Alternating Direction Implicit Scheme Research Articles

A generalized curvilinear solver for spherical shell Rayleigh-Bénard convection

Summary A three-dimensional finite-difference solver has been developed and implemented for Boussinesq convection in a spherical shell. The solver transforms any complex curvilinear domain into an equivalent Cartesian domain using Jacobi transformation and solves the governing equations in the latter. This feature enables the solver to account for the effects of the non-spherical shape of the convective regions of planets and stars. Apart from parallelization using MPI, implicit treatment of the viscous terms using a pipeline alternating direction implicit scheme and HYPRE multigrid accelerator for pressure correction makes the solver efficient for high-fidelity direct numerical simulations. We have performed simulations of Rayleigh-Bénard convection at two Rayleigh numbers Ra = 105 and 107 while keeping the Prandtl number fixed at unity (Pr = 1). The average radial temperature profile and the Nusselt number match very well, both qualitatively and quantitatively, with the existing literature. Closure of the turbulent kinetic energy budget, apart from the relative magnitude of the grid spacing compared to the local Kolmogorov scales, ensures sufficient spatial resolution.

Optimizing heat transfer efficiency with injection in ventilated cavity in the presence of internal thermal effects

Venturing into unexplored territory, the study explores the dynamics of mixed convection flow within a square cavity, influenced by a ventilated channel and an injection port. By conducting numerical simulations, the impact of internal heat generation/absorption and viscous dissipation on heat transfer efficiency is thoroughly examined. Employing a finite difference method alongside an alternate direction implicit scheme and iterative successive under-relaxation (SUR) technique, the study systematically solves the governing equations. The effect of physical parameters, namely the Reynolds number ( 50 ≤ Re ≤ 500 ), Richardson number ( 0.01 ≤ Ri ≤ 10 ), volumetric internal heat generation ( △ * > 0 )/absorption ( △ * < 0 ), Eckert number ( 0 ≤ Ec ≤ 0.3 ), and especially variation of injection strength, on the streamlines, isotherms, and average Nusselt number are discussed graphically. The findings demonstrate a significant improvement in heat transfer efficiency, ranging from 17% to 73% with injection, underscoring its substantial implications for advancing thermal management across diverse engineering domains.

Fast and stable dimension splitting simulations for the hydrodynamically coupled three-component conserved Allen–Cahn phase field model

In this study, an efficient dimension splitting method is developed for solving the coupled system of three-component conserved Allen–Cahn model and incompressible Navier–Stokes equation. The proposed method completely decouples numerical discrete system and transforms the computing of high-dimensional problems into one-dimensional problems, which greatly minimizes high computational costs in two- and three-dimensions. Via employing the direction splitting pressure correction method and alternate direction implicit (ADI) scheme on MAC staggered grids, the practical computing can be accomplished by matrix multiplication, which further allows us to employ parallel operations for computing acceleration. To guarantee the long time simulation stability, a novel post-processing method is introduced to force the physical properties, such as the maximum bound principle (MBP), mass conservation, and no volume leakage conditions, of the numerical solution. Numerical test results show that the proposed numerical algorithm can provide stable and second-order solution. A large number of simulations are performed to verify the effectiveness of the proposed method in simulating and exploring the behavior of three-phase flows.

Efficient computation of heat distribution of processed workpiece with coupling to bottom water vapor in water-jet guided laser machining.

In this paper, we present a novel approach for calculating the heat distribution within a processed workpiece subjected to laser irradiation while accounting for the influence of bottom water vapor. A comprehensive mathematical model is introduced and numerical techniques using difference approximation are employed. Initially, the three-dimensional heat equation, originally defined in the rectangular coordinate system, is transformed into a corresponding model within the cylindrical coordinate system, incorporating a nonlinear boundary condition to account for coupling effects. Subsequently, leveraging the axial symmetry of the heat distribution, the three-dimensional model is simplified into a two-dimensional one. This simplified model is solved using the alternating direction implicit scheme coupled with the Crank-Nicolson method. Moreover, we develop a high-precision numerical treatment for the nonlinear boundary condition within the cylindrical coordinate system. To validate our methodology, simulation experiments are conducted on three distinct samples. Our comparative results demonstrate the feasibility and efficiency of the proposed approach in the context of water-jet guided laser processing.

