Tridiagonal Systems Of Equations Research Articles

A Five-Step Block Method Coupled with Symmetric Compact Finite Difference Scheme for Solving Time-Dependent Partial Differential Equations

In this article, we present a five-step block method coupled with an existing fourth-order symmetric compact finite difference scheme for solving time-dependent initial-boundary value partial differential equations (PDEs) numerically. Firstly, a five-step block method has been designed to solve a first-order system of ordinary differential equations that arise in the semi-discretisation of a given initial boundary value PDE. The five-step block method is derived by utilising the theory of interpolation and collocation approaches, resulting in a method with eighth-order accuracy. Further, characteristics of the method have been analysed, and it is found that the block method possesses A-stability properties. The block method is coupled with an existing fourth-order symmetric compact finite difference scheme to solve a given PDE, resulting in an efficient combined numerical scheme. The discretisation of spatial derivatives appearing in the given equation using symmetric compact finite difference scheme results in a tridiagonal system of equations that can be solved by using any computer algebra system to get the approximate values of the spatial derivatives at different grid points. Two well-known test problems, namely the nonlinear Burgers equation and the FitzHugh-Nagumo equation, have been considered to analyse the proposed scheme. Numerical experiments reveal the good performance of the scheme considered in the article.

Infiltration analysis of perforated storm sewer: Finite difference modelling versus field tests

Due to urbanization, the ground in cities is often covered with impervious structures. Rainfall infiltration is then limited to cause the prolonged draught-like condition. Combined sanitary and storm sewers need to satisfy the requirements of rainfall runoff, such that urban flooding can occur easily due to insufficient discharge capacity of sewers during heavy rainfalls. Sanitary and storm sewers should be constructed separately, and perforation in storm sewer can enable the occurrence of partial rainfall infiltration to: (1) ease the burden of rainfall runoff, and (2) help the groundwater recharge. In this investigation, the alternating direction implicit method (ADI) is adopted to simulate the infiltration process of stormwater in the ground. Nonlinear second order differential equations for governing equation, initial and boundary conditions are established, and converted into tridiagonal systems of equations. The Thomas algorithm is employed to solve the flow conditions at all discretized nodes. The efficacy of the proposed finite difference method is assessed by comparing against field measurements of cumulative infiltration. Upon the successful calibration of the model, parametric analyses are conducted for design optimization of drainage holes. The opening ratio is suggested at 0.25%, such that the drainage performance of perforated sewer can be optimized.

Parallel Thomas approach development for solving tridiagonal systems in GPU programming − steady and unsteady flow simulation

The solution of tridiagonal system of equations using graphic processing units (GPU) is assessed. The parallel-Thomas-algorithm (PTA) is developed and the solution of PTA is compared to two known parallel algorithms, i.e. cyclic-reduction (CR) and parallel-cyclic-reduction (PCR). Lid-driven cavity problem is considered to assess these parallel approaches. This problem is also simulated using the classic Thomas algorithm that runs on a central processing unit (CPU). Runtimes and physical parameters of the mentioned GPU and CPU algorithms are compared. The results show that the speedup of CR, PCR and PTA against the CPU runtime is 4.4x,5.2x and 38.5x, respectively. Furthermore, the effect of coalesced and uncoalesced memory access to GPU global memory is examined for PTA, and a 2x-speedup is achieved for the coalesced memory access. Additionally, the PTA performance in a time dependent problem, the unsteady flow over a square, is assessed and a 9x-speedup is obtained against the CPU.

Simulation of boundary layer under laminar and turbulent modes of newtonian fluid motion in a flexible pipeline

The article deals with the modeling of boundary layer parameters for Newtonian fluids under laminar and turbulent modes of motion. Based on the system of Prandtl equations and initial boundary conditions under laminar motion, using the Gallorkin method, a tri-diagonal system of equations is formed, which connects the values of functions at the node of nets n+1 across the boundary layer. The numerical method uses the Thomas algorithm to calculate values Ujn+. The velocity value Vjn+1 is determined from the continuity equation by integration across the boundary layer. The Navier-Stokes equation in dimensionless form was used to model the turbulent boundary layer, given the velocity U is an independent variable. The differential equation system was solved using the numerical Dorodnicin method. The results of modeling the velocity distribution in the boundary layer, the thickness of the boundary layer in the section of the flexible pipeline 0.8-1.5 m from the beginning of the fluid entering the pipeline at the expense up to 0.1 kg/s are presented. Keywords: boundary layer, turbulent mode, velocity, Prandtl equation

Fourth-order central compact scheme for the numerical solution of incompressible Navier–Stokes equations

