Alternating Direction Implicit Algorithm Research Articles

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.

Error analysis of fast L1 ADI finite difference/compact difference schemes for the fractional telegraph equation in three dimensions

This article proposes the fast L1 alternating direction implicit (ADI) finite difference and compact difference schemes to solve the fractional telegraph equation in three-dimensional space. The fully-discrete fast L1 ADI finite difference scheme can be established via the fast L1 formula for the approximation of mixed Caputo fractional derivatives and the central difference formula for the approximation of the spatial derivative term, then from which an ADI algorithm is designed to reduce three-dimensional problems to a series of one-dimensional problems. We add the corresponding compact operators in all directions of the space to get the fully-discrete L1 ADI compact difference scheme. Then the convergence in L2 and H1 norms of two ADI schemes is derived via energy method. Eventually, numerical experiments are carried out to verify the theoretical estimates.

Fast BDF2 ADI methods for the multi-dimensional tempered fractional integrodifferential equation of parabolic type

In this paper, we consider the numerical solutions of the multi-dimensional tempered fractional integrodifferential equation. First, the second-order backward differentiation formula (BDF2) and the second-order convolution quadrature rule are utilized for the temporal discretization, and the finite difference method (FDM) and alternating direction implicit (ADI) technique are used for the spatial discretization, in which the ADI algorithm is mainly employed to reduce the computational cost of high-dimensional non-local problems. Then, the fully discrete BDF2 ADI difference scheme for multi-dimensional tempered non-local problems can be obtained. After that, the stability and convergence of the BDF2 ADI difference scheme in multi-dimensional cases are proved by means of the energy method. In the numerical experiments, some examples validate the theoretical analysis.

A fast dynamic model for a large scale heat pipe embedded latent heat thermal energy storage system for optimal sizing and control

One of the challenges to design and control phase change material (PCM) based latent heat thermal energy storage (LHTES) systems is to develop fast models to accurately represent their transient and nonlinear behaviors. In this paper, a fast explicit modeling approach for a multi-dimensional finned PCM system, called alternating front propagating, is proposed based on the concept of the alternating-direction implicit algorithm for multi-dimensional parabolic and elliptic differential equations. The proposed modeling method is employed to develop a fast explicit dynamic model for a large scale LHTES system, whose heat transfer is enhanced by imbedded circular finned heat pipes. The developed fast model has negligible computational cost and is of sufficient accuracy compared to high-fidelity numerical solutions. The fast explicit model of the LHTES system was employed in a model predictive control framework to determine the HTF flowrate so that its outlet temperature meets a target value. The controlled HTF flowrate can increase the usage efficiency of the stored heat in a LHTES system.

Novel higher order convolutional perfectly matched layer based on complex envelope ADI‐FDTD algorithm for truncating the unmagnetized plasma

An improved higher order convolutional perfectly matched layer (HO-CPML) algorithm is proposed, which combines the complex envelope (CE) method and the alternating direction implicit (ADI) method. This algorithm is solved by the auxiliary differential equation method in the unmagnetized plasma, so as to simulate the unconditionally stable finite-difference-time-domain (FDTD). The novel HO-CPML with fractional decomposition method different from the previous one is adopted to truncate the unmagnetized plasma, making full use of the characteristics of the HO-CPML. The numerical example in plasma simulation shows the characteristics of the algorithm directly. According to the simulation results, the algorithm is effective, and the absorption effect is better than the conventional ADI (CL-ADI) algorithm and the first order CPML (FO-CPML) algorithm.

The formally second-order BDF ADI difference/compact difference scheme for the nonlocal evolution problem in three-dimensional space

This work formulates two kinds of alternating direction implicit (ADI) schemes for the parabolic-type three-dimensional evolution equation with a weakly singular kernel. The second-order backward differentiation formula (BDF2) and the second-order convolution quadrature (CQ) technique are applied to the discretization of the time derivative and the Riemann-Liouville (R-L) integral, respectively. Then, the fully-discrete BDF2 difference scheme and BDF2 compact difference scheme are constructed via the general centered difference and compact difference method, respectively. Meanwhile, the ADI algorithms are designed reasonably for two schemes to reduce the computational cost. The stability and convergence of two ADI schemes are derived via the energy method. Finally, several numerical examples are provided and tested to validate the theoretical analysis.

