Alternating Direction Implicit Research Articles

An alternating-direction hybrid implicit-explicit finite-difference time-domain method for the Schrödinger equation

This paper proposes a novel hybrid FDTD method for solving the time-dependent Schrödinger equation, which is fundamental for modeling materials and designing nanoscale devices. The wave function is propagated on nonuniform grids by applying explicit updates in part of the grid and implicit updates elsewhere. The latter are based on the Alternating-Direction Implicit (ADI) scheme while the former are constructed with a central difference for the time derivative. A rigorous stability analysis proves that spatial steps can be selectively removed from the stability criterion thus combining the unconditional stability of the ADI scheme with fast explicit calculations. The scheme excels in its flexibility by efficiently discretizing and balancing explicit with implicit updates, as such expediting the computations. Moreover, it retains the linear complexity of explicit schemes with respect to space and time, making it especially scalable to numerically large problems. Several numerical experiments, including a laterally tunnel-coupled quantum wire and a nanowire double-barrier resonant-tunneling diode, show the validity of the scheme by demonstrating its high accuracy and decreased CPU time compared to traditional methods.

Computationally Efficient Optimal Control for Unstable Power System Models

In this article, the focus is mainly on gaining the optimal control for the unstable power system models and stabilizing them through the Riccati-based feedback stabilization process with sparsity-preserving techniques. We are to find the solution of the Continuous-time Algebraic Riccati Equations (CAREs) governed from the unstable power system models derived from the Brazilian Inter-Connected Power System (BIPS) models, which are large-scale sparse index-1 descriptor systems. We propose the projection-based Rational Krylov Subspace Method (RKSM) for the iterative computation of the solution of the CAREs. The novelties of RKSM are sparsity-preserving computations and the implementation of time-convenient adaptive shift parameters. We modify the Low-Rank Cholesky-Factor integrated Alternating Direction Implicit (LRCF-ADI) technique-based nested iterative Kleinman–Newton (KN) method to a sparse form and adjust this to solve the desired CAREs. We compare the results achieved by the Kleinman–Newton method with that of using the RKSM. The applicability and adaptability of the proposed techniques are justified numerically with MATLAB simulations. Transient behaviors of the target models are investigated for comparative analysis through the tabular and graphical approaches.

Transient Three-Dimensional Natural Convection in Confined Porous Media with Time-Periodic Boundary Conditions

Transient three-dimensional natural convection in confined fluid-saturated porous media had been investigated numerically through this work. The geometry selected is a box with time-periodic temperature variation at the vertical sides and constant wall temperature at the top and bottom. In this investigation, the momentum equation of flow through fluid-saturated porous media had been transformed to a vector potential form and solved numerically using the Successive Over Relaxation method while the energy equation is solved using the three-dimensional Alternating Direction Implicit method. The values of the Rayleigh number under investigation are (150, 200, 250 and 300). The results are presented in a form of contour maps for the temperature and a velocity vector maps for the velocity. The results reveal that the temperature within the box is increased as the time or Rayleigh number increase.The Nusselt number varies inversely with the time and directly with the Rayleigh number. Two cell convective patterns are obtained in the y-z

A novel two-step streamline-diffusion FEM for singularly perturbed 2D parabolic PDEs

In this article, a class of singularly perturbed 2D parabolic convection-diffusion-reaction initial-boundary-value problem is numerically solved by the alternating direction implicit type operator splitting streamline-diffusion finite element method. The proposed scheme alleviates the computational complexity and the high storage requirement for higher-dimensional problems. The overall stability of the two-step method is established. A piecewise-uniform Shishkin mesh is used for the spatial domain discretization. ε-uniform error estimate has been derived with an appropriate choice of the stabilization parameter. The parameter is shown to depend on the time step-length which is necessary for the stability of the method. Some numerical simulations are carried out to validate the theoretical error estimate.

