Stability Analysis of Difference Methods for Parabolic Initial Value Problems

A decomposition of the numerical solution can be defined by the normal mode representation, that generalizes further the spatial eigenmode decomposition of the von Neumann analysis by taking into account the boundary conditions which are not periodic. In this paper we present some new theoretical results on normal mode analysis for a linear and parabolic initial value problem. Furthermore we suggest an algorithm for the calculation of stability regions based on the normal mode theory.     

Optimal Error Estimates of Two Mixed Finite Element Methods for Parabolic Integro-Differential Equations with Nonsmooth Initial Data

In the first part of this article, a new mixed method is proposed and analyzed for parabolic integro-differential equations (PIDE) with nonsmooth initial data. Compared to the standard mixed method for PIDE, the present method does not bank on a reformulation using a resolvent operator. Based on energy arguments combined with a repeated use of an integral operator and without using parabolic type duality technique, optimal $L^2$ L 2 -error estimates are derived for semidiscrete approximations, when the initial condition is in $L^2$ L 2 . Due to the presence of the integral term, it is, further, observed that a negative norm estimate plays a crucial role in our error analysis. Moreover, the proposed analysis follows the spirit of the proof techniques used in deriving optimal error estimates for finite element approximations to PIDE with smooth data and therefore, it unifies both the theories, i.e., one for smooth data and other for nonsmooth data. Finally, we extend the proposed analysis to the standard mixed method for PIDE with rough initial data and provide an optimal error estimate in $L^2,$ L 2 , which improves upon the results available in the literature.     

Spectral Element Method for Mixed Inhomogeneous Boundary Value Problems of Fourth Order

In this paper, we investigate spectral element method for fourth order problems with mixed inhomogeneous boundary conditions. Some results on the composite Legendre quasi-orthogonal approximation are established, which play important roles in spectral element method with non-uniform meshes and non-uniform approximation modes. As an example of applications, the spectral element scheme is provided for a model problem, with the convergence analysis. Numerical results demonstrate its spectral accuracy, and coincide with the analysis well. In particular, the suggested method is convenient for local mesh refinement and local mode increment, and so it works well even for the solutions changing rapidly, oscillating seriously, or behaving differently in different subdomains.     

Mass- and Momentum Conservation of the Least-Squares Spectral Element Method for the Stokes Problem

The opinion that least-squares methods are not useful due to their poor mass conserving property should be revised. It will be shown that least-squares spectral element methods perform poorly with respect to mass conservation, but this is compensated with a superior momentum conservation. With these new insights, one can firmly state that the least-squares spectral element method remains an interesting alternative for the commonly used Galerkin spectral element formulation     

A Spectral Element Projection Scheme for Incompressible Flow with Application to the Unsteady Axisymmetric Stokes Problem

This paper presents a modified Goda scheme in the simulation of unsteady incompressible Navier–Stokes flows in cylindrical geometries. The study is restricted to the case of axisymmetric flows. For the justification of the robustness of our scheme some computational test cases are investigated. It turns out that by adopting the new approach, a significant accuracy improvement on both pressure and velocity can be obtained relative to the classical Goda scheme.     

A Least-Squares Spectral Element Formulation for the Stokes Problem

Least-squares spectral element methods seem very promising since they combine the generality of finite element methods with the accuracy of the spectral methods and also the theoretical and computational advantages in the algorithmic design and implementation of the least-squares methods. The new element in this work is the choice of spectral elements for the discretization of the least-squares formulation for its superior accuracy due to the high-order basis-functions. The main issue of this paper is the derivation of a least-squares spectral element formulation for the Stokes equations and the role of the boundary conditions on the coercivity relations. The numerical simulations confirm the usual exponential rate of convergence when p-refinement is applied which is typical for spectral element discretization.     

