Fourth-Order Splitting Methods for Time-Dependant Differential Equations

This study was suggested by previous work on the simulation of evolution equations with scale-dependent processes,e.g.,wave-propagation or heat-transfer,that are modeled by wave equations or heat equations.Here,we study both parabolic and hyperbolic equations.We focus on ADI (alternating direction implicit) methods and LOD (locally one-dimensional) methods,which are standard splitting methods of lower order,e.g.second-order.Our aim is to develop higher-order ADI methods,which are performed by Richardson extrapolation,Crank-Nicolson methods and higher-order LOD methods,based on locally higher-order methods.We discuss the new theoretical results of the stability and consistency of the ADI methods.The main idea is to apply a higher- order time discretization and combine it with the ADI methods.We also discuss the dis- cretization and splitting methods for first-order and second-order evolution equations. The stability analysis is given for the ADI method for first-order time derivatives and for the LOD (locally one-dimensional) methods for second-order time derivatives.The higher-order methods are unconditionally stable.Some numerical experiments verify our results.     

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 class of one-parameter alternating direction implicit methods for two-dimensional wave equations with discrete and distributed time-variable delays

For solving the initial-boundary value problem of two-dimensional wave equations with discrete and distributed time-variable delays, in the present paper, we first construct a class of basic one-parameter methods. In order to raise the computational efficiency of this class methods, we remold the methods as one-parameter alternating direction implicit (ADI) methods. Under the suitable conditions, the remolded methods are proved to be stable and convergent of second order in both of time and space. With several numerical experiments, the computational effectiveness and theoretical exactness of the methods are confirmed. Moreover, it is illustrated that the proposed one-parameter ADI method has the better advantage in computational efficiency than the basic one-parameter methods.     

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.     

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     

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     

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.     

A fourth order ADI method for semidiscrete parabolic equations

A fourth order fourstep ADI method is described for solving the systems of ordinary differential equations which are obtained when a (nonlinear) parabolic initial-boundary value problem in two dimensions is semi-discretized. The local time-discretization error and the stability conditions are derived. By numerical experiments it is demonstrated that the (asymptotic) fourth order behaviour does not degenerate if the time step increases to relatively large values. Also a comparison is made with the classical ADI method of Peaceman and Rachford showing the superiority of the fourth order method in the higher accuracy region, particularly in nonlinear problems.     

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，…     

The modified alternating direction preconditioning method for the numerical solution of the elliptic self-adjoint second order and biharmonic equations

This paper introduces a generalised ADI method called the Modified Alternating Direction Preconditioning method (MADP method) for the numerical solution of the elliptic self-adjoint second order and biharmonic equations in a rectangle. The related theory of the MADP method is developed in detail and optimum values for the involved parameters are determined for both cases.     

三维热传导方程的紧交替方向差分格式

本文研究了三维热传导方程的紧交替方向隐式差分格式.利用算子方法导出了紧交替方向隐式差分格式,并利用Fourier分析方法证明了差分格式的收敛性和绝对稳定性,Richardson外推法外推一次得到具有O(T3+h6)阶精度的近似解.本文方法是对二维热传导方程问题的推广,同样适用于多维的情形.     

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     

Avoiding complex arithmetic in the low-rank ADI method efficiently

We present a new reformulation of the low-rank ADI method for solving large-scale Lyapunov equations which uses only real arithmetic operation and storage in the presence of complex shift parameters. This makes the method applicable on computing environments where complex computations and storage are not supported or not efficiently available. For generalized Lyapunov equations it is significantly more efficient than the older completely real formulation of the low-rank ADI as confirmed by numerical examples. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Comparing time integrators for parabolic equations in two space dimensions with a mixed derivative

A numerical comparison is made between three integration methods for semi-discrete parabolic partial differential equations in two space variables with a mixed derivative. Linear as well as non-linear equations are considered. The integration methods are the well-known one-step line hopscotch method, a four-step line hopscotch method, and a stabilized, explicit Runge-Kutta method.     