A unified formulation of splitting-based implicit time integration schemes

Splitting-based time integration approaches such as fractional step, alternating direction implicit, operator splitting, and locally one dimensional methods partition the system of interest into components, and solve individual components implicitly in a cost-effective way. This work proposes a unified formulation of splitting time integration schemes in the framework of general-structure additive Runge–Kutta (GARK) methods. Specifically, we develop implicit-implicit (IMIM) GARK schemes, provide the order conditions for this class, and explain their application to partitioned systems of ordinary differential equations. We show that classical splitting methods belong to the IMIM GARK family, and therefore can be studied in this unified framework. New IMIM-GARK splitting methods are developed and tested using parabolic systems.

Atmospheric parabolic equation with Galerkin discretization and boundary fitted grid for modeling infrasound propagation over topography

Studies utilizing the parabolic equation (PE) to model infrasound propagation over topography have traditionally implemented them as 2D or Nx2D PE models. These methods, to one extent or another, trade-off accuracy for enhanced computational speed. Three-dimensional parabolic equation (3DPE) models are seldom seen in atmospheric acoustic propagation studies compared to their 2D and Nx2D counterparts, despite the fact that theta coupling and proper modeling of horizontal refraction and diffraction in a 3DPE model should enable greater accuracy. In this work a 3DPE in cylindrical coordinates is developed under the alternating direction implicit scheme, with a Galerkin discretization and boundary fitted grid, such that the PE is capable of handling arbitrary and irregular topography features. Finally, 2D, Nx2D, and 3D implementations of the aforementioned PE model are examined for cases of propagation over a simple hill in homogeneous, downward refracting, and upward refracting atmospheres. Insights are drawn with regards to the accuracy at a variety of positions and ranges, and the computational speeds of the various methods.

Numerical simulation of two-dimensional fractional neutron diffusion model describing dynamical behaviour of sodium-cooled fast reactor

This article proposes an efficient numerical technique for solving the two-dimensional fractional neutron diffusion (FND) model with time-dependent reactivity which describes neutron transport in a finite cylindrical nuclear reactor core. In this technique, the time discretization is based on the L1 scheme and the space discretization is based on Alternating Direction Implicit (ADI) difference scheme. The neutron flux distribution in the reactor is calculated. The effects of the order of fractional derivative and simulation time on the neutron flux profile are examined. The numerical results of fractional model are compared with those of the P1 model and the normal diffusion model. The elapsed computational time of the proposed numerical algorithm is provided to show the computational efficiency of the method. It is shown that the present method is of O(Δt+hr2+hz2) convergence, where Δt is the temporal step size and hr and hz are the spatial step sizes.

A pencil distributed direct numerical simulation solver with versatile treatments for viscous term

We present an efficient second- or fourth-order finite difference direct numerical simulation (DNS) solver using pencil-like domain decomposition parallel strategy, with the ability to handle different boundary conditions. The viscous term is treated implicitly, partial implicitly, or explicitly. The runtimes for different viscous treatments and different boundary conditions are evaluated quantitatively, which can help us to get a fast computational speed for specific flow cases. FFT-based method is used for solving Pressure Poisson equation, and alternating direction implicit approach is adopted for Helmholtz equations during the implicit viscous treatment. The numerical results achieve good match with those obtained from spectral methods. In addition, the resulting solver exhibits quite good parallel efficiency, and demonstrates good computational speedup and versatility than other pencil-like DNS solvers reported in the literature. The source code is freely available at https://github.com/GongZheng-Justin/Channel3d.

Fourth-order alternating direction implicit difference scheme to simulate the space-time Riesz tempered fractional diffusion equation

The current paper proposes a new high-order finite difference scheme with low computational complexity to solve the space-time fractional tempered diffusion equation. At the first stage, the time derivative has been approximated by a difference scheme with second-order accuracy. Furthermore, in the next step, a compact operator has been employed to discretize the space derivative with fourth-order accuracy. After deriving the time-discrete scheme, its stability is analysed. So, a suitable term is added to the main difference scheme. By adding this term, we could construct the main ADI scheme. In the final stage, the convergence order of the full-discrete scheme based upon the ADI formulation is proved. The convergence order of the constructed technique is . The numerical results show the efficiency of the new technique.