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 fast ADI algorithm for nonlinear Poisson equation in heterogeneous dielectric media

Recently, a nonlinear Poisson equation has been introduced to model nonlinear and nonlocal hyperpolarization effects in electrostatic solute-solvent interaction for biomolecular solvation analysis. Due to a strong nonlinearity associated with the heterogeneous dielectric media, this Poisson model is difficult to solve numerically, particularly for large protein systems. A new pseudo-transient continuation approach is proposed in this paper to efficiently and stably solve the nonlinear Poisson equation. A Douglas type alternating direction implicit (ADI) method is developed for solving the pseudo-time dependent Poisson equation. Different approximations to the dielectric profile in heterogeneous media are considered in the standard finite difference discretization. The proposed ADI scheme is validated by considering benchmark examples with exact solutions and by solvation analysis of real biomolecules with various sizes. Numerical results are in good agreement with the theoretical prediction, experimental measurements, and those obtained from the boundary value problem approach. Since the time stability of the proposed ADI scheme can be maintained even using very large time increments, it is efficient for electrostatic analysis involving hyperpolarization effects.

Improved Alternating Direction Implicit Method

The alternating direction implicit method (ADI) is a common classical numerical method that was first introduced to solve the heat equation in two or more spatial dimensions and can also be used to solve parabolic and elliptic partial differential equations as well. In this paper, We introduce an improvement to the alternating direction implicit (ADI) method to get an equivalent scheme to Crank-Nicolson differences scheme in two dimensions with the main feature of ADI method. The new scheme can be solved by similar ADI algorithm with some modifications. A numerical example was provided to support the theoretical results in the research.

Three-Dimensional Higher Order PML Based on Alternating Direction Implicit Algorithm

Based on the alternating direction implicit finite-difference time-domain (ADI-FDTD) algorithm, a higher order perfectly matched layer (HO-PML) is proposed to solve the problems in a three-dimensional computational domain. The proposed scheme can not only take advantage of the HO-PML for improving the absorbing performance but also has the advantage of the ADI-FDTD algorithm in terms of the unconditional stability. A dielectric resonator antenna example is carried out to validate its effectiveness. The results show that the Courant–Friedrich–Levy limit can be eliminated and the absorbing performance can be further enhanced.

An inexact alternating direction method of multipliers for the solution of linear complementarity problems arising from free boundary problems

A large number of free boundary problems can be formulated as linear-complementarity problems. In this paper, we propose an inexact alternating direction method of multipliers for solving linear complementarity problem arising from free boundary problems by using the special structure of these problems. The convergence of our proposed method is proved. Numerical results show that the proposed method is feasible and effective, and it is significantly faster than modified alternating direction implicit algorithm and many other methods, especially when dimension of the problem being solved is large.

Unconditionally Stable Divergence-Free Vector-Based Meshless Method for Transient Electromagnetic Analysis

A vector-based meshless method or vector meshless method, which is proven to be divergence-free, was recently proposed for transient electromagnetic analysis. In this method, an explicit finite-difference scheme is used for approximating the time derivatives in Maxwell’s equations. Therefore, the stability criterion constraints the maximum time-step. This paper presents the alternating-direction implicit (ADI) formulation of the vector meshless method based on a different form of the ADI algorithm named the leapfrog ADI scheme. The computational cost of the leapfrog ADI scheme is less than the conventional ADI algorithm and algebraically equivalent. In the proposed vector meshless method, Courant–Friedrichs–Lewy limit does not constrain the time-step due to its implicit formulation. Moreover, the stability condition of the conventional vector meshless method and the proposed ADI vector meshless method are investigated through their amplification matrices in this paper. The results show that the proposed method has two features: unconditional stability and divergence-free property. The first one is obtained due to the incorporation of the leapfrog ADI algorithm, and the second one is achieved by the use of vector basis functions. The divergence-free property of the proposed method can lead to obtaining more accurate solutions than the conventional meshless method. In addition, some numerical examples are presented to verify the properties and the efficiency of the proposed method.