A Spectral Method for the Time Evolution in Parabolic Problems

Numerical time propagation of linear parabolic problems is commonly performed by Taylor expansion based schemes, such as Runge–Kutta. However, explicit schemes of this type impose a stringent stability restriction on the time step when the space discretization matrix is poorly conditioned. Thus the computational work required for integration over a long and fixed time interval is controlled by stability rather than by accuracy of the scheme. We develop an improved time evolution scheme based on a new Chebyshev series expansion for solving time-dependent inhomogeneous parabolic initial-boundary value problems in which the stability condition is relaxed. Spectral accuracy of the time evolution scheme is achieved. Additionally, the approximation derived here can be useful for solving quasi-linear parabolic evolution problems by exponential time differencing methods     

Analysis of a Discontinuous Least Squares Spectral Element Method

This paper addresses the development of a Discontinuous Spectral Least-Squares method. Based on pre-multiplication with a mesh-dependent function a discontinuous functional can be set up. Coercivity of this functional will be established. An example of the approximation to a continuous solution and a solution in which a jump is prescribed will be presented. The discontinuous least-squares method preserves symmetry and positive definiteness of the discrete system.     

High Accuracy Difference Formulae For A Fourth Order Quasi-Linear Parabolic Initial Boundary Value Problem Of First Kind

In this paper, new three level implicit finite difference methods of O(k^2+h^2) and O(k^2+h^4) are proposed for the numerical solution of fourth order quasi-linear parabolic partial differential equations in one space variable, where k\gt 0 and h\gt 0 are grid sizes in time and space coordinates respectively. In both cases, we use only nine grid points. The numerical solution of \partial u/\partial x is obtained as a by-product of the method. The characteristic equation for a model problem is established. Application to a linear singular equation is also discussed in detail. Four examples illustrate the utility of the new difference methods.     

Wrapping Function of the Initial Value Problem for ODE: Applications

We discuss applications of the concept of wrapping function in studying the wrapping effect associated with validated (interval) methods for numerical solution of the initial value problem for ordinary differential equations. Initial value problems are characterized with respect to the occurrence of wrapping effect using that there is no wrapping effect if and only if the wrapping function equals the optimal interval enclosure of the solution. Particular attention is paid to linear systems of ODEs where the functions quantifying the wrapping effect can be presented in a simpler form. A necessary condition for no wrapping effect is proved for the general case.     

An Object-Oriented Toolbox for Spectral Element Analysis

A common bottleneck for numericists is the complexity of the implementation programs. The usual procedural programming approach demands time and effort to program, develop, and test new formulations. This article addresses a particularly involved subject area, that of spectral element methods with mortars for large-scale applications. It is shown that the implementation burden can be alleviated by resorting to an object-oriented design approach. A toolbox consisting of a set of object-oriented classes is discussed. In order to solve a particular problem at hand, the user proceeds by creating an application where he/she loosely activates objects of the classes. When an operation exceeds the functionalities of the classes, the user can enrich these classes or create new ones. Practical examples are provided. Issues concerning computational efficiency and concurrent execution are addressed.     

Dispersion Analysis for Discontinuous Spectral Element Methods

A framework for studying the amplitude and phase errors for discontinuous spectral element methods applied to wave propagation problems is presented. In this framework, boundary conditions can be accounted for and the spatial distribution of the errors within individual elements can be obtained. This is of importance for spectral element discretizations, for which it might be convenient to have the element size larger than the wavelength. When applied to multiple element discretizations, this allows identification of criteria for reducing the errors. While these criteria depend in general on the particular application and the discretization itself, an attempt is made to obtain optimal methods for the case when the wave propagation takes place over a large number of elements. Unfortunately, such optimization leads in the most general case to full mass matrices, and hence is useful mainly for linear problems.     

An Efficient Double Legendre Spectral Method for Parabolic and Elliptic Partial Differential Equations

We present a double Legendre spectral methods that allow the efficient approximate solution for the parabolic partial differential equations in a square subject to the most general inhomo-geneous mixed boundary conditions. The differential equations with their boundary and initial conditions are reduced to systems of ordinary differential equations for the time-dependent expansion coefficients. These systems are greatly simplified by using tensor matrix algebra, and are solved by using the step-by-step method. One numerical application of how to use these methods is described. Numerical results obtained compare favorably with those of the analytical solution. Accurate double Legendre spectral approximations for Poisson' and Helmholtz' equations are also noted.     