ABSTRACTThis paper provides an implicit central compact scheme for the numerical solution of incompressible Navier–Stokes equations. The solution procedure is based on the artificial compressibility method that transforms the governing equations into a hyperbolic-parabolic form. A fourth-order central compact scheme with a sixth-order numerical filtering is used for the discretization of convective terms and fourth-order central compact scheme for the viscous terms. Dual-time stepping approach is applied to time discretization with backward Euler difference scheme to the pseudo-time derivative, and three point second-order backward difference scheme to the physical time derivative. An approximate factorization-based alternating direction implicit scheme is used to solve the resulting block tridiagonal system of equations. The accuracy and efficiency of the proposed numerical method is verified by simulating several two-dimensional steady and unsteady benchmark 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.

A Difference Scheme Using Spline for a Class of Singular Boundary Value Problem

In this paper, a finite difference scheme using cubic splines has been discussed to find the numerical solution for a class of singular two point boundary value problems for certain ordinary differential equations. The cubic spline approximation leads to the tridiagonal system of equations, which can be solved using Newton’s method. Second order convergence of the method has been established for quite general conditions. Two numerical examples are given to demonstrate the method and verify the second order accuracy.

SLADI: a semi-Lagrangian alternating-direction implicit method for the numerical solution of advection–diffusion problems with application to electricity storage valuations

In this paper, an efficient and novel methodology for numerically solving advection-diffusion problems is presented: a semi-Lagrangian approach for hyperbolic problems of advection is combined with an alternating-direction implicit method for parabolic problems involving diffusion. This is used to value a four-dimensional "storage option" (linked to storing electricity) involving three space variables and time. Efficiency is obtained by solving (only) tridiagonal systems of equations at every time step by incorporating the alternating-direction methodology. Extensive numerical experimentation indicates that the method is stable and accurate; three variants of the scheme are assessed and excellent numerical convergence can be observed. Further, a methodology for determining and results for optimal storage operation arepresented.

A numerically stable rational approximant for the split-step Padé propagator

The split-step Pade algorithm due to Collins [J. Acoust. Soc. Am. 93, 1736-1742 (1993)] provides a fast and accurate method for solving the parabolic equation (PE). The formal solution to the PE propagator, which involves the pseudo-differential operator (1+X)^1/2, is replaced by an [n/n]-Pade rational approximant. This approximant can be expanded either as a product or a sum of rational-linear terms, each term leading to a tridiagonal system of equations in X which is readily solved numerically. To ensure adequate suppression of undesirable contributions from the evanescent part of the spectrum (X<-1), stability constraints must be imposed. In contrast to this approach, we follow the suggestion of Lu and Ho [Optics Lett. 27, 683-685 (2002)] and examine the use of an [n-1/n]-Pade approximant that inherently dampens these evanescent components of the spectrum. An algorithm for generating the necessary coefficients is described. Transmission losses computed using both rational approximants are compared for a typical shallow-water configuration.

Parallelization of 2-D IADE-DY Scheme on Geranium Cadcam Cluster for Heat Equation

A parallel implementation of the Iterative Alternating Direction Explicit method by D’Yakonov (IADE-DY) for solving 2-D heat equation on a distributed system of Geranium Cadcam cluster (GCC) using the Message Passing Interface (MPI) is presented. The implementation of the scheduling of n tri-diagonal system of equations with the above method was used to show improvement on speedup, effectiveness, and efficiency. The Master/Worker paradigm and Single Program Multiple Data (SPMD) model is employed to manage the whole computation based on the use of domain decomposition. The completion of the execution can need task recovery and favorable configuration. The above mentioned details consist of a main report about the numerical validation of the parallelization through simulation to demonstrate the proposed method effectiveness on the cluster system. It was found that the rate of convergence decreases as the number of processors increases. The result of this paper suggests that the 2-D IADE-DY scheme is a good approach to solving problems, particularly when it is simulation with more processors.

Exponential Compact Higher Order Scheme for Nonlinear Steady Convection-Diffusion Equations

AbstractThis paper presents an exponential compact higher order scheme for Convection-Diffusion Equations (CDE) with variable and nonlinear convection coefficients. The scheme is for one-dimensional problems and produces a tri-diagonal system of equations which can be solved efficiently using Thomas algorithm. For two-dimensional problems, the scheme produces an accuracy over a compact nine point stencil which can be solved using any line iterative approach with alternate direction implicit procedure. The convergence of the iterative procedure is guaranteed as the coefficient matrix of the developed scheme satisfies the conditions required to be positive. Wave number analysis has been carried out to establish that the scheme is comparable in accuracy with spectral methods. The higher order accuracy and better rate of convergence of the developed scheme have been demonstrated by solving numerous model problems for one and two-dimensional CDE, where the solutions have the sharp gradient at the solution boundary.