One‐step leapfrog alternating direction implicit procedure for left‐handed material in open region problems with enhanced absorption

AbstractBy incorporating higher order formulation and one‐step leapfrog alternating direction implicit (ADI) procedure, unconditionally stable perfectly matched layer implementation is proposed for terminating frequency‐dependent left‐handed material in finite‐difference time‐domain lattice. To be more specific, frequency‐dependent left‐handed material can be calculated by the piecewise linear recursive convolution method. The proposed scheme shows advantages of higher order formulation and one‐step leapfrog ADI procedure in terms of enhancing absorbing performance, improving computational efficiency and simulating open region problems. The effectiveness is demonstrated through the computational domain with dipole antenna model above the left‐handed material slab. From results, it can be concluded that the proposed formulation can obtain considerable absorption, outstanding efficiency, and improved entire performance. Meanwhile, the proposed scheme also keeps its stability when time step far exceeds the Courant‐Friedrich‐Levy condition.

A novel three-dimensional implicit numerical model of a borehole field heat exchanger that accounts for seasonal fluctuations of the soil temperature

This study proposes a new three-dimensional implicit numerical model to simulate a borehole field heat exchanger that accounts for seasonal fluctuations of the soil temperature. The model uses an alternating-direction implicit finite difference approach to perform the simulation. The effect of seasonal changes in the soil temperature is considered using an internal heat source term in the governing equations. The model is validated against several published in-situ tests as well as a laboratory experimental study and the results obtained from a TRNSYS version of the duct storage (DST) model, demonstrating an average error of less than 3%.The validated model is then applied to analyse the impact of accounting for seasonal soil temperature fluctuations at various borehole lengths and climate zones. This analysis compares the results obtained from a model that uses constant temperature boundary conditions against those from a model that contains the internal heat source term and uses adiabatic boundary conditions. A significant discrepancy between the outputs of the two models has been observed for shallow borehole heat exchangers (H < 60 m) and at locations with a high annual amplitude of the surface temperature variations (ΔTa > 5°C). Thus, the assumption of constant temperature at the ground surface is only appropriate for deep boreholes (H > 60 m) or at locations with a relatively low annual amplitude of the surface temperature variations (ΔTa < 5°C).

A class of new successive permutation iterative algorithms for diffusion equation

Based on the discretization of full-implicit difference schemes, we construct a new class of successive permutation iterative algorithms for one-dimensional and two-dimensional diffusion equations in this paper. The stability of the new algorithms is guaranteed, and the form of growth matrices are provided in the paper. Compared with the numerical results of the full-implicit difference method, the Gauss–Seidel iterative method and Alternating direction implicit (ADI) method, it is shown that the new algorithms not only have the same accuracy as the full-implicit difference method but also are more accurate and fast than the Gauss–Seidel iterative method and ADI method. Furthermore, the presented algorithms can avoid solving large linear equations, hence improve the calculation speed dramatically.

Hydrodynamic features and water ecology improvement of Yuehai Lake based on ecological water diversion

To mitigate lake eutrophication, this paper collects the relevant data and monitors the water of Yuehai Lake, the seat of Ningxia Hui Autonomous Region, China, and constructs an eco-dynamics model of the lake water. Then, the alternating direction implicit (ADI) method was adopted to discretely solve the hydraulic equation and the convective diffusion equation, and the model parameters were calibrated and verified against the monitoring data of hydrology, meteorology, and water quality from May 2018 to October 2019. After that, the laws of nutrient transmission, diffusion, and spatiotemporal distribution were obtained through simulation with such factors as chemical oxygen demand (COD), ammonia nitrogen, total nitrogen, total phosphorus, and chlorophyll a; the simulation results agree well with the measured data. Based on the actual situation of the lake, a scheme was prepared to change the hydrodynamic condition from three aspects: replenished water flow, rainfall, and wind-driven circulation. The results show that the flow field of Yuehai Lake is not greatly affected by the first two factors, but determined by wind-driven circulation. The proposed scheme for ecological water diversion is expected to effectively improve the water quality of Yuehai Lake.

