Efficient Alternating Direction Implicit Research Articles

BDF2 ADI orthogonal spline collocation method for the fractional integro-differential equations of parabolic type in three dimensions

In this paper, we are concerned with constructing a fast and an efficient alternating direction implicit (ADI) scheme for the fractional parabolic integro-differential equations (FPIDE) with a weakly singular kernel in three dimensions (3D). Our constructed scheme is based on a second-order backward differentiation formula (BDF2) for temporal discretization, orthogonal spline collocation (OSC) method for spatial discretization and a second-order fractional quadrature rule proposed by Lubich for the Riemann-Liouville fractional integral. The stability and convergence of the constructed numerical scheme are derived. Finally, some numerical examples are given to illustrate the accuracy and validity of the BDF2 ADI OSC method. Based on the obtained results, the numerical results are in line with the theoretical ones.

Reduction of edge beading defects on rectangular substrates using a rotation about an axis perpendicular to the spin axis

Here, we consider the coating dynamics of a binary liquid on a rectangular substrate that both spins and rotates about an axis perpendicular to the spin axis. This rotation causes not only a force parallel to the substrate but a large normal force on the horizontal region of the substrate, known as elevated gravity. This force not only smooths out any irregularities in the film, it also causes an increase in the evaporation rate, resulting in a higher viscosity, which in turn decreases the thinning of the coating due to centrifugal forces. We derive the lubrication form of the governing equations for such a geometry that includes surface tension and body forces acting on the liquid due to the spinning and rotation of the substrate, as well as viscosity changes and surface tension gradients due to evaporation of the volatile component. The resultant coupled nonlinear differential equations are solved numerically using an efficient alternating direction implicit method. The numerical results indicate that the rotation can reduce edge beading defects for parameters typically employed by the spin coating industry. It is particularly effective for small average coating thicknesses, where it can almost eliminate edge beading. Surface tension gradients, which act in the post rotation dry phase, can also further affect edge beading, particularly in the corner regions.

An efficient ADI difference scheme for the nonlocal evolution equation with multi-term weakly singular kernels in three dimensions

The paper constructs a fast efficient numerical scheme for the nonlocal evolution equation with three weakly singular kernels in three-dimensional space. In the temporal direction, We apply the backward Euler (BE) alternating direction implicit (ADI) method for the time derivative, simultaneously the first-order convolution quadrature formula is employed to deal with Riemann-Liouville (R-L) fractional integral term. In order to obtain a completely discrete implicit difference scheme, we use the standard central finite difference method (FDM) in space. The stability and convergence of the BE ADI difference scheme are proved rigorously with the convergence order in which h and τ are corresponding on the step size of space and time respectively. The ADI algorithm greatly reduces the computational cost of the three-dimensional problem. At last, several numerical results are given to verify that the numerical results are in agreement with our theoretical analysis.

The efficient ADI Galerkin finite element methods for the three-dimensional nonlocal evolution problem arising in viscoelastic mechanics

In this article, we investigate and analyze new methods for the numerical solution of the three-dimensional nonlocal evolution problem arising in viscoelastic mechanics. Then these methods combine Galerkin finite element methods (FEMs) for the spatial discretization with corresponding alternating direction implicit (ADI) algorithms, based on the backward Euler (BE) method and Crank-Nicolson (CN) method, respectively, from which, the Riemann-Liouville (R-L) integral term is approximated by relevant convolution quadrature rules. The $ L^2 $-norm stability and convergence of two ADI Galerkin schemes are proved by the energy argument. Numerical results confirm the predicted space-time convergence orders.

