High-order Compact Alternating Direction Implicit Research Articles

Numerical approximation of a two-dimensional fractional neutron diffusion model describing dynamics of neutron flux in a nuclear reactor

The goal of this work is to develop a high-order numerical technique to solve a two-dimensional fractional neutron diffusion (FND) model describing dynamical behavior of a lead-cooled fast reactor. This method is based on L1 technique for temporal discretization and a high-order compact Alternating Direction Implicit (ADI) finite difference scheme for the spatial discretization. Numerical simulations are conducted to demonstrate the applicability of the method. We calculate the neutronic flux in the core and examine how the order of the temporal-fractional derivative influences the neutronic flux. The results obtained with FND model for different values of anomalous diffusion coefficient are compared against those obtained with P1 model and classical neutron diffusion equation. In addition, the computational time (CPU time) is provided in order to justify the computational efficiency of the proposed method.

On high-order compact schemes for the multidimensional time-fractional Schr\xf6dinger equation

In this article, some high-order compact finite difference schemes are presented and analyzed to numerically solve one- and two-dimensional time fractional Schrödinger equations. The time Caputo fractional derivative is evaluated by the L1 and L1-2 approximation. The space discretization is based on the fourth-order compact finite difference method. For the one-dimensional problem, the rates of the presented schemes are of order O(tau ^{2-alpha }+h^{4}) and O(tau ^{3-alpha }+h^{4}), respectively, with the temporal step size τ and the spatial step size h, and alpha in (0,1). For the two-dimensional problem, the high-order compact alternating direction implicit method is used. Moreover, unconditional stability of the proposed schemes is discussed by using the Fourier analysis method. Numerical tests are performed to support the theoretical results, and these show the accuracy and efficiency of the proposed schemes.

Sharp error estimate of a compact L1-ADI scheme for the two-dimensional time-fractional integro-differential equation with singular kernels

A high-order compact alternating direction implicit scheme is considered to solve the two-dimensional time-fractional integro-differential equation with weak singularity near the initial time in this paper. The L1 formula and trapezoidal PI rule on nonuniform meshes, which greatly improve the temporal accuracy compared to the method on uniform grids, are adopted to approximate the Caputo derivative and the Riemann-Liouville integral, respectively. With the help of a modified discrete fractional Grönwall inequality and some crucial skills, the stability and convergence of the proposed scheme are analyzed. Numerical results confirm the sharpness of the error analysis.

A high-order compact alternating direction implicit method for solving the 3D time-fractional diffusion equation with the Caputo–Fabrizio operator

In this paper, a high-order compact finite difference method (CFDM) with an operator-splitting technique for solving the 3D time-fractional diffusion equation is considered. The Caputo–Fabrizio time operator is evaluated by the $$L_1$$ approximation, and the second-order space derivatives are approximated by the compact CFDM to obtain a discrete scheme. Alternating direction implicit method (ADI) is used to split the problem into three separate one-dimensional problems. The local truncation error analysis is discussed. Moreover, the convergence and stability of the numerical method are investigated. Finally, some numerical examples are presented to demonstrate the accuracy of the compact ADI method.

Exponential compact ADI method for a coupled system of convection-diffusion equations arising from the 2D unsteady magnetohydrodynamic (MHD) flows

In this paper, an exponential high-order compact alternating direction implicit (EHOC-ADI) difference method is developed for solving the coupled equations representing the unsteady incompressible, viscous magnetohydrodynamic (MHD) flow through a straight pipe of rectangular section. The method, in which the Crank-Nicolson scheme is used for the time discretization and an original EHOC difference scheme established for the steady 1D coupled system of convection-diffusion equations is used for the spatial discretization, is second order accurate in time and fourth-order accurate in space and requires only a regular five-point 2D stencil similar to that in the standard second-order methods and the three-point stencil for each 1D operator. A distinguishing desirable property of the proposed method is combining the computational efficiency of the lower order methods with superior accuracy inherent in high order approximations. The unconditional stable character of the method is verified by means of the discrete Fourier (or von Neumann) analysis. Numerical examples are carried out to illustrate and assess the performance and the accuracy of the method proposed currently. Computational results of the MHD flow in the 2D square-channel problems are presented for Hartmann numbers ranging from 0 to 106 and compared with the exact solutions and those obtained using other available methods in the literatures.