Application of a Spectral Element Method to Two-Dimensional Forward-Facing Step Flow

In the present study, we investigate the two-dimensional laminar flow through a one-sided constriction of a plane channel with a ratio of h:H=1:4 (where h is the step height and H is the channel height). The computational approach employed is based on a mixed implicit/explicit time discretization scheme together with a highly accurate spatial discretization using a P _N –P _N–2 spectral-element method. It is well known that this so-called forward-facing step (FFS) flow exhibits a singularity in the pressure and the velocity derivatives at the corner point. We account for this singularity by a geometric mesh refinement strategy that was proposed in a hp-FEM context. A detailed numerical study of the FFS flow reveals that length and height of the recirculation zone in front of the step are almost constant for creeping flow. In the limit of high Reynolds numbers the length and height of the recirculation zone increase proportional to Re ^0.6 and Re ^0.2, respectively.     

Stable Computer Method for Solving Initial Value Problems with Engineering Applications

Engineering and applied mathematics disciplines that involve differential equations in general, and initial value problems in particular, include classical mechanics, thermodynamics, electromagnetism, and the general theory of relativity. A reliable, stable, efficient, and consistent numerical scheme is frequently required for modelling and simulation of a wide range of real-world problems using differential equations. In this study, the tangent slope is assumed to be the contra-harmonic mean, in which the arithmetic mean is used as a correction instead of Euler’s method to improve the efficiency of the improved Euler’s technique for solving ordinary differential equations with initial conditions. The stability, consistency, and efficiency of the system were evaluated, and the conclusions were supported by the presentation of numerical test applications in engineering. According to the stability analysis, the proposed method has a wider stability region than other well-known methods that are currently used in the literature for solving initial-value problems. To validate the rate convergence of the numerical technique, a few initial value problems of both scalar and vector valued types were examined. The proposed method, modified Euler explicit method, and other methods known in the literature have all been used to calculate the absolute maximum error, absolute error at the last grid point of the integration interval under consideration, and computational time in seconds to test the performance. The Lorentz system was used as an example to illustrate the validity of the solution provided by the newly developed method. The method is determined to be more reliable than the commonly existing methods with the same order of convergence, as mentioned in the literature for numerical calculations and visualization of the results produced by all the methods discussed, Mat Lab-R2011b has been used.     

Spectral Data Modeling for a Lighting Application

In computer graphics, image generation uses the RGB colorimetric system. However, this system does not produce an accurate simulation of the spectral characteristics of both light and material, due to the fact that it is device dependent. Indeed, to get realistic images, the image calculation process must deal with spectral characteristics of lights and materials and with the problem of sampling the wavelength domain. Methods have been shown that are a good way to solve the two problems mentioned above, however, these approaches do not take into account complex spectrums (discontinuous). In this paper, we propose a method which removes the constraints imposed by the current methods. Our method is based upon an algorithm of spectrum analysis.     

Hybrid Multigrid/Schwarz Algorithms for the Spectral Element Method

We study the performance of the multigrid method applied to spectral element (SE) discretizations of the Poisson and Helmholtz equations. Smoothers based on finite element (FE) discretizations, overlapping Schwarz methods, and point-Jacobi are considered in conjunction with conjugate gradient and GMRES acceleration techniques. It is found that Schwarz methods based on restrictions of the originating SE matrices converge faster than FE-based methods and that weighting the Schwarz matrices by the inverse of the diagonal counting matrix is essential to effective Schwarz smoothing. Several of the methods considered achieve convergence rates comparable to those attained by classic multigrid on regular grids.     

Exponentially Accurate Spectral Element Method for Fourth Order Elliptic Problems

In this paper, a fully non-conforming least-squares spectral element method for fourth order elliptic problems on smooth domains is presented. The proposed method works for a general fourth order elliptic operator with non-homogeneous data in two dimensions and gives exponentially accurate solutions. We derive differentiability estimates and prove our main stability estimate theorem using a non-conforming spectral element method. We then formulate a numerical scheme using a block diagonal preconditioner. Error estimates are also proven for the proposed method. We provide the computational complexity of our method and present results of numerical simulations that have been performed to validate the theory.