Analysis of a compact multi-step ADI method for linear parabolic equation

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

In this paper, a novel compact alternating direction implicit (ADI) scheme is proposed for solving the time-fractional subdiffusion equation in two space dimensions. The established scheme is based on the modified L1 method in time and the compact finite difference method in space. The unique solvability, unconditionally stability and convergence of the scheme are proved. The derived compact ADI scheme is coincident with the one for 2D integer order parabolic equation when the , where is the order of the Riemann–Liouville derivative operator. In addition, the novel ADI scheme is used to solve the 2D modified fractional diffusion equation, and the corresponding stability and convergence results are also given. Numerical results are provided to verify the theoretical analysis.

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. AMS subject classifications: 35J60, 35J65, 65M99, 65N12, 65Z05, 74S10, 76R50, 80A20, 80M25

Compact alternating direction implicit method for two-dimensional time fractional diffusion equation

We propose a multigrid Alternating-Direction Implicit (ADI) method for solving Maxwell's equations in this paper. This method is based on the interpretation of the ADI method as an iterative solver of the Crank-Nicolson (CN) scheme. By introducing a special solving procedure for the residual equation within the ADI framework, multigrid schemes are incorporated into the iterative ADI method. The accuracy and efficiency of the proposed multigrid ADI method are further demonstrated by numerical examples.

Compact difference scheme for a class of fractional-in-space nonlinear damped wave equations in two space dimensions

In this paper, we study the high efficient numerical methods to solve a class of variable coefficient delay parabolic differential equations. We propose two types of multistep finite difference schemes while prove the solvability, convergence and stability of both schemes. The convergence orders are and , respectively, in the sense of -norm. By some new analytical transformations, we extend our schemes to solve a more general variable coefficient delay convection–diffusion–reaction equations and also apply them to the higher dimensional cases. Furthermore, multistep finite difference method and compact multistep finite difference method for two dimensional variable coefficient diffusion–reaction equations are proposed and the suitable alternate direction implicit (ADI) technique is constructed for the multistep finite difference scheme. At last, several numerical experiments are carried out to illustrate the effectiveness of both schemes.

The approximate-factorisation-splitting (CNAFS) method as an efficient implementation of the Crank-Nicolson scheme for solving the three-dimensional Maxwell equations in the time domain, using much less CPU time and memory than a direct implementation, is presented. At each time step, the CNAFS method solves tridiagonal matrices successively instead of solving a huge sparse matrix. It is shown that CNAFS is unconditionally stable and has much smaller anisotropy than the alternating-direction implicit (ADI) method, though the numerical dispersion is the same as in the ADI method along the axes. In addition, for a given mesh density, there will be one value of the Courant number at which the CNAFS method has zero anisotropy, whereas the Crank-Nicolson scheme always has anisotropy. Analysis shows that both ADI and CNAFS have time step-size limits to avoid numerical attenuation, although both are still unconditionally stable beyond their limit.

ABSTRACTThis paper presents a novel one‐parameter linearized compact alternating direction implicit (ADI) scheme for solving the 2D and 3D non‐linear parabolic equations with distributed delay, where the ADI technique is used to reduce the computational cost greatly. The unique solvability, error estimates in and norms and stability of the proposed scheme are studied, and it is shown under some appropriate conditions that the scheme is convergent of order two in temporal direction and order four in spatial direction. Some numerical experiments are conducted to validate the theoretical results as well as the effectiveness and accuracy of the derived schemes.

We consider the initial boundary value problem for a fractional evolution integro-differential equation with a weakly singular kernel arising from viscoelastic rods and membranes in a bounded smooth domain. In our numerical scheme, an arbitrary-order orthogonal spline collocation (OSC) method is used for spatial approximation and combined with the Crank-Nicolson (CN) alternating direction implicit (ADI) method for time approximation, and the integral term is approximated via the product integration (PI) rule. The detailed stability and convergence of the CN ADI OSC method in norm are proved. Finally, some numerical simulations are provided to illustrate our theoretical error estimates. In addition, the comparison between the non-uniform mesh and the uniform mesh shows the effectiveness of the non-uniform mesh when the solution is not sufficiently smooth.

Contaminant transport in a soil formation is described by advection dispersion equation. In this study, a horizontal and transversal contaminant transport along transient groundwater flow under non-linear sorption is solved numerically to examine the contaminant distribution profiles in finite soil media. The horizontal and transversal pore-water seepage velocities are defined as varying exponentially with time. Dispersion is considered directly proportional to the first power of the pore-water velocity. Axial input sources varying exponentially with time are assigned along the two-dimensional coordinate axes. For approximating the governing solute transport equation into algebraic equations, Crank–Nicolson (CN) and alternating direction implicit (ADI) methods are used. Both the numerical solutions are illustrated graphically with the help of computer software for various hydrological input data. In a special case, the numerical solutions are also compared with an approximate solution obtained by PDEtool. The comparison is performed with the help of contour plots. The CN method gives more accurate result than ADI method for the present model problem.

Numerical investigation of the parabolic mixed derivative diffusion equation via Alternating Direction Implicit methods

A 2D higher‐order alternating‐direction‐implicit (ADI) finite‐difference time‐domain method based on a compact scheme is presented in this paper. This compact ADI method improves the efficiency of computation by reducing the bandwidth of the matrix to be inversed from seven to five for the 4th‐order scheme. The method is proved to be unconditionally stable and its dispersion analysis is also presented. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 44: 273–275, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20609

A linearized compact ADI numerical method for the two-dimensional nonlinear delayed Schrödinger equation

A compact ADI scheme for two-dimensional fractional sub-diffusion equation with Neumann boundary condition