Armadillo generation distributed system with geranium Cadcam cluster for solving 2D telegraphic problem

In this paper, the parallel implementation of the iterative alternating direction explicit method by D'Yakonov (IADE-DY) and the Mitchell and Fairweather double sweep method (DS-MF) compared with the alternative direction implicit (ADI) method for solving the 2D telegraphic problem on a distributed system of Armadillo cluster and geranium Cadcam cluster using the parallel virtual machine and message passing interface is presented. We implement the scheduling of n tridiagonal system of equations with the above methods to show improvement on speedup and efficiency with parallel strategies across the two platforms. The single program multiple data model was employed for the implementation. The IADE-DY, DS-MF and ADI are developed by the splitting of the implicit equation using the finite difference discretization on the 2D telegraphic problem. The implementation is discussed in relation to means of the performance parallel strategies and analysis. The model enhances overlap communication and computation to avoid unnecessary synchronization, hence, the method yields significant speedup by the use of the non-blocking communication. We present some analyses that are helpful for speedup and efficiency.

Analysis of flows past airfoils at very low Reynolds numbers

This article presents the special features of airfoil flows at very low Reynolds numbers, which are of interest for unmanned micro-air-vehicles. Airfoil flow solutions at low Reynolds numbers are obtained with an efficient numerical analysis based on a pseudo-time integration method using artificial compressibility for solving accurately the Navier—Stokes equations. The flow problem is solved in a rectangular computational domain obtained by a coordinate transformation from the physical flow domain around the airfoil at incidence. This method uses a second-order central differencing approach on a stretched staggered grid. A special decoupling procedure using the continuity equation reduces the problem to the solution of scalar tridiagonal systems of equations, which enhances substantially the computational efficiency of the method. The pressure distribution, lift, and drag coefficients are presented for several NACA airfoils at various incidences and low Reynolds numbers between 400 and 6000. Streamline contours for the flow with separations past several airfoils at low Reynolds numbers have also been generated and compared. For a better understanding of the complex flow separation phenomena in the viscous flows past airfoils at very low Reynolds numbers, the onset of separation and reattachment locations have been calculated, and a detailed study is presented on the influence of the Reynolds number, angle of attack, relative thickness and camber, and the maximum camber position along the chord.

Analysis of Unsteady Confined Viscous Flows With Variable Inflow Velocity and Oscillating Walls

The inflow velocities in various components of many engineering systems often display variations in time (fluctuations) during the operation cycle, which may substantially affect the flow-induced vibrations and instabilities of these systems. For this reason, the aeroelasticity study of these systems should include the effect of the inflow velocity variations, which until now has not been taken into account. This paper presents a fluid-dynamic analysis of the unsteady confined viscous flows generated by the variations in time of the inflow velocities and by oscillating walls, which is required for the study of flow-induced vibration and instability of various engineering systems. The time-accurate solutions of the Navier–Stokes equations for these unsteady flows are obtained with a finite-difference method using artificial compressibility on a stretched staggered grid, which is a second-order method in space and time. A special decoupling procedure, based on the utilization of the continuity equation, is used in conjunction with a factored alternate direction scheme to substantially enhance the computational efficiency of the method by reducing the problem to the solution of scalar tridiagonal systems of equations. This method is applied to obtain solutions for the benchmark unsteady confined flow past a downstream-facing step, generated by harmonic variations in time of the inflow velocity and by an oscillating wall, which display multiple flow separation regions on the upper and lower walls. The influence of the Reynolds number and of the oscillation frequency and the amplitudes of the inflow velocity and oscillating wall on the formation of the flow separation regions are thoroughly analyzed in this paper. It was found that for certain values of the Reynolds number and oscillation frequency and amplitudes, the flow separation at the upper wall is present only during a portion of the oscillatory cycle and disappears for the rest of the cycle, and that for other values of these parameters secondary flow separations may also be formed.

Variant of the Thomas Algorithm for opposite‐bordered tridiagonal systems of equations

AbstractTo solve tridiagonal systems of linear equations, the Thomas Algorithm is a much more efficient method than, for instance, Gaussian elimination. The algorithm uses a series of elementary row operations and can solve a system of n equations in 𝒪(n) operations, instead of 𝒪(n3) . Many variations of the Thomas Algorithm have been developed over the years to solve very specific near‐tridiagonal matrix. However, none of these methods address the situation of a system of linear equations that could easily be solved if elementary operations on columns are applied, instead of elementary operations on rows. The present paper proposes an efficient method that allows the use of elementary column operations to solve linear systems of equations using vector multiplication techniques, such as the one proposed by Thomas. Copyright © 2008 John Wiley &amp; Sons, Ltd.

