A New Implementation of GMRES Using Generalized Purcell Method

In this paper, a new method based on the generalized Purcell method is proposed to solve the usual least-squares problem arising in the GMRES method. The theoretical aspects and computational results of the method are provided. For the popular iterative method GMRES, the decomposition matrices of the Hessenberg matrix is obtained by using a simple recursive relation instead of Givens rotations. The other advantages of the proposed method are low computational cost and no need for orthogonal decomposition of the Hessenberg matrix or pivoting. The comparisons for ill-conditioned sparse standard matrices are made. They show a good agreement with available literature

Numerical solution of linear time delay systems using Chebyshev-tau spectral method

In this paper, a hybrid method based on method of steps and a Chebyshev-tau spectral method for solving linear time delay systems of differential equations is proposed. The method first converts the time delay system to a system of ordinary dierential equations by the method of steps and then employs Chebyshev polynomials to construct an approx- imate solution for the system. In fact, the solution of the system is expanded in terms of orthogonal Chebyshev polynomials which reduces the solution of the system to the solution of a system of algebraic equations. Also, we transform the coefficient matrix of the algebraic system to a block quasi upper triangular matrix and the latter system can be solved more efficiently than the first one. Furthermore, using orthogonal Chebyshev polynomials enables us to apply fast Fourier transform for calculating matrix-vector multiplications which makes the proposed method to be more efficient. Consistency, stability and convergence analysis of the method are provided. Numerous numerical examples are given to demonstrate efficiency and accuracy of the method. Comparisons are made with available literature

L2-stability analysis of novel ETD scheme for Kuramoto–Sivashinsky equations

AbstractThe aim of this paper is to study the stability analysis of novel ETD scheme proposed by the authors based on spectral methods, the exponential time differencing and Taylor expansion. Stability issue of the proposed numerical scheme is related to an analysis of the stability of the corresponding ODE system for time marching approach. It is proved that the novel scheme is L2-stable in solving the Kuramoto–Sivashinsky model problems. The truncation error and the stability region for the novel scheme are provided. Comparisons with available literature are made

Numerical Solution of Some Nonlinear Volterra Integral Equations of the First Kind

In this paper, the solving of a class of the nonlinear Volterra integral equations (NVIE) of the first kind is investigated. Here, we convert NVIE of the first kind to a linear equation of the second kind. Then we apply the operational Tau method to the problem and prove convergence of the presented method. Finally, some numerical examples are given to show the accuracy of the method

Single-Phase Flow of Non-Newtonian Fluids in Porous Media

The study of flow of non-Newtonian fluids in porous media is very important and serves a wide variety of practical applications in processes such as enhanced oil recovery from underground reservoirs, filtration of polymer solutions and soil remediation through the removal of liquid pollutants. These fluids occur in diverse natural and synthetic forms and can be regarded as the rule rather than the exception. They show very complex strain and time dependent behavior and may have initial yield-stress. Their common feature is that they do not obey the simple Newtonian relation of proportionality between stress and rate of deformation. Non-Newtonian fluids are generally classified into three main categories: time-independent whose strain rate solely depends on the instantaneous stress, time-dependent whose strain rate is a function of both magnitude and duration of the applied stress and viscoelastic which shows partial elastic recovery on removal of the deforming stress and usually demonstrates both time and strain dependency. In this article the key aspects of these fluids are reviewed with particular emphasis on single-phase flow through porous media. The four main approaches for describing the flow in porous media are examined and assessed. These are: continuum models, bundle of tubes models, numerical methods and pore-scale network modeling.Comment: 94 pages, 12 figures, 1 tabl

An explicit spectral collocation method for the drug release coronary stents

This research aims to solve a comprehensive one-dimensional model of drug release from cardiovascular stents in which the drug binding is saturable and reversible. We used the Lagrange collocation method for space dimension and the modified Euler method for time discretization. The existence and uniqueness of the solution, are provided. The consistency, stability, and convergence analysis of the proposed scheme are provided, to show that numerical simulations are valid. Numerical results accurate enough and efficient just by using fewer mesh

NUMERICAL SIMULATION OF DRUG RELEASE CORONARY STENTS USING A SEMI DISCRETE SPECTRAL COLLOCATION METHOD

Abstract: Cardiovascular diseases which include atherosclerosis, are one of the main cause of death in the industrialized world. The most common treatment method for these diseases is a cardiovascular stent. The problem is governed by a set of linear partial differential equations with appropriate boundary conditions. A semi-discrete Chebyshev spectral collocation method aims to find numerical solutions for the reduced unsteady 2-dimension problem of drug release from the stent. The scheme uses the Chebyshev spectral collocation method to approximate the space derivatives and an analytical solution for temporal space. Numerical solutions were carried out on the concentration of the drug in the wall of the tissue. The drug release profile provides important information about its effect on the delivery of therapeutic agents to the vessel wall. For simplicity, one shape of stent and their surrounding normal tissues are selected, and Fick law was employed. The results suggest that the profile of the drug release from the stent has a 2 dimensional hyperbolic shape. Numerical analysis of the error and the rate of convergence of the scheme are also discussed. The proposed scheme is simple to set up, efficient to implement and requires less computational costs than other methods available

Viscoelastic flow in an undulating tube using spectral methods

The flow of a viscoelastic fluid through an undulating tube is considered. The fluid is modelled using the UCM constitutive equation. The governing set of equations is solved using a time-splitting technique. This is based on separate treatments of the convection and generalised Stokes operators. The spatial discretisation is based on a spectral discretisation in which the radial basis functions satisfy the conditions along the axis of symmetry of the tube explicitly. Compatible approximation spaces are chosen for velocity, pressure and extra-stress. The convection problem is solved in a type-sensitive manner using a high-order Runge–Kutta method. The weak formulation of the generalised Stokes problem is discretised and solved using a nested conjugate gradient technique. The effect of elasticity on flow resistance is investigated and comparisons made with other results in the literature

Implicit extended discontinuous Galerkin scheme for solving singularly perturbed Burgers' equations

We present the implicit-modal discontinuous Galerkin scheme for solving the coupled viscous and singularly perturbed Burgers’ equations. This scheme overcomes overshoot and undershoots phenomena in the singularly perturbed Burgers’ equations. We present the stability analysis and obtain suitable ranges for penalty terms and time steps. Also, we gain the constant of trace inequality for the approximate function and its first derivatives based on Legendre basis functions. The numerical results have good agreement with the analytical and available approximate solutions