Single-term and multi-term nonuniform time-stepping approximation methods for two-dimensional time-fractional diffusion-wave equation

The aim of this work is to propose two efficient schemes to handle the accuracy near the singularity at t=0 in solving two-dimensional time-fractional diffusion-wave equation (TFDWE). The considered time fractional derivative is in the Caputo sense of order α(1<α<2). For the approximation of the time-fractional Caputo derivative (TFCD), we use Nonuniform L1 method (single-step) and Nonuniform Crank-Nicolson L1−2 method (multi-step). The L1 method has order of convergence (OC) min⁡(3−α,γα), where γ is the mesh grading parameter used in construction of the nonuniform mesh, and L1−2 method has second OC. We consider nonuniform time mesh to compensate the lack of smoothness caused by the presence of singularity in TFCD at t=0. After that, we adopt these two methods to approximate TFCD and apply the central difference operator for the space direction derivative approximations to get the system of equations for considered model. Then, we use the Alternating Direction Implicit (ADI) approach to develop two kinds of fully discrete schemes under the regularity conditions to solve the TFDWE. Further, we prove the stability analysis of these two schemes. Two numerical examples are given for one-dimensional (1D) and two-dimensional (2D) TFDWE with smooth and non-smooth exact solutions to indicate the accuracy of ADI schemes. The illustrated examples show that both schemes have second-order accuracy in space direction, and in temporal direction the schemes achieve min⁡(3−α,γα) and second order convergence, respectively for all 1<α<2. The corresponding absolute error is plotted to see the advantage of nonuniform time meshes at the initial singularity t=0.

Efficient Numerical Simulation of Biochemotaxis Phenomena in Fluid Environments.

A novel dimension splitting method is proposed for the efficient numerical simulation of a biochemotaxis model, which is a coupled system of chemotaxis-fluid equations and incompressible Navier-Stokes equations. A second-order pressure correction method is employed to decouple the velocity and pressure for the Navier-Stokes equations. Then, the alternating direction implicit scheme is used to solve the velocity equation, and the operator with dimension splitting effect is used instead of the traditional elliptic operator to solve the pressure equation. For the chemotactic equation, the operator splitting method and extrapolation technique are used to solve oxygen and cell density to achieve second-order time accuracy. The proposed dimension splitting method splits the two-dimensional problem into a one-dimensional problem by splitting the spatial derivative, which reduces the computation and storage costs. Finally, through interesting experiments, we show the evolution of the cell plume shape during the descent process. The effect of changing specific parameters on the velocity and plume shape during the descent process is also studied.

Stefan problem coupled with natural convection: An application to dissolution process

This paper investigates the process of dissolution of a solute in a solvent placed in an horizontal concentric cylinder. The theoretical investigation solves a Stefan problem with phase transition due to natural convective flow. To realize the objective, the governing equation for the concentration distribution, stream function–vorticity form of the Navier–Stokes equation for the flow field, and a Stefan condition for calculating the timescale evolution of the front are coupled together with different parameters. These non-linear equations are solved using a stable and second-order accurate boundary-fitted alternating direction implicit scheme with first-order upwind difference approximation for convective terms. The numerical scheme is validated initially by applying it to solve a natural convection problem with no phase transition, for which benchmark solutions are available. The validated scheme is then applied to the chosen problem followed by a refinement study to obtain a reliable solution. The obtained results are used to analyze the effect of physical parameters such as the Stefan number (Ste), geometric aspect ratio of solute to fluid, the Rayleigh number (Ra) and the Schmidt (Sc) number on dissolution rates as well as the flow patterns. It is observed that the solute dissolution, without the temperature influence, mainly depends on the annulus gap width (L) and the convection rate. Additionally, it is also observed that, for the Rayleigh numbers greater than 105, the unit circular-shaped solute initially dissolves uniformly from the outer surface, but as the time progresses, due to the influence of laminar boundary layer flow around the solute, it changes into an egg-shape.