Efficiently energy-dissipation-preserving ADI methods for solving two-dimensional nonlinear Allen-Cahn equation

Over the years, energy-dissipation-preserving (EDP) numerical methods (EDP-NMs) for the nonlinear system of energy dissipation have attracted considerable attention because they are good at long-term computations. More recently, the invariant energy quadratization methods (IEQMs) have been proposed by Xiaofeng Yang and his collaborators to construct linearized EDP-NMs for the nonlinear system of energy dissipation, such as gradient flows. However, the system of the linear algebraic equations corresponding to the obtained EDP-NMs involves the variable coefficient matrices in general, thus resulting in complex computations. Besides, most of efficient alternating direction implicit (ADI) methods are not able to preserve the structures of continuous problem, such as, energy dissipation, energy conservation. To overcome these shortcomings, taking two-dimensional (2D) nonlinear Allen-Cahn equation for example, two energy-dissipation-preserving ADI finite difference methods (FDMs) have been developed by using IEQMs. Numerical analyses of them, such as, the discrete energy-dissipation laws, error estimations and stabilities, have been derived in detail. Numerical results support the theoretical analysis, and confirm the efficiency and accuracy of the proposed algorithms.

Efficient ADI schemes and preconditioning for a class of high-dimensional spatial fractional diffusion equations with variable diffusion coefficients

In this paper, alternating direction implicit (ADI) finite difference method and preconditioned Krylov subspace method are combined to solve a class of high-dimensional spatial fractional diffusion equations with variable diffusion coefficients. We prove the unconditional stability and convergence rate of the ADI finite difference method provided that the diffusion coefficients satisfy the given conditions. For the linear system in each spatial direction, we establish a circulant approximate inverse preconditioner to accelerate the Krylov subspace method. In addition, we also use matrix-free algorithms and fast Fourier transforms (FFT) to speed up the solution of linear systems. Numerical experiments show the utility of the ADI method and the preconditioner.

Second-order accurate, robust and efficient ADI Galerkin technique for the three-dimensional nonlocal heat model arising in viscoelasticity

This paper adopts an accurate and robust algorithm for the nonlocal heat equation having a weakly singular kernel in three dimensions. The proposed method approximates the unknown solution in two steps. First, the temporal discretization is achieved through a Crank-Nicolson finite difference with the nonuniform temporal meshes, which are applied to tackle the singularity behavior for the exact solution when t=0. Second, the spatial discretization is derived by means of the Galerkin finite element. Additionally, the alternating direction implicit (ADI) approach is adopted to decrease computational burden. The proposed method in terms of the convergence and stability analysis is studied by using the discrete energy method in detail with optimal rates of convergence O(k2+hr+1), r≥1 based on some appropriate assumptions, where k and h represent step sizes in the temporal and spatial directions, respectively. Numerical results highlight the effectiveness of the proposed method and the correctness of the theoretical prediction.

Efficient alternating direction implicit numerical approaches for multi-dimensional distributed-order fractional integro differential problems

Efficient alternating direction implicit numerical approaches for multi-dimensional distributed-order fractional integro differential problems

The efficient alternating direction implicit Galerkin method for the nonlocal diffusion-wave equation in three dimensions

The efficient alternating direction implicit Galerkin method for the nonlocal diffusion-wave equation in three dimensions

An efficient alternating direction implicit finite difference scheme for the three-dimensional time-fractional telegraph equation

In this work, an efficient alternating direction implicit (ADI) finite difference scheme is proposed to solve the three-dimensional time-fractional telegraph equation. The fully-discrete scheme is established via the L1 discrete formula in temporal direction and finite difference method in spatial direction. In order to reduce the computational cost of solving three-dimensional problems, an ADI algorithm is designed and applied. Then, via the energy method, we prove the stability and convergence of the scheme in L2 and H1 norms, respectively. Eventually, several numerical examples are given to validate the theoretical results.

