Direction Implicit Algorithm Research Articles

A mass-conservative higher-order ADI method for solving unsteady convection\u2013diffusion equations

In the paper, a high-order alternating direction implicit (ADI) algorithm is presented to solve problems of unsteady convection and diffusion. The method is fourth- and second-order accurate in space and time, respectively. The resulting matrix at each ADI computation can be obtained by repeatedly solving a penta-diagonal system which produces a computationally cost-effective solver. We prove that the proposed scheme is mass-conserved and unconditionally stable by means of discrete Fourier analysis. Numerical experiments are performed to validate the mass conservation and illustrate that the proposed scheme is accurate and reliable for convection-dominated problems.

An Artificial Compressibility Method for 3D Phase-Field Model and its Application to Two-Phase Flows

In this work, a numerical scheme based on artificial compressibility formulation of a phase-field model is developed for simulating two-phase incompressible flow problems. The coupled nonlinear systems composed of the incompressible Navier–Stokes equations and volume preserving Allen–Cahn-type phase-field equation are recast into conservative form with source terms, which are suited to implement high-resolution schemes originally developed for hyperbolic conservation laws. The Boussinesq approximation is used to account for the buoyancy effect in flow with small density difference. The fifth-order weighted essentially nonoscillatory (WENO) scheme is used for discretizing the convective terms while dual-time stepping (DTS) technique is used for obtaining time accuracy at each physical time step. Beam–Warming approximate factorization scheme is utilized to obtain block tridiagonal system of equations in each spatial direction. The alternating direction implicit (ADI) algorithm is used to solve the resulting system of equations. The performance of the method is demonstrated by its application to some 2D and 3D benchmark viscous two-phase flow problems.

APPLICATIONS OF ALTERNATING DIRECTION SOLVER FOR SIMULATIONS OF TIME-DEPENDENT PROBLEMS

This paper deals with application of Alternating Direction solver (ADS) to nonstationarylinear elasticity problem solved with isogeometric FEM. Employingtensor product B-spline basis in isogeometric analysis under some restrictionsleads to system of linear equations with matrix possessing tensor product structure.Alternating Direction Implicit algorithm is a direct method that exploitsthis structure to solve the system in O (N ), where N is a number of degreesof freedom (basis functions). This is asymptotically faster than state-of-theartgeneral purpose multi-frontal direct solvers. In this paper we also presentthe complexity analysis of ADS incorporating dependence on order of B-splinebasis.

A novel numerical model for billet reheating furnace

A numerical model of billet reheating furnace is proposed, which includes heat fluxes calculation around four billet surfaces and two-dimensional conduction calculation inside billet. Radiation and convection heat fluxes on top and bottom surfaces are calculated simultaneously, based on quartic and linear difference between furnace gas and billet surface temperatures, while furnace gas temperature is determined according to thermocouple values along furnace length together with billet surface temperature. Lateral fluxes are also calculated considering angle factor on billets interval. Two-dimensional partial differential equation is acquired for billet conduction to determine temperature distribution, which is discretised and solved by Alternating Direction Implicit and TriDiagonal Matrix Algorithm. Two embedded thermocouple experiments were carried out to verify furnace gas temperature, the effect of billet interval on lateral heat flux calculation as well as billet temperature. It met agreement well with experiments on billet temperature, which could be a better prerequisite for further reheating furnace automatic control.

EXPONENTIAL COMPACT HIGHER-ORDER SCHEMES AND THEIR STABILITY ANALYSIS FOR UNSTEADY CONVECTION-DIFFUSION EQUATIONS

Exponential compact higher-order schemes have been developed for unsteady convection-diffusion equation (CDE). One of the developed scheme is sixth-order accurate which is conditionally stable for the Péclet number 0 ≤ Pe ≤ 2.8 and the other is fourth-order accurate which is unconditionally stable. Schemes for two-dimensional (2D) problems are made to use alternate direction implicit (ADI) algorithm. Example problems are solved and the numerical solutions are compared with the analytical solutions for each case.

Application of Alternating Direction Implicit (ADI) Algorithm to Staggered-grid PSTD Modeling of Electromagnetic Waves

By applying the fast Fourier transform (FFT) and inverse transform to represent the spatial derivatives, the pseudo-spectral time-domain (PSTD) method has achieved a spatial grid of only two points per wavelength while maintaining a high accuracy. Its computational speed can be further accelerated by applying alternating-direction implicit (ADI), which makes PSTD works even when the time step exceeds the maximum time step allowed under the Courant- Friedrich-Levy (CFL) condition. However, due to the unsuitability of FFT on non-continuou condition, conventional ADI-PSTD turns to be unstable in media with high property contrast. We have expanded current ADI-PSTD into staggered-grid ADI-PSTD. After applying the staggered- grid approach into our algorithm by shifting the spatial derivatives halfway between 2 adjacent nodes, the stability of the difierentiation operators is enhanced. We will discuss the staggered- grid ADI-PSTD method in detail and apply it to the simulation of a half-space model. The analysis and discussion of the results show its advantages and further potential.

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.

An Adaptive Spline Wavelet ADI (SW-ADI) Method for Two-Dimensional Reaction–Diffusion Equations

We study a spline wavelet alternative direction implicit (SW-ADI) algorithm for solving two-dimensional reaction diffusion equations. This algorithm is based on a collocation method for PDEs with a specially designed spline wavelet for the Sobolev spaceH2(I)on a closed intervalI. By using the tensor product nature of adaptive wavelet meshes, we propose a SW-ADI method for two-dimensional problems. The proposed SW-ADI method is an efficient time-dependent adaptive method with second-order accuracy for solutions with localized phenomena, such as in flame propagations. Issues like new boundary wavelets for more accurate boundary conditions, adaptive strategy for two-dimensional meshes, data structure and storage and implementation details, and numerical results will be discussed.

A PARALLEL ADI ALGORITHM FOR HIGH-ORDER FINITE-DIFFERENCE SOLUTION OF THE UNSTEADY HEAT CONDUCTION EQUATION, AND ITS IMPLEMENTATION ON THE CM-5

We describe a parallel alternate direction implicit (ADI) algorithm for the solution of the unsteady heat conduction equation discretized with fourth-order accuracy. A novel parallel pentadiagonal line-inversion procedure based on a divide-and-conquer strategy is used in conjunction with a domain-decomposition technique. The algorithm has been implemented on the CM-5 in the MIMD mode, and its performance for varying grid sizes and number of processors is investigated.