A new high-order compact ADI finite difference scheme for solving 3D nonlinear Schr\xf6dinger equation

In this paper, firstly, we solve the linear 3D Schrödinger equation using Douglas–Gunn alternating direction implicit (ADI) scheme and high-order compact (HOC) ADI scheme, which have the order O(tau^{2}+h^{2}) and O(tau^{2}+h^{4}), respectively. Secondly, a fourth-order compact ADI scheme, based on the Douglas–Gunn ADI scheme combined with second-order Strang splitting technique, is proposed for solving 3D nonlinear Schrödinger equation. Stability analysis has demonstrated that these schemes are unconditionally stable. Finally, numerical results show that these schemes preserve the conservation laws and provide accurate and stable solutions for the 3D linear and nonlinear Schrödinger equations.

Effect of the nodes near boundary points on the stability analysis of sixth-order compact finite difference ADI scheme for the two-dimensional time fractional diffusion-wave equation

In this paper, the aim is to present a high-order compact alternating direction implicit (ADI) scheme for the two-dimensional time fractional diffusion-wave (FDW) equation. The time fractional derivative which has been described in the Caputo’s sense is approximated by a scheme of order O(τ3−α), 1<α<2 and the space derivatives are discretized with a sixth-order compact procedure. The solvability, stability and H1 norm of the scheme are proved. Numerical results are provided to verify the accuracy and efficiency of the proposed method of solution. The sixth-order accuracy in the space directions has not been achieved in previously studied schemes.

High-order compact ADI method using predictor–corrector scheme for 2D complex Ginzburg–Landau equation

In this article, a high-order compact alternating direction implicit (HOC-ADI) finite difference scheme is applied to numerical solution of the complex Ginzburg–Landau (GL) equation in two spatial dimensions with periodical boundary conditions. The GL equation has been used as a mathematical model for various pattern formation systems in mechanics, physics, and chemistry. The proposed HOC-ADI method has fourth-order accuracy in space and second-order accuracy in time. To avoid solving the nonlinear system and to increase the accuracy and efficiency of the method, we proposed the predictor–corrector scheme. Validation of the present numerical solutions has been conducted by comparing with the exact and other methods results and evidenced a good agreement.

The study of a fourth-order multistep ADI method applied to nonlinear delay reaction–diffusion equations

In this paper, a high-order compact alternating direction implicit (HOC ADI) method, which combines fourth-order compact difference approximation to spatial derivatives and second order backward differentiation formula (BDF2) for temporal integration, is derived for nonlinear delay reaction–diffusion equations. By the discrete energy method, its optimal error estimates in L2- and H1-norms are constructively obtained. Then, a class of Richardson extrapolation algorithms (REAs) are established to improve computational efficiency. Besides, a modified HOC ADI solver is devised to reduce time cost as delay is very small. Numerical results confirm the theoretical results and performance of our algorithms.

A New High-Order Compact ADI Method for 3-D Unsteady Convection-Diffusion Problems with Discontinuous Coefficients

In this article, a fourth-order compact alternating direction implicit (ADI) method, based on the clasical Douglas-Gunn ADI method combined with Richardson's extrapolation technique, is proposed for solving 3-D unsteady convection-diffusion equations with discontinuous coefficients. To achieve fourth-order temporal accuracy, a correction term that ensures the realization of extrapolation and reduces the splitting error is introduced. The scheme has unconditional stability and can be solved by the Thomas algorithm. Numerical experiments, including continuous/discontinuous coefficients cases, are conducted to verify the robustness and the high accuracy of this new method.

Compact alternating direction implicit method to solve two-dimensional nonlinear delay hyperbolic differential equations

In this article, a high-order compact alternating direction implicit method combined with a Richardson extrapolation technique is developed to solve a class of two-dimensional nonlinear delay hyperbolic differential equations. The solvability, stability and convergence of the method are analysed simultaneously in L2- and H1-norms by the discrete energy method. Numerical experiments are provided to demonstrate the accuracy and efficiency of the schemes.

New High-Order Compact ADI Algorithms for 3D Nonlinear Time-Fractional Convection-Diffusion Equation

Numerical approximations of the three-dimensional (3D) nonlinear time-fractional convection-diffusion equation is studied, which is firstly transformed to a time-fractional diffusion equation and then is solved by linearization method combined with alternating direction implicit (ADI) method. By using fourth-order Padé approximation for spatial derivatives and classical backward differentiation method for time derivative, two new high-order compact ADI algorithms with ordersO(τmin(1+α,2−α)+h4)andO(τ2−α+h4)are presented. The resulting schemes in each ADI solution step corresponding to a tridiagonal matrix equation can be solved by the Thomas algorithm which makes the computation cost effective. Numerical experiments are shown to demonstrate the high accuracy and robustness of two new schemes.