Adaptive ADI difference solution of quenching problems based on the 3D convection–reaction–diffusion equation

Abstract The 3D quenching problem reflecting solid-burn scene based on convection–reaction–diffusion equation is creatively concerned in this work. The spatial derivatives of original equation are discretized by Taylor series and the temporal derivatives are approximated by the Crank–Nicolson (CN) method. After the discrete schemes are arranged, an alternating direction implicit (ADI) scheme on adaptive grid is constructed to interpret quenching phenomena of the three-dimension (3D) equation with singularity source. Quenching time, quenching domain, and characteristics relative to temperature as well as variation of temperature over time are achieved via scientific experiment and analysis. Comparing with the 1D or 2D problem, it is harder for the 3D problem to produce quenching phenomena. Regardless of different convection functions, it can form quenching behaviors through experiments when only the elements which include degeneracy parameter, convection parameters, and domain sizes are configured properly. We hope all this can offer references for the 3D engineering problem. At the same time, it will offer support to research the relationship between quenching phenomena and degeneracy parameter, convection parameters, and domain sizes in the future, respectively.

Pointwise error estimate of an alternating direction implicit difference scheme for two-dimensional time-fractional diffusion equation

An alternating direction implicit (ADI) difference method is adopted to solve the two-dimensional time-fractional diffusion equation with Dirichlet boundary condition whose solution has some weak singularity at initial time. L1 scheme on uniform mesh is used to discretize the temporal Caputo fractional derivative. Pointwise-in-time error estimate is given for the fully discrete ADI scheme, where the error bound does not blowup when α (the order of fractional derivative) approaches 1−. It is shown both in theoretically and numerically that the temporal convergence order of the ADI scheme is O(τ2α+τtnα−1) at time t=tn; hence the scheme is globally O(τα) accurate in temporal direction, but it is O(τmin⁡{2α,1}) when t is away from 0.

SVD-Krylov based Sparsity-preserving Techniques for Riccati-based Feedback Stabilization of Unstable Power System Models

We propose an efficient sparsity-preserving reduced-order modelling approach for index-1 descriptor systems extracted from large-scale power system models through two-sided projection techniques. The projectors are configured by utilizing Gramian based singular value decomposition (SVD) and Krylov subspace-based reduced-order modelling. The left projector is attained from the observability Gramian of the system by the low-rank alternating direction implicit (LR-ADI) technique and the right projector is attained by the iterative rational Krylov algorithm (IRKA). The classical LR-ADI technique is not suitable for solving Riccati equations and it demands high computation time for convergence. Besides, in most of the cases, reduced-order models achieved by the basic IRKA are not stable and the Riccati equations connected to them have no finite solution. Moreover, the conventional LR-ADI and IRKA approach not preserves the sparse form of the index-1 descriptor systems, which is an essential requirement for feasible simulations. To overcome those drawbacks, the fitting of LR-ADI and IRKA based projectors from left and right sides, respectively, desired reduced-order systems attained. So that, finite solution of low-rank Riccati equations, and corresponding feedback matrix can be executed. Using the mechanism of inverse projection, the Riccati-based optimal feedback matrix can be computed to stabilize the unstable power system models. The proposed approach will maintain minimized ℌ2 -norm of the error system for reduced-order models of the target models.

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

We develop a linearized compact alternating direction implicit (ADI) numerical method to solve the nonlinear delayed Schrödinger equation in two-dimensional space. By discrete energy estimate method, we analyse the convergence of the fully-discrete numerical method, and show that the numerical scheme is of order O(Δt2+h4) with time stepsize Δt and space stepsize h. At last, we present several numerical examples to confirm theoretical analyses.

Numerical methods for time-fractional convection-diffusion problems with high-order accuracy

Abstract In this paper, we consider the numerical method for solving the two-dimensional time-fractional convection-diffusion equation with a fractional derivative of order α \alpha ( 1 &lt; α &lt; 2 1\lt \alpha \lt 2 ). By combining the compact difference approach for spatial discretization and the alternating direction implicit (ADI) method in the time stepping, a compact ADI scheme is proposed. The unconditional stability and H 1 {H}^{1} norm convergence of the scheme are proved rigorously. The convergence order is O ( τ 3 − α + h 1 4 + h 2 4 ) O\left({\tau }^{3-\alpha }+{h}_{1}^{4}+{h}_{2}^{4}) , where τ \tau is the temporal grid size and h 1 {h}_{1} , h 2 {h}_{2} are spatial grid sizes in the x x and y y directions, respectively. It is proved that the method can even attain ( 1 + α ) \left(1+\alpha ) order accuracy in temporal for some special cases. Numerical results are presented to demonstrate the effectiveness of theoretical analysis.

