A comparative study of ADI splitting methods for parabolic equations in two space dimensions

The main purpose of the paper is a numerical comparison of three integration methods for semi-discrete parabolic partial differential equations in two space variables. Linear as well as nonlinear,equations are considered. The integration methods are the well-known ADI method of Peaceman and Rachford, a global extrapolation scheme of the classical ADI method to order four and a fourth order, four-step ADI splitting method.     

A high‐order ADI finite difference scheme for a 3D reaction‐diffusion equation with neumann boundary condition

In this article, we extend the fourth‐order compact boundary scheme in Liao et al. (Numer Methods Partial Differential Equations 18 (2002), 340–354) to a 3D problem and then combine it with the fourth‐order compact alternating direction implicit (ADI) method in Gu et al. (J Comput Appl Math 155 (2003), 1–17) to solve the 3D reaction‐diffusion equation with Neumann boundary condition. First, the reaction‐diffusion equation is solved with a compact fourth‐order finite difference method based on the Padé approximation, which is then combined with the ADI method and a fourth‐order compact scheme to approximate the Neumann boundary condition, to obtain fourth order accuracy in space. The accuracy in the temporal dimension is improved to fourth order by applying the Richardson extrapolation technique, although the unconditional stability of the numerical method is proved, and several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed new algorithm. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

An exponential high‐order compact ADI method for 3D unsteady convection–diffusion problems

In this article, we develop an exponential high order compact alternating direction implicit (EHOC ADI) method for solving three dimensional (3D) unsteady convection–diffusion equations. The method, which requires only a regular seven‐point 3D stencil similar to that in the standard second‐order methods, is second order accurate in time and fourth‐order accurate in space and unconditionally stable. The resulting EHOC ADI scheme in each alternating direction implicit (ADI) solution step corresponding to a strictly diagonally dominant matrix equation can be solved by the application of the one‐dimensional tridiagonal Thomas algorithm with a considerable saving in computing time. Numerical experiments for three test problems are carried out to demonstrate the performance of the present method and to compare it with the classical Douglas–Gunn ADI method and the Karaa's high‐order compact ADI method. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

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     

Convergence of compact ADI method for solving linear Schrödinger equations

A compact ADI scheme of second‐order in time and fourth‐order in space is proposed for solving linear Schrödinger equations with periodic boundary conditions. By using the recently suggested discrete energy method, it is shown that the stable compact ADI method is unconditionally convergent in the maximum norm. Numerical experiments, including the comparisons with the second‐order ADI scheme and the time‐splitting Fourier pseudospectral method, are presented to support the theoretical results and show the effectiveness of our method. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012     

Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations

Alternating direction implicit (ADI) schemes are computationally efficient and widely utilized for numerical approximation of the multidimensional parabolic equations. By using the discrete energy method, it is shown that the ADI solution is unconditionally convergent with the convergence order of two in the maximum norm. Considering an asymptotic expansion of the difference solution, we obtain a fourth‐order, in both time and space, approximation by one Richardson extrapolation. Extension of our technique to the higher‐order compact ADI schemes also yields the maximum norm error estimate of the discrete solution. And by one extrapolation, we obtain a sixth order accurate approximation when the time step is proportional to the squares of the spatial size. An numerical example is presented to support our theoretical results. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

A fourth-order compact ADI method for solving two-dimensional unsteady convection–diffusion problems

In this article, an exponential high-order compact (EHOC) alternating direction implicit (ADI) method, in which the Crank–Nicolson scheme is used for the time discretization and an exponential fourth-order compact difference formula for the steady-state 1D convection–diffusion problem is used for the spatial discretization, is presented for the solution of the unsteady 2D convection–diffusion problems. The method is temporally second-order accurate and spatially fourth order accurate, which requires only a regular five-point 2D stencil similar to that in the standard second-order methods. The resulting EHOC ADI scheme in each ADI solution step corresponds to a strictly diagonally dominant tridiagonal matrix equation which can be inverted by simple tridiagonal Gaussian decomposition and may also be solved by application of the one-dimensional tridiagonal Thomas algorithm with a considerable saving in computing time. The unconditionally stable character of the method was verified by means of the discrete Fourier (or von Neumann) analysis. Numerical examples are given to demonstrate the performance of the method proposed and to compare mostly it with the high order ADI method of Karaa and Zhang and the spatial third-order compact scheme of Note and Tan.     