A parallel implementation of Wang’s method for solving tridiagonal systems

Abstract In this paper, a parallel implementation of Wang’s method for solving tridiagonal system of equations on the multiprocessor machine using occam language is presented. The parallel algorithm has been designed for shared and distributed memory machine that support data parallel and message passing. The over all performance of this implementation on 9 each of processors is given. The communication times are very important and any improvement on this communication would have a significant performance of the implementation. The significance of these results are discussed.

Numerical Simulation of the 3D Laminar Viscous Flow on a Horizontal-axis Wind Turbine Blade

The aim of this paper is to describe the methodology followed in order to determine the viscous effects of a uniform wind on the blades of small horizontal-axis wind turbines that rotate at a constant angular speed. The numerical calculation of the development of the three-dimensional boundary layer on the surface of the blades is carried out under laminar conditions and considering flow rotation, airfoil curvature and blade twist effects. The adopted geometry for the twisted blades is given by cambered thin blade sections conformed by circular are airfoils with constant chords. The blade is working under stationary conditions at a given tip speed ratio, so that an extensive laminar boundary layer without flow separation is expected. The boundary layer growth is determined on a non-orthogonal curvilinear coordinate system related to the geometry of the blade surface. Since the thickness of the boundary layer grows from the leading edge of the blade and also from the tip to the blade root, a domain transformation is proposed in order to solve the discretized equations in a regular computational 3D domain. The non-linear system of partial differential coupled equations that governs the boundary layer development is numerically solved applying a finite difference technique using the Krause zig-zag scheme. The resulting coupled equations of motion are linearized, leading to a tridiagonal system of equations that is iteratively solved for the velocity components inside the viscous layer applying the Thomas algorithm, procedure that allows the subsequent numerical determination of the shear stress distribution on the blade surface.

An accurate integral‐based scheme for advection–diffusion equation

AbstractThis paper proposes an accurate integral‐based scheme for solving the advection–diffusion equation. In the proposed scheme the advection–diffusion equation is integrated over a computational element using the quadratic polynomial interpolation function. Then elements are connected by the continuity of first derivative at boundary points of adjacent elements. The proposed scheme is unconditionally stable and results in a tridiagonal system of equations which can be solved efficiently by the Thomas algorithm. Using the method of fractional steps, the proposed scheme can be extended straightforwardly from one‐dimensional to multi‐dimensional problems without much difficulty and complication. To investigate the computational performances of the proposed scheme five numerical examples are considered: (i) dispersion of Gaussian concentration distribution in one‐dimensional uniform flow; (ii) one‐dimensional viscous Burgers equation; (iii) pure advection of Gaussian concentration distribution in two‐dimensional uniform flow; (iv) pure advection of Gaussian concentration distribution in two‐dimensional rigid‐body rotating flow; and (v) three‐dimensional diffusion in a shear flow. In comparison not only with the QUICKEST scheme, the fully time‐centred implicit QUICK scheme and the fully time‐centred implicit TCSD scheme for one‐dimensional problem but also with the ADI‐QUICK scheme, the ADI‐TCSD scheme and the MOSQUITO scheme for two‐dimensional problems, the proposed scheme shows convincing computational performances. Copyright © 2001 John Wiley &amp; Sons, Ltd.

The block alternating group explicit method (blage) to solve 2D-elliptic differential equations

In this paper, we present the Block Alternating Group Explicit Method (BLAGE) to solve the block tridiagonal linear system of equations derived from the discretisation of 2D-elliptic boundary value problems. The convergence of the method is proved and the case for optimum acceleration parameters is also studied. The BLAGE method is used to solve two partial differential equations for which it is required to solve block tridiagonal linear system involving the solution of simpler tridiagonal system of equations. The method is accurate and flexible.

Application of an improved self-starter to geoacoustic inversion

The self-starter is improved using the operator of the split-step Pade solution. In addition to providing greater stability and being applicable closer to the source, the improved self-starter is an efficient forward model for geoacoustic inversion. It is necessary to solve only O(10) tridiagonal systems of equations to obtain the acoustic field on a vertical array located O(10) wavelengths from a source. This experimental configuration is effective for geoacoustic inverse problems involving unknown parameters deep in the ocean bottom. For problems involving depth-dependent acoustic parameters, the improved self-starter can be used to solve nonlinear inverse problems involving O(10) unknown sediment parameters in less than a minute on the current generation of workstations.