difference scheme and the fourth-order scheme based on Richardson extrapolation. Our interest is concentrated on a system of ten parabolic partial differential equations in air pollution modeling. We analyze numerical experiments to compare the two approaches with respect to accuracy, computational complexity, nonnegativity preserving, etc. The sixth-order approximation based on the fourth-order compact difference scheme combined with Richardson extrapolation is also discussed numerically.

High Order Compact Multigrid Solver for Implicit Solvation Models.

The electrostatic problem defined by the continuum solvation models used in molecular mechanics and ab initio molecular dynamics is solved in real space through multiscale methods. First, the Poisson equation is rewritten as a stationary convection-diffusion equation and discretized by a general mesh size fourth-order compact difference scheme. Then, the linear system associated with such a discrete version of the elliptic partial differential equation is solved by a parallel (geometric) multigrid solver whose convergence rates and robustness are improved by an iterant recombination technique in which the multigrid acts as a preconditioner of a Krylov subspace method. The numerical tests performed on ideal and physical systems described by linear Poisson equations under different boundary conditions show the good performance of this accelerated multigrid solver. Furthermore, nonlinear Poisson equations, like the regular modified Poisson-Boltzmann equation, can also be solved by using in addition simple iterative schemes.

Maximum norm error estimates of fourth-order compact difference scheme for the nonlinear Schr\xf6dinger equation involving a quintic term

A compact finite difference (CFD) scheme is presented for the nonlinear Schrödinger equation involving a quintic term. The two discrete conservative laws are obtained. The unconditional stability and convergence in maximum norm with order O({tau }^{2}+h^{4}) are proved by using the energy method. A numerical experiment is presented to support our theoretical results.

Collocation method based on shifted Chebyshev and radial basis functions with symmetric variable shape parameter for solving the parabolic inverse problem

ABSTRACTThis work introduces a new numerical solution to the inverse parabolic problem with source control parameter that has important applications in large fields of applied science. We expand the approximate solution of the inverse problem in terms of shifted Chebyshev polynomials in time and radial basis functions with symmetric variable shape parameter in space, with unknown coefficients. Unknown coefficient matrix determined using the collocation technique. Sample results show that the proposed method is very accurate. Moreover, the proposed method is compared with two other methods, fourth-order compact difference scheme and method of lines. Finally, we examine the stability of our method for the case where there is additive noise in input data.

Application of High-Order Compact Difference Scheme in the Computation of Incompressible Wall-Bounded Turbulent Flows

In the present work, a highly efficient incompressible flow solver with a semi-implicit time advancement on a fully staggered grid using a high-order compact difference scheme is developed firstly in the framework of approximate factorization. The fourth-order compact difference scheme is adopted for approximations of derivatives and interpolations in the incompressible Navier–Stokes equations. The pressure Poisson equation is efficiently solved by the fast Fourier transform (FFT). The framework of approximate factorization significantly simplifies the implementation of the semi-implicit time advancing with a high-order compact scheme. Benchmark tests demonstrate the high accuracy of the proposed numerical method. Secondly, by applying the proposed numerical method, we compute turbulent channel flows at low and moderate Reynolds numbers by direct numerical simulation (DNS) and large eddy simulation (LES). It is found that the predictions of turbulence statistics and especially energy spectra can be obviously improved by adopting the high-order scheme rather than the traditional second-order central difference scheme.