Analysis of a Peaceman-Rachford ADI scheme for Maxwell equations in heterogeneous media

The Peaceman-Rachford alternating direction implicit (ADI) scheme for linear time-dependent Maxwell equations is analyzed on a heterogeneous cuboid. Due to discontinuities of the material parameters, the solution of the Maxwell equations is less than H2-regular in space. For the ADI scheme, we prove a rigorous time-discrete error bound with a convergence rate that is half an order lower than the classical one. Our statement imposes only assumptions on the initial data and the material parameters, but not on the solution. To establish this result, we analyze the regularity of the Maxwell equations in detail in an appropriate functional analytical framework. The theoretical findings are complemented by a numerical experiment indicating that the proven convergence rate is indeed observable and optimal.

An Implicit Numerical Method for the Riemann–Liouville Distributed-Order Space Fractional Diffusion Equation

This paper investigates a two-dimensional Riemann–Liouville distributed-order space fractional diffusion equation (RLDO-SFDE). However, many challenges exist in deriving analytical solutions for fractional dynamic systems. Efficient and reliable methods need to be explored for solving the RLDO-SFDE numerically. We develop an alternating direction implicit scheme and prove that the numerical method is unconditionally stable and convergent with an accuracy of O(σ2+ρ2+τ+hx+hy). After employing an extrapolated technique, the convergence order is improved to second order in time and space. Furthermore, a fast algorithm is constructed to reduce computational costs. Two numerical examples are presented to verify the effectiveness of the numerical methods. This study may provide more possibilities for simulating diffusion complexities by fractional calculus.

A generalized alternating direction implicit method for transient thermal simulation of 2-D structures with locally refined grids

In this article, a generalized alternating direction implicit (ADI) method for transient thermal simulation of two-dimensional structures with locally refined grids is proposed. First, a flexible approach is employed to generate the semistructured grids with local refinement. The generalized ADI method is then derived based on these semistructured grids, in which two new directions are added to deal with the grid points in the transition regions. By this means, the ADI scheme is successfully extended to handle the transient simulation of objects with locally refined grids while maintaining the property of linear complexity in computation. The accuracy and high efficiency of the proposed method are verified by several typical examples.

High-order orthogonal spline collocation ADI scheme for a new complex two-dimensional distributed-order fractional integro-differential equation with two weakly singular kernels

We present an alternating direction implicit scheme for the numerical solution of the distributed-order fractional integro-differential equation with two weakly singular kernels. Orthogonal spline collocation with piecewise Hermite bicubics is used for spatial discretization. The weighted and shifted Grünwald difference formula is employed to discretize the distributed-order time-fractional derivative combined with the second-order quadrature convolution rule introduced by Lubich for the Riemann–Liouville fractional integral. We prove the stability of the proposed scheme and derive error estimates. We also describe an efficient implementation of the scheme and show the numerical results demonstrating the accuracy and convergence rates in norm.

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.

Force-Extension Curve of a Polymer Chain Entangled with a Static Ring-Shaped Obstacle.

The way to theoretically approach dynamic and static topological constraints of polymer entanglements still presents a great challenge in polymer physics. So far, only the problem of static entanglement with multiple simple objects has been solved in theory by a superspace approach in our previous work. This work is devoted to extending the superspace approach to study a polymer chain entangled with a relatively complicated object-a ring-shaped object with genus one. Taking advantage of the axial symmetry of the model setup, the 3D diffusion equations in the superspace can be numerically solved within the 2D coordinates using a specially designed alternating-direction implicit (ADI) scheme. A series of numerical calculations reveal that the topological entanglement effect of the ring will exert a topological entropy attractive force on the linear chain, which can be used to explain the viscosity-increase phenomenon observed in recent simulations and experiments. Furthermore, the influences of the ring size and the entangling modes on the topological entropy force are also investigated by examining the corresponding force-extension curves. This work, together with our previous work, might pave the path toward the complete formulation of static topological constraints.