交错网格紧致差分格式和满足等价性的压力Poisson方程

1．引言随着电子计算机的发展，越来越多的实际问题数值模拟成为现实，但还有很多非线性问题数值计算时间太长，内存要求过大．数值方法的改进可使计算量和存储量大大减少，例如，对二维非定常问题，要使误差达到N－4量级，二阶格式计算点数为（N2）3，（包括时间方向），而四阶格式计算点数仅为N3，差N3倍！而计算量差的倍数更多．当N＝16时N3＝4096，当N＝256时，N31678万．紧致差分格式具有精度高，差分式基点少，＜线性）稳定性好，对高频波分辨率高，边界差分点少等优点【’，‘，’，’。’，’，‘’］，本文中的格式基点数为3，…     

A high order finite difference method with Richardsonextrapolation for 3D convection diffusion equation

In this paper, we extend the Sun and Zhang’s [24] work on high order finite difference method, which is based on the Richardson extrapolation technique and an operator interpolation scheme for the one and two dimensional steady convection diffusion equations to the three dimensional case. Firstly, we employ a fourth order compact difference scheme to get the fourth order accurate solution on the fine and the coarse grids. Then, we use the Richardson extrapolation technique by combining the two approximate solutions to get a sixth order accurate solution on coarse grid. Finally, we apply an operator interpolation scheme to achieve the sixth order accurate solution on the fine grid. During this process, we use alternating direction implicit (ADI) method to solve the resulting linear systems. Numerical experiments are conducted to verify the accuracy and effectiveness of the present method.     

A fast numerical solution of the 3D nonlinear tempered fractional integrodifferential equation

In this paper, we investigate the numerical solution of the three-dimensional (3D) nonlinear tempered fractional integrodifferential equation which is subject to the initial and boundary conditions. The backward Euler (BE) method in association with the first-order convolution quadrature rule is employed to discretize this equation for time, and the Galerkin finite element method is applied for space, which is combined with an alternating direction implicit (ADI) algorithm, in order to reduce the computational cost for solving the three-dimensional nonlocal problem. Then a fully discrete BE ADI Galerkin finite element scheme can be obtained by linearizing the non-linear term. Thereafter we prove a positive-type lemma, from which the stability and convergence of the proposed numerical scheme are derived based on the energy method. Numerical experiments are performed to verify the effectiveness of the proposed approach.     

An improved numerical method for balanced truncation for symmetric second-order systems

We consider balanced truncation model order reduction for symmetric second-order systems. The occurring large-scale generalized and structured Lyapunov equations are solved with a specially adapted low-rank alternating directions implicit (ADI) type method. Stopping criteria for this iteration are investigated, and a new result concerning the Lyapunov residual within the low-rank ADI method is established. We also propose a goal-oriented stopping criterion which tries to incorporate the balanced truncation approach already during the ADI iteration. The model reduction approach using the ADI method with different stopping criteria is evaluated on several test systems.     

Nonlinear evolution equations for describing waves in bubbly liquids with viscosity and heat transfer consideration

Nonlinear evolution equations of the fourth order and its partial cases are derived for describing nonlinear pressure waves in a mixture liquid and gas bubbles. Influence of viscosity and heat transfer is taken into account. Exact solutions of nonlinear evolution equation of the fourth order are found by means of the simplest equation method. Properties of nonlinear waves in a liquid with gas bubbles are discussed.     

Compact ADI method for solving parabolic differential equations

A compact alternating direction implicit (ADI) method has been developed for solving two‐dimensional parabolic differential equations. In this study, the second‐order derivatives with respect to space are discretized using the high‐order compact finite differences. The Peaceman‐Rachford ADI method is then used for developing a new ADI scheme. It is shown by the discrete Fourier analysis that this new ADI scheme is unconditionally stable. The method can be generalized to the three‐dimensional case and an unconditionally stable compact Douglas ADI scheme is obtained. The method is illustrated by numerical examples. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 129–142, 2002; DOI 10.1002/num.1037     

计算高维带弱奇异核发展型方程的交替方向隐式欧拉方法

本文主要研究高维带弱奇异核的发展型方程的交替方向隐式(ADI)差分方法.向后欧拉(Euler)方法联立一阶卷积求积公式处理时间方向的离散,有限差分方法处理空间方向的离散,并进一步构造了ADI全离散差分格式.然后将二维问题延伸到三维问题,构造三维空间问题的ADI差分格式.基于离散能量法,详细证明了全离散格式的稳定性和误差分析.随后给出了2个数值算例,数值结果进一步验证了时间方向的收敛阶为一阶,空间方向的收敛阶为二阶,和理论分析结果一致.     