A numerical method for two-dimensional multi-term time-space fractional nonlinear diffusion-wave equations

Recently, numerous numerical schemes have been developed for solving single-term time-space fractional diffusion-wave equations. Among them, some popular methods were constructed by using the graded meshes due to the solution with low temporal regularity. In this paper, we present an efficient alternating direction implicit (ADI) scheme for two-dimensional multi-term time-space fractional nonlinear diffusion-wave equations. Firstly, the considered problem is equivalently transformed into its partial integro-differential form with the Riemann-Liouville integral and multi-term Caputo derivatives. Secondly, the ADI scheme is constructed by using the first-order approximations and L1 approximations to approximate the terms in time, and using the fractional centered differences to discretize the multi-term Riesz fractional derivatives in space. Furthermore, the fast implement of the proposed ADI scheme is discussed by the sum-of-exponentials technique for both Caputo derivatives and Riemann-Liouville integrals. Then, the solvability, unconditional stability and convergence of the proposed ADI scheme are strictly established. Finally, two numerical examples are given to support our theoretical results, and demonstrate the computational performances of the fast ADI scheme.

An alternating direction Galerkin method for a time-fractional partial differential equation with damping in two space dimensions

In this paper, we propose an efficient alternating direction implicit (ADI) Galerkin method for solving the time-fractional partial differential equation with damping, where the fractional derivative is in the sense of Caputo with order in (1,2). The presented numerical scheme is based on the L2-1_{sigma} method in time and the Galerkin finite element method in space. The unconditional stability and convergence of the numerical scheme are both carefully proved. Numerical results are displayed for supporting the theoretical analysis.

Computational solutions of three-dimensional advection-diffusion equation using fourth order time efficient alternating direction implicit scheme

To develop an efficient numerical scheme for three-dimensional advection diffusion equation, higher order ADI method was proposed. 2nd and fourth order ADI schemes were used to handle such problem. Von Neumann stability analysis shows that Alternating Direction Implicit scheme is unconditionally stable. The accuracy and efficiency of such schemes was depicted by two test problems. Numerical results for two test problems were carried out to establish the performance of the given method and to compare it with the others Typical methods. Fourth order ADI method were found to be very efficient and stable for solving three dimensional Advection Diffusion Equation. The proposed methods can be implemented for solving non-linear problems arising in engineering and physics.

An Efficient 3-D Semivectorial Horizontally Wide-Angle Beam-Propagation Method

An efficient 3-D semivectorial horizontally wide-angle beam-propagation method (WA-BPM) based on the alternating direction implicit (ADI) scheme is proposed, where only the paraxial term in the vertical direction is considered. The present formulation reduces the splitting error to the second order and shows good compatibility with previously developed WA-BPMs. A block matrix decomposition process and a simulation window-cutting scheme are utilized to further improve the efficiency of the WA-BPM. Then, by applying Pade approximation to the resulting operator, an efficient ADI scheme that can be solved by the Thomas algorithm is constructed. Alternatively, the resulting operator can be approximated by the Denman–Beavers iterative scheme, leading to an ADI scheme with faster convergence rate and reasonable efficiency. Results from several numerical examples show the effectiveness of the present WA-BPM.

A three‐level linearized compact difference scheme for the Ginzburg–Landau equation

A high‐order finite difference method for the two‐dimensional complex Ginzburg–Landau equation is considered. It is proved that the proposed difference scheme is uniquely solvable and unconditionally convergent. The convergent order in maximum norm is two in temporal direction and four in spatial direction. In addition, an efficient alternating direction implicit scheme is proposed. Some numerical examples are given to confirm the theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 876–899, 2015

Numerical solutions for thin film flow down the outside and inside of a vertical cylinder

We consider a model for thin film flow down the outside and inside of a vertical cylinder. Our focus is to study the effect that the curvature of the cylinder has on the gravity-driven instability of the advancing contact line and to simulate the resulting fingering patterns that form due to this instability. The governing partial differential equation is fourth order with a nonlinear degenerate diffusion term that represents the stabilising effect of surface tension. We present numerical solutions obtained by implementing an efficient alternating direction implicit scheme. When compared to the problem of flow down a vertical plane, we find that increasing substrate curvature tends to increase the fingering instability for flow down the outside of the cylinder, whereas flow down the inside of the cylinder substrate curvature has the opposite effect. Further, we demonstrate the existence of nontrivial travelling wave solutions which describe fingering patterns that propagate down the inside of a cylinder at constant speed without changing form. These solutions are perfectly analogous to those found previously for thin film flow down an inclined plane.