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.

Hybrid nanofluid magnetohydrodynamic mixed convection in a novel W-shaped porous system

PurposeThis study aims to numerically examine the influence of various geometric parameters of a novel W-shaped porous cavity undergoing hybrid nanofluid-based magnetohydrodynamic mixed convection. The W-shaped cavity is modified from the classical trapezoidal cavity by constructing a triangular shape at its bottom. This cavity is isothermally active at the bottom, with different numbers and heights of the triangular peak (or undulation). The heated hybrid nanofluid (Cu–Al2O3–H2O) flow is cooled through the translating top wall. Inclined sidewalls are thermally insulated. To compare the impacts of change in geometric parameters, a square cavity under similar boundary conditions is also simulated. This study is carried out systematically addressing the various influences from a range of parameters like side angles (γ), number (m) and height (λ) of the bottom undulation, Reynolds number (Re), Richardson number (Ri), Darcy number (Da), Hartmann number (Ha), hybrid nanoparticles volume fraction (φ) on the overall thermal performance of the cavity.Design/methodology/approachApplying the finite volume approach, the transport equations involving multiphysical conditions like porous substance, hybrid nanofluid, magnetic field and shearing force are solved numerically by using a written FORTRAN-based code following the SIMPLE algorithm. The algebraic equations are solved over all the control volumes in an iterative process using the alternate direction implicit scheme and the tri-diagonal matrix algorithm. The converged solution of the iterative process is obtained when the relative error levels satisfy the convergence criterion of 10–8 and 10–10 for the maximum residuals and the mass defect, respectively.FindingsIt is revealed that an increase in the bottom undulation height always improves the thermal energy transfer despite the reduction of fluid volume. Thermal energy transfer significantly depends on the heating and cooling surface lengths, fluid volume in the cavity and the magnitude of the bottom undulation height of the W-shaped cavity. With the increase in bottom undulation height, effective heating length increases by ∼28%, which leads to a ∼15% reduction in the effective volume of the working fluid and a gain in heat transfer by ∼56.48%. In general, the overall thermal energy transport is improved by increasing Re, Ri and Da; whereas it is suppressed by increasing Ha.Research limitations/implicationsThere are many opportunities for future research experimentally or numerically, considering different curvature effects, orientations of the geometry, working fluids, boundary conditions, etc. Furthermore, this study could be extended by considering unsteady flow or turbulent flow.Practical implicationsIn many modern systems/processes pertaining to materials processing, continuous casting, food processing, chemical reactors, biomedical applications, etc. fine control in the transport process is a major concern. The findings of this analysis can effectively be useful for other applications for getting more control features in terms of achieving the operational objectives. The approach of the system analysis (considering geometrical size parameters to delve into the underlying transport physics) and the obtained simulated results presented in the work can usefully be applicable to similar thermal systems/devices such as materials processing, thermal mixing, chemical reactors, heat exchangers, etc.Originality/valueFrom the well-documented and vast pool of literature survey, it is understood that there exists no such investigation on the considered geometry and study. This study contributes a lot to understanding magnetic field moderated thermofluid flow of a hybrid nanofluid in a porous medium filled W-shaped cavity, in consideration of different geometrical shape parameters (undulation peak numbers at bottom wall, peak heights, side angles and heating and cooling length). Findings brought by this study provide great insights into the design and operation under various ranges of multiphysical thermofluid-flow processing phenomena.

