ADI spectral collocation methods for parabolic problems

We discuss the Crank–Nicolson and Laplace modified alternating direction implicit Legendre and Chebyshev spectral collocation methods for a linear, variable coefficient, parabolic initial-boundary value problem on a rectangular domain with the solution subject to non-zero Dirichlet boundary conditions. The discretization of the problems by the above methods yields matrices which possess banded structures. This along with the use of fast Fourier transforms makes the cost of one step of each of the Chebyshev spectral collocation methods proportional, except for a logarithmic term, to the number of the unknowns. We present the convergence analysis for the Legendre spectral collocation methods in the special case of the heat equation. Using numerical tests, we demonstrate the second order accuracy in time of the Chebyshev spectral collocation methods for general linear variable coefficient parabolic problems.     

二维广义非线性Sine-Gordon方程的一个ADI格式

本文对二维广义非线性Sine-Gordon方程提出了一个带参数的ADI格式,其精度为O(τ2 h1),有效的降低了计算量,并证明了格式的稳定性与收敛性,最后通过参数的不同选取给出了数值算例,结果表明本文的格式是有效的和可靠的.     

A Study of Infrared Photonic Crystal Devices Using New ADI-FDTD Algorithm

To investigate the infrared photonic crystal devices numerically, the traditional finite-difference time-domain (FDTD) method has been modified by combining with a new alternating direction implicit (ADI) algorithm. An improvement of two-five in speed over previous FDTD methods can be obtained by calculating the envelope rather than the fast-varying field, and the numerical errors are minimized. Consider the isolated localized coupled-cavity modes, the phenomenon of eigenmode splitting has been observed when the coupled-cavity structures in two dimension triangular dielectric photonic crystals are simulated. The results are in good agreement with experiments.     

A minimum residual algorithm for solving linear systems

A minimum residual algorithm for solving a large linear system (I+S)x=b, with b∈ℂ ⁿ and S∈ℂ ^n×n being readily invertible, is proposed. For this purpose Krylov subspaces are generated by applying S and S ^-1 cyclically. The iteration can be executed with every linear system once the coefficient matrix has been split into the sum of two readily invertible matrices. In case S is a translation and a rotation of a Hermitian matrix, a five term recurrence is devised. In memory of Germund Dahlquist (1925–2005).AMS subject classification (2000) 65F10     

Numerical solution of three‐dimensional velocity–vorticity Navier–Stokes equations by finite difference method

This paper describes the finite difference numerical procedure for solving velocity–vorticity form of the Navier–Stokes equations in three dimensions. The velocity Poisson equations are made parabolic using the false‐transient technique and are solved along with the vorticity transport equations. The parabolic velocity Poisson equations are advanced in time using the alternating direction implicit (ADI) procedure and are solved along with the continuity equation for velocities, thus ensuring a divergence‐free velocity field. The vorticity transport equations in conservative form are solved using the second‐order accurate Adams–Bashforth central difference scheme in order to assure divergence‐free vorticity field in three dimensions. The velocity and vorticity Cartesian components are discretized using a central difference scheme on a staggered grid for accuracy reasons. The application of the ADI procedure for the parabolic velocity Poisson equations along with the continuity equation results in diagonally dominant tri‐diagonal matrix equations. Thus the explicit method for the vorticity equations and the tri‐diagonal matrix algorithm for the Poisson equations combine to give a simplified numerical scheme for solving three‐dimensional problems, which otherwise requires enormous computational effort. For three‐dimensional‐driven cavity flow predictions, the present method is found to be efficient and accurate for the Reynolds number range 100?Re?2000. Copyright © 2004 John Wiley & Sons, Ltd.     

A multi-block ADI finite-volume method for incompressible Navier–Stokes equations in complex geometries

An efficient second-order accurate finite-volume method is developed for a solution of the incompressible Navier–Stokes equations on complex multi-block structured curvilinear grids. Unlike in the finite-volume or finite-difference-based alternating-direction-implicit (ADI) methods, where factorization of the coordinate transformed governing equations is performed along generalized coordinate directions, in the proposed method, the discretized Cartesian form Navier–Stokes equations are factored along curvilinear grid lines. The new ADI finite-volume method is also extended for simulations on multi-block structured curvilinear grids with which complex geometries can be efficiently resolved. The numerical method is first developed for an unsteady convection–diffusion equation, then is extended for the incompressible Navier–Stokes equations. The order of accuracy and stability characteristics of the present method are analyzed in simulations of an unsteady convection–diffusion problem, decaying vortices, flow in a lid-driven cavity, flow over a circular cylinder, and turbulent flow through a planar channel. Numerical solutions predicted by the proposed ADI finite-volume method are found to be in good agreement with experimental and other numerical data, while the solutions are obtained at much lower computational cost than those required by other iterative methods without factorization. For a simulation on a grid with O(10⁵) cells, the computational time required by the present ADI-based method for a solution of momentum equations is found to be less than 20% of that required by a method employing a biconjugate-gradient-stabilized scheme.     

Fourth-order alternating direction implicit compact finite difference schemes for two-dimensional Schrödinger equations

In this paper, alternating direction implicit compact finite difference schemes are devised for the numerical solution of two-dimensional Schrödinger equations. The convergence rates of the present schemes are of order O(h⁴+τ²). Numerical experiments show that these schemes preserve the conservation laws of charge and energy and achieve the expected convergence rates. Representative simulations show that the proposed schemes are applicable to problems of engineering interest and competitive when compared to other existing procedures.     

An ADI Petrov–Galerkin method with quadrature for parabolic problems

We propose and analyze a fully discrete Laplace modified alternating direction implicit quadrature Petrov–Galerkin (ADI‐QPG) method for solving parabolic initial‐boundary value problems on rectangular domains. We prove optimal order convergence results for a restricted class of the associated elliptic operator and demonstrate accuracy of our scheme with numerical experiments for some parabolic problems with variable coefficients.© 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Efficient high-order immersed interface methods for heat equations with interfaces

An efficient high-order immersed interface method （IIM） is proposed to solve two-dimensional （2D） heat problems with fixed interfaces on Cartesian grids, which has the fourth-order accuracy in the maximum norm in both time and space directions. The space variable is discretized by a high-order compact （HOC） difference scheme with correction terms added at the irregular points. The time derivative is integrated by a Crank-Nicolson and alternative direction implicit （ADI） scheme. In this case, the time accuracy is just second-order. The Richardson extrapolation method is used to improve the time accuracy to fourth-order. The numerical results confirm the convergence order and the efficiency of the method.     

A backward Euler alternating direction implicit difference scheme for the three‐dimensional fractional evolution equation

A backward Euler alternating direction implicit (ADI) difference scheme is formulated and analyzed for the three‐dimensional fractional evolution equation. In our method, the Riemann‐Liouville fractional integral term is treated by means of first order convolution quadrature suggested by Lubich. Meanwhile, an ADI technique is adopted to reduce the multidimensional problem to a series of one‐dimensional problems. A fully discrete difference scheme is constructed with space discretization by finite difference method. Two new inner products and corresponding norms are defined to analyze the scheme. The verification of stability and convergence is based on the nonnegative character of the real quadratic form associated with the convolution quadrature. Numerical experiments are reported to demonstrate the efficiency of our scheme.