Tridiagonal Scheme Research Articles

Tension spline approach to singularly perturbed delay differential equations involving large delay

To solve the singlularly perturbed delay differential equations of convection type, spline in tension technique is used. Taylor series approximation is used to tackle the first order derivative term involved in the problem and finally a tridiagonal scheme is obtained, which is then solved by Thomas algorithm. An example is illustrated to show the method's applicability and it is solved numerically for different values of perturbation parameter. The maximum absolute errors and rate of convergence are given in tables and it is seen that our method is of first order convergent.

Fitted Numerical Scheme for Second-Order Singularly Perturbed Differential-Difference Equations with Mixed Shifts

This paper presents the study of singularly perturbed differential-difference equations of delay and advance parameters. The proposed numerical scheme is a fitted fourth-order finite difference approximation for the singularly perturbed differential equations at the nodal points and obtained a tridiagonal scheme. This is significant because the proposed method is applicable for the perturbation parameter which is less than the mesh size , where most numerical methods fail to give good results. Moreover, the work can also help to introduce the technique of establishing and making analysis for the stability and convergence of the proposed numerical method, which is the crucial part of the numerical analysis. Maximum absolute errors range from 10 − 03 up to 10 − 10 , and computational rate of convergence for different values of perturbation parameter, delay and advance parameters, and mesh sizes are tabulated for the considered numerical examples. Concisely, the present method is stable and convergent and gives more accurate results than some existing numerical methods reported in the literature.

Solution of Singularly Perturbed Boundary Value Problems with Singularity Using Variable Mesh Finite Difference Method

We use non-polynomial spline with variable mesh to establish a numerical scheme for the solution of boundary value problem with singularity. The discrete equation of the problem is developed based on the condition of the class C 1 of non-polynomial spline at the inner nodes and it is not valid for singularity. At singularity t = 0, the problem is modified in order to have a three term relationship. The method’s tridiagonal scheme is analyzed using the well-known discrete imbedding invariant algorithm. We discuss the error analysis of the scheme and two examples with layer at one end of the boundary are consider to demonstrate the practical utility of the scheme. Maximum absolute errors are present in tabular form to show the efficiency of the proposed method.

Solution of convection-diffusion problems with singularity using variable mesh spline of third order

The paper deals with a third order finite difference approach with variable mesh using the non-polynomial spline for the solution of a problems with singularity in convection-diffusion equation. The problem's discretization equation is constructed using the continuity condition at the inner nodes for the derivatives of first order of the non-polynomial spline, which is not valid at singularity. At the singularity, the problem is modified in order to have a three-term relationship. The method's tridiagonal scheme is interpreted by means of discrete invariant imbedding algorithm. Error analysis of the method is analyzed and the maximum absolute error in the solution is tabulated. Layer behaviour is picturized in graphs.

Solution of convection-diffusion problems with singularity using variable mesh spline of third order

The paper deals with a third order finite difference approach with variable mesh using the non-polynomial spline for the solution of a problems with singularity in convection-diffusion equation. The problem's discretization equation is constructed using the continuity condition at the inner nodes for the derivatives of first order of the non-polynomial spline, which is not valid at singularity. At the singularity, the problem is modified in order to have a three-term relationship. The method's tridiagonal scheme is interpreted by means of discrete invariant imbedding algorithm. Error analysis of the method is analyzed and the maximum absolute error in the solution is tabulated. Layer behaviour is picturized in graphs.

A Computational technique for solving Singularly Perturbed Boundary Value Problems

This paper presents a computational technique for solving singularly perturbed boundary value problems with the boundary layer at one end. This technique is a non-iterative on small deviating argument which converts the original second order boundary value problem to the first order differential equation with the deviating argument. We use the trapezoidal method on the first order differential equation; tridiagonal scheme is obtained and is solved efficiently. Numerical examples show the validity of the present method.