多体系统动力学微分／代数方程组的一类新的数值分析方法

本文讨论了多体系统动力学微分／代数混合方程组的数值离散问题．首先把参数t并入广义坐标讨论，简化了方程组及其隐含条件的结构，并将其化为指标1的方程组．然后利用方程组的特殊结构，引入一种局部离散技巧并构造了相应的算法．算法结构紧凑，易于编程，具有较高的计算效率和良好的数值性态，且其形式适合于各种数值积分方法的的实施．文末给出了具体算例．     

On inexact alternating direction implicit iteration for continuous Sylvester equations

In this paper, we study the alternating direction implicit (ADI) iteration for solving the continuous Sylvester equation AX + XB = C , where the coefficient matrices A and B are assumed to be positive semi‐definite matrices (not necessarily Hermitian), and at least one of them to be positive definite. We first analyze the convergence of the ADI iteration for solving such a class of Sylvester equations, then derive an upper bound for the contraction factor of this ADI iteration. To reduce its computational complexity, we further propose an inexact variant of the ADI iteration, which employs some Krylov subspace methods as its inner iteration processes at each step of the outer ADI iteration. The convergence is also analyzed in detail. The numerical experiments are given to illustrate the effectiveness of both ADI and inexact ADI iterations.     

A New Integrator for Special Third Order Differential Equations With Application to Thin Film Flow Problem

In recent time, Runge-Kutta methods that integrate special third order ordinary differential equations (ODEs) directly are proposed to address efficiency issues associated with classical Runge-Kutta methods. Albeit, the methods require evaluation of three set of equations to proceed with the numerical integration. In this paper, we propose a class of multistep-like Runge-Kutta methods (hybrid methods), which integrates special third order ODEs directly. The method is completely derivative-free. Algebraic order conditions of the method are derived. Using the order conditions, a four-stage method is presented. Numerical experiment is conducted on some test problems. The method is also applied to a practical problem in Physics and engineering to ascertain its validity. Results from the experiment show that the new method is more accurate and efficient than the classical Runge-Kutta methods and a class of direct Runge-Kutta methods recently designed for special third order ODEs.     

An LDLT factorization based ADI algorithm for solving large-scale differential matrix equations

Large-scale differential matrix equations appear in many applications like optimal control of partial differential equations, balanced truncation model order reduction of linear time varying systems etc. Here, we will focus on matrix Riccati differential equations (RDE). Solving such matrix valued ordinary differential equations (ODE) is a highly storage and time consuming process. Therefore, it is necessary to develop efficient solution strategies minimizing both. We present an LDL^T factorization based ADI method for solving algebraic Lyapunov equations (ALE) arising in the innermost iteration during the application of Rosenbrock ODE solvers to RDEs. We show that the LDL^T-type decomposition avoids complex arithmetic, as well as cancellation effects arising from indefinite right hand sides of the ALEs appearing in the classic ZZ^T based approach. Additionally, a certain number of linear system solves can be saved within the ADI algorithm by reducing the number of column blocks in the right hand sides while the full accuracy of the standard low-rank ADI is preserved. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

AN ADI GALERKIN METHOD WITH MOVING FINITE ELEMENT SPACES FOR A CLASS OF SECOND-ORDER HYPERBOLIC EQUATIONS

1　 IntroductionADI Galerkin methods were first formulated for the solution of nonlinear parabolic andlinear second-order hyperbolic problems on rectangular regions by Douglas and Dupont[1 ] .These methods combine alternating-direction method and Galerkin finite element methodtogether.So,they have the advantage of reducing the solution of a multidimensional problemto the solution of sets of independent one-dimensional problems,decreasing the amount ofcalculation,natural parallelism and highe…     

On the ADI method for Sylvester equations

This paper is concerned with the numerical solution of large scale Sylvester equations AX−XB=C, Lyapunov equations as a special case in particular included, with C having very small rank. For stable Lyapunov equations, Penzl (2000) [22] and Li and White (2002) [20] demonstrated that the so-called Cholesky factor ADI method with decent shift parameters can be very effective. In this paper we present a generalization of the Cholesky factor ADI method for Sylvester equations. An easily implementable extension of Penz’s shift strategy for the Lyapunov equation is presented for the current case. It is demonstrated that Galerkin projection via ADI subspaces often produces much more accurate solutions than ADI solutions.