Convergence analysis of high-order compact alternating direction implicit schemes for the two-dimensional time fractional diffusion equation

High-order compact finite difference method with operator-splitting technique for solving the two dimensional time fractional diffusion equation is considered in this paper. The Caputo derivative is evaluated by the L1 approximation, and the second order derivatives with respect to the space variables are approximated by the compact finite differences to obtain fully discrete implicit schemes. Alternating Direction Implicit (ADI) method is used to split the original problem into two separate one dimensional problems. One scheme is given by replacing the unknowns by the values on the previous level directly and a correction term is added for another scheme. Theoretical analysis for the first scheme is discussed. The local truncation error is analyzed and the stability is proved by the Fourier method. Using the energy method, the convergence of the compact finite difference scheme is proved. Numerical results are provided to verify the accuracy and efficiency of the two proposed algorithms. For the order of the temporal derivative lies in different intervals $\left(0,\frac{1}{2}\right)$ or $\left[\frac{1}{2},1\right)$ , corresponding appropriate scheme is suggested.

A rational high-order compact ADI method for unsteady convection–diffusion equations

Based on a fourth-order compact difference formula for the spatial discretization, which is currently proposed for the one-dimensional (1D) steady convection–diffusion problem, and the Crank–Nicolson scheme for the time discretization, a rational high-order compact alternating direction implicit (ADI) method is developed for solving two-dimensional (2D) unsteady convection–diffusion problems. The method is unconditionally stable and second-order accurate in time and fourth-order accurate in space. The resulting scheme in each ADI computation step corresponds to a tridiagonal matrix equation which can be solved by the application of the 1D tridiagonal Thomas algorithm with a considerable saving in computing time. Three examples supporting our theoretical analysis are numerically solved. The present method not only shows higher accuracy and better phase and amplitude error properties than the standard second-order Peaceman–Rachford ADI method in Peaceman and Rachford (1959) [4], the fourth-order ADI method of Karaa and Zhang (2004) [5] and the fourth-order ADI method of Tian and Ge (2007) [23], but also proves more effective than the fourth-order Padé ADI method of You (2006) [6], in the aspect of computational cost. The method proposed for the diffusion–convection problems is easy to implement and can also be used to solve pure diffusion or pure convection problems.

High-order compact ADI (HOC-ADI) method for solving unsteady 2D Schrödinger equation

In this paper, a high-order compact (HOC) alternating direction implicit (ADI) method is proposed for the solution of the unsteady two-dimensional Schrödinger equation. The present method uses the fourth-order Padé compact difference approximation for the spatial discretization and the Crank–Nicolson scheme for the temporal discretization. The proposed HOC-ADI method has fourth-order accuracy in space and second-order accuracy in time. The resulting scheme in each ADI computation step corresponds to a tridiagonal system which can be solved by using the one-dimensional tridiagonal algorithm with a considerable saving in computing time. Numerical experiments are conducted to demonstrate its efficiency and accuracy and to compare it with analytic solutions and numerical results established by some other methods in the literature. The results show that the present HOC-ADI scheme gives highly accurate results with much better computational efficiency.

A high-order ADI method for parabolic problems with variable coefficients

A high-order compact alternating direction implicit (ADI) method is proposed for solving two-dimensional (2D) parabolic problems with variable coefficients. The computational problem is reduced to sequence one-dimensional problems which makes the computation cost-effective. The method is easily extendable to multi-dimensional problems. Various numerical tests are performed to test its high-order accuracy and efficiency, and to compare it with the standard second-order Peaceman–Rachford ADI method. The method has been applied to obtain the numerical solutions of the lid-driven cavity flow problem governed by the 2D incompressible Navier–Stokes equations using the stream function-vorticity formulation. The solutions obtained agree well with other results in the literature.

A high-order compact ADI method for solving three-dimensional unsteady convection-diffusion problems

We derive a high-order compact alternating direction implicit (ADI) method for solving three-dimentional unsteady convection-diffusion problems. The method is fourth-order in space and second-order in time. It permits multiple uses of the one-dimensional tridiagonal algorithm with a considerable saving in computing time and results in a very efficient solver. It is shown through a discrete Fourier analysis that the method is unconditionally stable in the diffusion case. Numerical experiments are conducted to test its high order and to compare it with the standard second-order Douglas-Gunn ADI method and the spatial fourth-order compact scheme by Karaa. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006