Drop formation in a magnetic fluid coating a horizontal cylinder carrying an axial electric current

We consider the effect of a magnetic field generated by a current flowing in the axial direction, in a cylindrical conducting medium on the evolution of a thin magnetic liquid film coating the outside of the cylinder. We first derive the new governing equations when a body force, due to the interaction of the magnetic dipoles in the liquid with the magnetic field due to the current in the cylinder, is included with surface tension and gravitational forces. The equations are discretized and solved numerically using an alternating direction implicit algorithm. Simulations demonstrate that the transition from a uniform coating to the formation of undulations to a final configuration of distinct drops follows a similar evolution for a wide range of cylinder radii and magnetic field strengths. Magnetic forces generally oppose the effects of gravity, and consequently the drainage toward the bottom of the cylinder is slowed, the characteristic time for drop formation is delayed, and the final coating is not confined to the cylinder bottom, but can wrap around the entire circumference of cylinder for sufficiently large magnetic fields. The range of dimensionless parameters considered are limited by the physicochemical properties of an ester ferrofluid, the maximum current density and radius of the conducting cylinder, and the susceptibility χ, where there is a linear relationship between H and M. Thus the results are applicable to actual physical situations.

Nonparametric Model Calibration for Derivatives

Consistently fitting vanilla option surface is an important issue in derivative modelling. In this paper, we consider three different models: local and stochastic volatility, local correlation, hybrid local volatility with stochastic rates, and address their exact, nonparametric calibration. This calibration process requires solving a nonlinear partial integro-differential equation. A modified alternating direction implicit algorithm is used, and its theoretical and numerical analysis is performed.

Comprehensive modeling of structural modification induced by a femtosecond laser pulse inside fused silica glass

A comprehensive theoretical model is proposed based on equations describing the nonlinear propagation of an ultrashort pulse inside transparent material, electron density evolution, non-Fourier heat conduction, and thermo-elasto plastic displacement which are respectively solved by various methods. These methods include the split-step finite difference technique and alternating-direction implicit algorithm, fourth-order Range–Kutta algorithm, hybrid finite-element method/finite-difference method, and finite-element method in both space and time to achieve refractive index changes. The whole chain of processes occurring in the interaction of a focused ultrashort laser pulse with fused silica glass in prevalent conditions of micromachining applications is numerically investigated. By optimizing the numerical method and by using an adaptive mesh approach, the execution time of the program is significantly reduced so that the calculations are done at each time step in a fraction of a second. Simulation results show that the energy and duration of the input pulse are very important parameters in induced changes, but the chirp of the input pulse is not an effective parameter. Consequently, by appropriate setting of those parameters one can design a desired refractive index profile.

Higher-order in time “quasi-unconditionally stable” ADI solvers for the compressible Navier–Stokes equations in 2D and 3D curvilinear domains

This paper introduces alternating-direction implicit (ADI) solvers of higher order of time-accuracy (orders two to six) for the compressible Navier-Stokes equations in two- and three-dimensional curvilinear domains. The higher-order accuracy in time results from 1) An application of the backward differentiation formulae time-stepping algorithm (BDF) in conjunction with 2) A BDF-like extrapolation technique for certain components of the nonlinear terms (which makes use of nonlinear solves unnecessary), as well as 3) A novel application of the Douglas-Gunn splitting (which greatly facilitates handling of boundary conditions while preserving higher-order accuracy in time). As suggested by our theoretical analysis of the algorithms for a variety of special cases, an extensive set of numerical experiments clearly indicate that all of the BDF-based ADI algorithms proposed in this paper are "quasi-unconditionally stable" in the following sense: each algorithm is stable for all couples $(h,\Delta t)$ of spatial and temporal mesh sizes in a problem-dependent rectangular neighborhood of the form $(0,M_h)\times (0,M_t)$. In other words, for each fixed value of $\Delta t$ below a certain threshold, the Navier-Stokes solvers presented in this paper are stable for arbitrarily small spatial mesh-sizes. The second order formulation has further been rigorously shown to be unconditionally stable for linear hyperbolic and parabolic equations in two-dimensional space. Although implicit ADI solvers for the Navier-Stokes equations with nominal second-order of temporal accuracy have been proposed in the past, the algorithms presented in this paper are the first ADI-based Navier-Stokes solvers for which second order or better accuracy has been verified in practice under non-trivial (non-periodic) boundary conditions.