Complex-envelope alternating-direction-implicit FDTD method for simulating active photonic devices with semiconductor/solid-state media

A complex-envelope (CE) alternating-direction-implicit (ADI) finite-difference time-domain (FDTD) approach to treat light-matter interaction self-consistently with electromagnetic field evolution for efficient simulations of active photonic devices is presented for the first time (to our best knowledge). The active medium (AM) is modeled using an efficient multilevel system of carrier rate equations to yield the correct carrier distributions, suitable for modeling semiconductor/solid-state media accurately. To include the AM in the CE-ADI-FDTD method, a first-order differential system involving CE fields in the AM is first set up. The system matrix that includes AM parameters is then split into two time-dependent submatrices that are then used in an efficient ADI splitting formula. The proposed CE-ADI-FDTD approach with AM takes 22% of the time as the approach of the corresponding explicit FDTD, as validated by semiconductor microdisk laser simulations.

Optimal and near–optimal advection–diffusion finite–difference schemes. VI Two–dimensional alternating directions

Computationally efficient alternating–direction implicit finite–difference schemes with high accuracy are derived for the numerical solution on a regular rectangular grid of the two–dimensional constant–coefficient diffusion equation with decay, advection, off–diagonal diffusivity and sources ∂ t c + λ c + u 1 ∂ x 1 c + u 2 ∂ x 2 c - k 11 ∂ x 1 2 c - 2 k 12 ∂ x 1 ∂ x 2 c - k 22 ∂ x 2 2 c = q . The 3×3×2 compact computational module involves three consecutive spatial points in each of the perpendicular spatial directions at two successive time levels. The schemes are tested for three variants of an exactly solvable test case, corresponding to the fractionation of an initial mixture of sinking, neutral and rising particles released together in a vertically well-mixed turbulent open-channel flow with mean current not aligned with either x 1 or x 2.

Efficient ADI and spline ADI methods for the steady‐state Navier‐Stokes equations

AbstractThe present paper provides an improved alternating direction implicit (ADI) technique as well as high‐order‐accurate spline ADI method for the numerical solution of steady two‐dimensional incompressible viscous flow problems. The vorticity‐stream function Navier‐Stokes equations are considered in a general curvilinear coordinate system, which maps an arbitrary two‐dimensional flow domain in the physical plane into a rectangle in the computational plane. The stream function equation is parabolized in time by means of a relaxation‐like time derivative and the steady state solution is obtained by a time‐marching ADI method requiring to solve only 2 × 2 block‐tridiagonal linear systems. The difference equations are written in incremental form; upwind differences are used for the incremental variables, for stability, whereas central differences approximate the non‐incremental terms, for accuracy, so that, at convergence, the solution is free of numerical viscosity and second‐order accurate. The high‐order‐accurate spline ADI technique proceeds in the same manner; in addition, at the end of each two‐sweep ADI cycle, the solution is corrected by means of a fifth‐order spline interpolating polynomial along each row and column of the computational grid, explicitly. The validity and the efficiency of the present methods are demonstrated by means of three test problems.

Numerical Technique for the Triple-Deck Problem

An efficient alternating direction implicit numerical technique is developed for solving the triple-deck fundamental problem for supersonic and subsonic flow. Flow past a parabolic hump on a flat plate is considered as a test case. Accuracy and reliability of the method are positively tested vs linearized equation results for a small hump height. Full nonlinear flow patterns are feasible, including mild separation. Numerical difficulties are encountered around the reattachment point for large hump heights.