Alternating direction implicit OSC scheme for the two-dimensional fractional evolution equation with a weakly singular kernel

In this paper, a new kind of alternating direction implicit (ADI) Crank-Nicolson-type orthogonal spline collocation (OSC) method is formulated for the two-dimensional fractional evolution equation with a weakly singular kernel arising in the theory of linear viscoelasticity. The novel OSC method is used for the spatial discretization, and ADI Crank-Nicolson-type method combined with the second order fractional quadrature rule are considered for the temporal component. The stability of proposed scheme is rigourously established, and nearly optimal order error estimate is also derived. Numerical experiments are conducted to support the predicted convergence rates and also exhibit expected super-convergence phenomena.     

Improved King’s methods with optimal order of convergence based on rational approximations

In this paper, we present three-point and four-point methods for solving nonlinear equations. The methodology is based on King’s family of fourth order methods [R.F. King, A family of fourth order methods for nonlinear equations, SIAM J. Numer. Anal. 10 (1973) 876–879] and further developed by using rational function approximations. The three-point method requires four function evaluations and has the order of convergence eight, whereas the four-point method requires five function evaluations and has the order of convergence sixteen. Therefore, the methods are optimal in the sense of Kung–Traub hypothesis. The proposed schemes are compared with closest competitors in a series of numerical examples. Moreover, theoretical order of convergence is verified in the examples.     

IIM-based ADI finite difference scheme for nonlinear convection–diffusion equations with interfaces

Based on Li’s immersed interface method (IIM), an ADI-type finite difference scheme is proposed for solving two-dimensional nonlinear convection–diffusion interface problems on a fixed cartesian grid, which is unconditionally stable and converges with two-order accuracy in both time and space in maximum norm. Correction terms are added to the right-hand side of standard ADI scheme at irregular points. The nonlinear convection terms are treated by Adams–Bashforth method, without affecting the stability of difference schemes. A new method for computing the correction terms is developed, in which the Adams–Bashforth method is employed. Thus we can get an explicit approximation for the computation of corrections, when the jump condition is solution-dependent. Three numerical experiments are displayed and analyzed. The numerical results show good agreement with the exact solutions and confirm the convergence order.     

Polynomial spline approach for solving second-order boundary-value problems with Neumann conditions

In this paper, a new difference scheme based on quartic splines is derived for solving linear and nonlinear second-order ordinary differential equations subject to Neumann-type boundary conditions. The scheme can achieve sixth order accuracy at the interior nodal points and fourth order accuracy at and near the boundary, which is superior to the well-known Numerov’s scheme with the accuracy being fourth order. Convergence analysis of the present method for linear cases is discussed. Finally, numerical results for both linear and nonlinear cases are given to illustrate the efficiency of our method.     

Fourth-order compact solution of the nonlinear Klein-Gordon equation

In this work we propose a fourth-order compact method for solving the one-dimensional nonlinear Klein-Gordon equation. We apply a compact finite difference approximation of fourth-order for discretizing spatial derivative and a fourth-order A-stable diagonally-implicit Runge-Kutta-Nyström (DIRKN) method for the time integration of the resulting nonlinear second-order system of ordinary differential equations. The proposed method has fourth order accuracy in both space and time variables and is unconditionally stable. Numerical results obtained from solving several problems possessing periodic, kinks, single and double-soliton waves show that the combination of a compact finite difference approximation of fourth order and a fourth-order A-stable DIRKN method gives an efficient algorithm for solving these problems.     

Galerkin-Legendre spectral schemes for nonlinear space fractional Schrödinger equation

In the paper, we first propose a Crank-Nicolson Galerkin-Legendre (CN-GL) spectral scheme for the one-dimensional nonlinear space fractional Schrödinger equation. Convergence with spectral accuracy is proved for the spectral approximation. Further, a Crank-Nicolson ADI Galerkin-Legendre spectral method for the two-dimensional nonlinear space fractional Schrödinger equation is developed. The proposed schemes are shown to be efficient with second-order accuracy in time and spectral accuracy in space which are higher than some recently studied methods. Moreover, some numerical results are demonstrated to justify the theoretical analysis.