A central compact scheme for numerical solution of two-phase incompressible flow using Allen–Cahn phase field model

This article presents a numerical method for the Allen–Cahn phase field model coupled with the incompressible Navier–Stokes equations for simulating two-phase incompressible flow. The numerical method is based on the artificial compressibility method with dual-time stepping technique applied to the governing equations written in conservative form. The convective terms are approximated by using an eighth-order central compact scheme with filtering while the viscous terms are approximated by a sixth-order central compact scheme. Approximate factorization-based alternating direction implicit algorithm is used to solve the discretized linear system. The effectiveness of the method is demonstrated by computing several benchmark two-phase incompressible flow problems.

Dynamics with Matrices Possessing Kronecker Product Structure

In this paper we present an application of Alternating Direction Implicit (ADI) algorithm for solution of non-stationary PDE-s using isogeometric finite element method. We show that ADI algorithm has a linear computational cost at every time step. We illustrate this approach by solving two example non-stationary three-dimensional problems using explicit Euler and Newmark time-stepping scheme: heat equation and linear elasticity equations for a cube. The stability of the simulation is controlled by monitoring the energy of the solution.

Real-time Heat Transfer Model Based on Variable Non-uniform Grid for Dynamic Control of Continuous Casting Billets

A real-time heat transfer model has been developed for dynamic control of continuous casting billets. In order to fulfill the real-time requirements for efficiency and accuracy, finite volume method (FVM) and alternating direction implicit algorithm (ADI) have been selected as the efficient numerical solution algorithm. And most important of all, variable non-uniform grid and variable time step have been adopted in the algorithm, accelerating the calculation of heat transfer model by 20–40 times. Further, the algorithm’s discretization parameters including the grid, time step and slice distance have been optimized by error control (< 3°C), improving the relative calculation time to 0.57. The real-time model has been calibrated by surface temperature measurements using a thermal infrared imager, and its online performance has been tested, within ±13°C of the measurements. After the calibration, the model has been applied to the dynamic control of secondary cooling and the dynamic control of the electric current of final electromagnetic stirring (FEMS), providing the surface temperatures and mushy zone radius at the installation position of FEMS. The latter has been validated by shell-thickness measurements using nail-shooting (relative error< 2.3%). The model-based dynamic control systems controlled the surface temperatures and the flow in mushy zone precisely with changing casting conditions, and have improved the billet quality obviously and reduced the fractures of rolling line during wire-drawing to 1/6 times per month in average from 2–3 times before.

A fast alternating direction implicit algorithm for geometric flow equations in biomolecular surface generation

In this paper, a new alternating direction implicit (ADI) method is introduced to solve potential driven geometric flow PDEs for biomolecular surface generation. For such PDEs, an extra factor is usually added to stabilize the explicit time integration. However, two existing implicit ADI schemes are also based on the scaled form, which involves nonlinear cross derivative terms that have to be evaluated explicitly. This affects the stability and accuracy of these ADI schemes. To overcome these difficulties, we propose a new ADI algorithm based on the unscaled form so that cross derivatives are not involved. Central finite differences are employed to discretize the nonhomogenous diffusion process of the geometric flow. The proposed ADI algorithm is validated through benchmark examples with analytical solutions, reference solutions, or literature results. Moreover, quantitative indicators of a biomolecular surface, including surface area, surface-enclosed volume, and solvation free energy, are analyzed for various proteins. The proposed ADI method is found to be unconditionally stable and more accurate than the existing ADI schemes in all tests. This enables the use of a large time increment in the steady state simulation so that the proposed ADI algorithm is very efficient for biomolecular surface generation.