The QR iteration method for Hermitian quasiseparable matrices of an arbitrary order

The QR iteration method for tridiagonal matrices is in the heart of one classical method to solve the general eigenvalue problem. In this paper we consider the more general class of quasiseparable matrices that includes not only tridiagonal but also companion, comrade, unitary Hessenberg and semiseparble matrices. A fast QR iteration method exploiting the Hermitian quasiseparable structure (and thus generalizing the classical tridiagonal scheme) is presented. The algorithm is based on an earlier work [Y. Eidelman and I. Gohberg, A modification of the Dewilde–van der Veen method for inversion of finite structured matrices, Linear Algebra Appl. 343–344 (2002) 419–450], and it applies to the general case of Hermitian quasiseparable matrices of an arbitrary order. The algorithm operates on generators (i.e., a linear set of parameters defining the quasiseparable matrix), and the storage and the cost of one iteration are only linear. The results of some numerical experiments are presented. An application of this method to solve the general eigenvalue problem via quasiseparable matrices will be analyzed separately elsewhere.

Implicit schemes with large time step for non‐linear equations: application to river flow hydraulics

AbstractIn this work, first‐order upwind implicit schemes are considered. The traditional tridiagonal scheme is rewritten as a sum of two bidiagonal schemes in order to produce a simpler method better suited for unsteady transcritical flows. On the other hand, the origin of the instabilities associated to the use of upwind implicit methods for shock propagations is identified and a new stability condition for non‐linear problems is proposed. This modification produces a robust, simple and accurate upwind semi‐explicit scheme suitable for discontinuous flows with high Courant–Friedrichs–Lewy (CFL) numbers.The discretization at the boundaries is based on the condition of global mass conservation thus enabling a fully conservative solution for all kind of boundary conditions.The performance of the proposed technique will be shown in the solution of the inviscid Burgers' equation, in an ideal dambreak test case, in some steady open channel flow test cases with analytical solution and in a realistic flood routing problem, where stable and accurate solutions will be presented using CFL values up to 100. Copyright © 2004 John Wiley & Sons, Ltd.

A Numerical Method for the Viscoelastic Melt-Spinning Model with Radial Resolutions of Temperature and Stress Fields

A numerical method to compute the viscoelastic melt-spinning model with radial resolutions of temperature and stress fields is formulated and applied to the low speed range. The starting framework is the reduction of the complete continuous model into both the perturbed two-dimensional model and the perturbed average model obtained from a first-order regular perturbation analysis available in the literature. The polymer rheology is described with the nonisothermal Phan-Thien and Tanner and Giesekus constitutive equations. By using the implicit tridiagonal scheme of finite differences coupled to the fourth-order Runge−Kutta method, an iterative numerical algorithm is proposed for the computation of the coupled balance equations. The temperature and stress fields in the filament as functions of axial and radial positions are obtained for a well-refined mesh and with high numerical precision. The numerical algorithm considers the appropriate interplay between axial and radial varying temperature and stress fields and the rigorous averaged balances of momentum and energy and averaged constitutive equation. The development of a skin-core structure is predicted with the two rheological models.

Moments expansion study of the Rabi Hamiltonian

The interaction of a single bosonic mode with a two-level fermion system is investigated. For the condensed-matter physicist the boson in question is typically a phonon whereas the fermion system is represented by a two-level electron structure. In this work we shall use both the connected moments expansion (CMX) and the alternate moments expansion (AMX) as well as the non-perturbative Lanczos tridiagonal scheme to study the ground-state spectrum of this system. Comparisons will be made with other approximation schemes as well as “exact” methods.

Ground state of a two-level system with phonon coupling

The ground state of a two-level system coupled to a dispersionless phonon bath is studied using both a connected moments expansion and a truncated Lanczos tridiagonal scheme. We consider the spin-boson Hamiltonian, $\ifmmode \hat{H}\else \^{H}\fi{}=\ensuremath{-}{\ensuremath{\delta}}_{o}{\ensuremath{\sigma}}_{x}+{\ensuremath{\sum}}_{\mathbf{k}}\ensuremath{\Elzxh}{\ensuremath{\omega}}_{\mathbf{k}}{a}_{\mathbf{k}}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{\mathbf{k}}+{\ensuremath{\sum}}_{\mathbf{k}}{g}_{\mathbf{k}}{(a}_{\mathbf{k}}^{\ifmmode\dagger\else\textdagger\fi{}}{+a}_{\mathbf{k}}){\ensuremath{\sigma}}_{z},$ where ${\ensuremath{\delta}}_{0}$ is the bare tunneling matrix element and ${g}_{\mathbf{k}}$ represents the coupling to the $\mathbf{k}$ phonon modes. Such systems have found relevance in applications to molecular polaron formation, exciton motion, and attenuation of sound in glasses. Our results are then compared to those of variational methods as well as an exact numerical diagonalization.