Magnetohydrodynamic thermal characteristics of water-based hybrid nanofluid-filled non-Darcian porous wavy enclosure: effect of undulation

PurposeThe aims of this study is to numerically investigate the thermal phenomena during magnetohydrodynamic (MHD) free convection in an oblique enclosure filled with porous media saturated with Cu–Al2O3/water hybrid nanofluid and heated at the left wavy wall. The thermophysical phenomena are explored thoroughly by varying the amplitude (λ) and undulation (n) of the wavy wall and the inclination of the enclosure (γ) along with other pertinent physical parameters. Darcy–Rayleigh number (Ram), Darcy number (Da), Hartmann number (Ha) and nanoparticle volumetric fraction (ϕ). The effect of all parameters has been analyzed and represented by using heatlines, isotherms, streamlines, average Nusselt number and local Nusselt number.Design/methodology/approachThe finite volume method is used to work out the transport equations coupled with velocity, pressure and temperature subjected to non-uniform staggered grid structure after grid-sensitivity analysis by an indigenous computing code and the semi-implicit method for pressure linked equations (SIMPLE) algorithm. The solution process is initiated following an iterative approach through the alternate direction implicit sweep technique and the tridiagonal matrix algorithm (TDMA) algorithm. The iterative process is continued until successive minimization of the residuals (<1e-8) for the governing equations.FindingsThis study reveals that the increase in the heating surface area does not always favor heat transfer. An increase in the undulation amplitude enhances the heat transfer; however, there is an optimum value of undulation of the wavy wall for this. The heat transfer enhancement because of the wall curvature is revealed at higher Ram, lower Da and Ha and lower volume fraction of nanoparticles. In general, this augmentation is optimum for four undulations of the wavy wall with an amplitude of λ = 0.3. The heat transfer enhancement can be more at the cavity inclination γ = 45°.Research limitations/implicationsThe technique of this investigation could be used in other multiphysical areas involving partial porous layers, conducting objects, different heating conditions, wall motion, etc.Practical implicationsThis study is to address MHD thermo-fluid phenomena of Cu–Al2O3/water-based hybrid nanofluid flow through a non-Darcian porous wavy cavity at different inclinations. The amplitude and number of undulations of the wavy wall, permeability of the porous medium, magnetic field intensity, nanoparticle volumetric fraction and inclinations of the enclosure play a significant role in the heat transfer process. This analysis and the findings of this work can be useful for the design and control of similar thermal systems/devices.Originality/valueMany researchers have examined the problem of buoyancy-induced free convection in a wavy-porous cavity packed with regular fluids or nanofluids. However, the effect of magnetic fields along with the amplitude (λ) at different undulations (n) of the heated wavy wall of an inclined enclosure is not attended so far to understand the transport mechanisms. Most often, the evolutions of the thermo-fluid phenomena in such complex geometries invoking different multiphysics are very intricate. Numerical implementations for simulations and subsequent post-processing of the results are also challenging.

On higher-order compact ADI schemes for the variable coefficient wave equation

We consider an initial-boundary value problem for the n-dimensional wave equation, n⩾2, with the variable sound speed with the nonhomogeneous Dirichlet boundary conditions. We construct and study three-level in time and compact in space three-point in each spatial direction alternating direction implicit (ADI) schemes having the approximation orders O(ht2+|h|4) and O(ht4+|h|4) on the uniform rectangular mesh. The study includes stability bounds in the strong and weak energy norms, the discrete energy conservation law and the error bound of the order O(ht2+|h|4) for the first scheme as well as a short justification of the approximation order O(ht4+|h|4) for the second scheme. We also present results of numerical experiments.