Soil Nature Effect Investigation on the Ground-to-Air Heat Exchanger for the Passive Cooling of Rooms

The building sector consumes much energy either for cooling or heating and is associated to greenhouse gas emissions. To meet energy and environmental challenges, the use of ground-to-air heat exchangers for preheating and cooling buildings has recently received considerable attention. They provide substantial energy savings and contribute to the improvement of thermal comfort in buildings. For these systems, the ground temperature plays the main role. The present work aims to investigate numerically the influence of the nature of soil on the thermal behavior of the ground-to-air heat exchanger used for building passive cooling. We have taken into account in this work the influence of the soil nature by considering three types of dry soil: clay soil, sandy-clay soil and sandy soil. The mixed convection equations governing the heat transfers in the earth-to-air heat exchanger have been presented and discretized using the finite difference method with an Alternate Direction Implicit (ADI) scheme. The resulting algebraic equations are then solved using the algorithm of Thomas combined with an iterative Gauss-Seidel procedure. The results show that the flow is dominated by forced convection. The examination of the sensitivity of the model to the type of soil shows that the distributions of contours of streamlines, isotherms, isovalues of moisture are less affected by the variations of the nature of soil through the variation of the diffusivity of the soil. However, it is observed that the temperature values obtained for the clay soil are higher while the sandy soil shows lower temperature values. The values of the ground-to-air heat exchanger efficiency are only slightly influenced by the nature of the soil. Nevertheless, we note a slightly better efficiency for the sandy soil than for the sandy-clayey silt and clayey soils. This result shows that a sandy soil would be more suitable for geothermal system installations.

High dimensional Riesz space distributed-order advection-dispersion equations with ADI scheme in compression format

&lt;abstract&gt;&lt;p&gt;A second order alternating direction implicit scheme for time-dependent Riesz space distributed-order advection-dispersion equations is applied to higher dimensions with the Tensor-Train decomposition technique. The solutions are solved in compressed format, the Tensor-Train format, and the errors accumulated due to compressions are analyzed to ensure convergence. Problems with low-rank data are tested, the results illustrated a steeper growth in the ranks of the numerical solutions than that in related works.&lt;/p&gt;&lt;/abstract&gt;

High-performance computation of pricing two-asset American options under the Merton jump-diffusion model on a GPU

This paper is concerned with fast, parallel and numerically accurate pricing of two-asset American options under the Merton jump-diffusion model, which gives rise to a two-dimensional partial integro-differential complementarity problem (PIDCP) with a nonlocal two-dimensional integral term. Following method-of-lines approach, the solution to the PIDCP can be computed quite accurately by a robust numerical technique that combines Ikonen–Toivanen splitting with an alternating direction implicit scheme. However, we observed that computing the numerical solution with this technique becomes extremely time consuming, mainly due to the handling of the integral term. In this paper we parallelize this technique by applying a parallel fast Fourier transformation algorithm to all matrix-vector multiplications involving the huge and dense integral approximation matrix by exploiting its block Toeplitz with Toeplitz block structure. We also parallelize other computationally intensive steps of this technique by applying a recently developed parallel cyclic reduction algorithm for pentadiagonal systems. Our solutions computed on a graphics processing unit (GPU) using CUDA® platform are compared for accuracy with those available in the literature. It is observed that by solving the PIDCP parallelly we could bring down the computational times from several hours to a few seconds in certain cases in our experiments.

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.

A second order accurate fixed-grid method for multi-dimensional Stefan problem with moving phase change materials

This paper proposes a front-tracking fixed grid method for solving Stefan problem with moving phase change materials in any arbitrary (irregular) bounded domains. To achieve this objective, the governing partial differential equations are transformed, using a curvilinear coordinate transformation, in to a fixed rectangular domain, at each time level and an alternate direction implicit (ADI) scheme is used to solve the resultant differential system. The proposed scheme is consistent, unconditionally stable and also produces second-order accurate results. Many numerical experiments are conducted, in both one and two phase environments, to validate the proposed method. All these validations confirmed the theoretical accuracy of the produced results which are presented in the form of error graphs and tables in each case. Importance is given to understand the influence of the physical parameters like Peclet number, Stefan number, material velocity, latent heat, and thermal conductivity, etc., on the rate of change of phase and observed a continuous enhancement of the same with in the chosen range of the parameters. Also noticed a faster movement of the interface with the increase in these parameters in the direction of melting or freezing.