Moments expansions for the correlation energy of an exactly solvable problem

In this work we study the ground-state properties of the model Hamiltonian H=12∑i=1N−d2dxi2+ω2xi2+∑i,jNg2xij2, which represents a set of N one-dimensional correlated harmonic oscillators. Here the parameter g describes the coupling between oscillators and may be either real (attractive case) or purely imaginary (repulsive case). We wish to study the correlation energy of the system by two methods: the Lanczos tridiagonal scheme and a newly developed moments expansion AMX (alternate moments expansion). Comparisons will be made with other moments expansions, in particular the CMX−LT(2) and CMX−LT(3) series.

Comparison between Lanczos and linked cluster methods for the Heisenberg antiferromagnet on a square lattice

The Lanczos tridiagonal scheme is applied to investigate the ground-state energy of the 2D anisotropic XXZ Heisenberg Model. Comparisons are made with connected moments and t-expansion methods as well as a recently deveoped plaquette expansion. A number of poles appear in each of the schemes, which has previously been unreported.

Modified oci scheme for two-point boundary-value problems

A sixth-order compact implicit tridiagonal scheme for solving a two-point boundary-value problem (TPBVP) is presented. This scheme is unconditionally stable. Numerical example and comparisons with other schemes are given.

Supra-Convergent Schemes on Irregular Grids

As Tikhonov and Samarskil showed for k = 2, it is not essential that k th-order compact difference schemes be centered at the arithmetic mean of the stencil's points to yield second-order convergence (although it does suffice). For stable schemes and even k, the main point is seen when the k th difference quotient is set equal to the value of the k th derivative at the middle point of the stencil; the proof is particularly transparent for k = 2. For any k, in fact, there is a ( k/2J -parameter family of symmetric averages of the values of the k th derivative at the points of the stencil which, when similarly used, yield second-order convergence. The result extends to stable compact schemes for equations with lower-order terms under general boundary conditions. Although the extension of Numerov's tridiagonal scheme (approximating D2y = f with third-order truncation error) yields fourth-order con- vergence on meshes consisting of a bounded number of pieces in which the mesh size changes monotonically, it yields only third-order convergence to quintic polynomials on any three- periodic mesh with unequal adjacent mesh sizes and fixed adjacent mesh ratios. A result of some independent interest is appended (and applied): it characterizes, simply, those functions of k variables which possess the property that their average value, as one translates over one period of an arbitrary periodic sequence of arguments, is zero; i.e., those bounded functions whose average value, as one translates over arbitrary finite sequences of arguments, goes to zero as the length of the sequences increases.

Supra-convergent schemes on irregular grids

As Tikhonov and Samarskiĭ showed for k = 2 k = 2 , it is not essential that kth-order compact difference schemes be centered at the arithmetic mean of the stencil’s points to yield second-order convergence (although it does suffice). For stable schemes and even k, the main point is seen when the kth difference quotient is set equal to the value of the kth derivative at the middle point of the stencil; the proof is particularly transparent for k = 2 k = 2 . For any k, in fact, there is a ⌊ k / 2 ⌋ \left \lfloor {k/2} \right \rfloor -parameter family of symmetric averages of the values of the kth derivative at the points of the stencil which, when similarly used, yield second-order convergence. The result extends to stable compact schemes for equations with lower-order terms under general boundary conditions. Although the extension of Numerov’s tridiagonal scheme (approximating D 2 y = f {D^2}y = f with third-order truncation error) yields fourth-order convergence on meshes consisting of a bounded number of pieces in which the mesh size changes monotonically, it yields only third-order convergence to quintic polynomials on any three-periodic mesh with unequal adjacent mesh sizes and fixed adjacent mesh ratios. A result of some independent interest is appended (and applied): it characterizes, simply, those functions of k variables which possess the property that their average value, as one translates over one period of an arbitrary periodic sequence of arguments, is zero; i.e., those bounded functions whose average value, as one translates over arbitrary finite sequences of arguments, goes to zero as the length of the sequences increases.