Showing papers on "Numerical analysis published in 1999"

Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction

Abstract: The Equations of Fluid Dynamics.- Notions on Hyperbolic Partial Differential Equations.- Some Properties of the Euler Equations.- The Riemann Problem for the Euler Equations.- Notions on Numerical Methods.- The Method of Godunov for Non#x2014 linear Systems.- Random Choice and Related Methods.- Flux Vector Splitting Methods.- Approximate#x2014 State Riemann Solvers.- The HLL and HLLC Riemann Solvers.- The Riemann Solver of Roe.- The Riemann Solver of Osher.- High#x2013 Order and TVD Methods for Scalar Equations.- High#x2013 Order and TVD Schemes for Non#x2013 Linear Systems.- Splitting Schemes for PDEs with Source Terms.- Methods for Multi#x2013 Dimensional PDEs.- Multidimensional Test Problems.- FORCE Fluxes in Multiple Space Dimensions.- The Generalized Riemann Problem.- The ADER Approach.- Concluding Remarks.

Variational iteration method – a kind of non-linear analytical technique: some examples

Abstract: In this paper, a new kind of analytical technique for a non-linear problem called the variational iteration method is described and used to give approximate solutions for some well-known non-linear problems. In this method, the problems are initially approximated with possible unknowns. Then a correction functional is constructed by a general Lagrange multiplier, which can be identified optimally via the variational theory. Being different from the other non-linear analytical methods, such as perturbation methods, this method does not depend on small parameters, such that it can find wide application in non-linear problems without linearization or small perturbations. Comparison with Adomian’s decomposition method reveals that the approximate solutions obtained by the proposed method converge to its exact solution faster than those of Adomian’s method.

Numerical methods for wave equations in geophysical fluid dynamics

Abstract: Introduction Basic Finite-Difference Methods Beyond Scalar Wave Equations Series-Expansion Methods Finite Volume Methods 6 Semi-Lagrangian Methods Physically Insignificant Fast Waves Non-reflecting Boundary conditions Appendix: Derivations of two fundamental theorems.

The Mortar finite element method with Lagrange multipliers

Abstract: The present paper deals with a variant of a non conforming domain decomposition technique: the mortar finite element method. In the opposition to the original method this variant is never conforming because of the relaxation of the matching constraints at the vertices (and the edges in 3D) of subdomains. It is shown that, written under primal hybrid formulation, the approximation problem, issued from a discretization of a second order elliptic equation in 2D, is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates with respect to natural norms. Finally the parallelization advantages consequence of this variant are also addressed.

On the Approximation of Complicated Dynamical Behavior

Abstract: We present efficient techniques for the numerical approximation of complicated dynamical behavior. In particular, we develop numerical methods which allow us to approximate Sinai--Ruelle--Bowen (SRB)-measures as well as (almost) cyclic behavior of a dynamical system. The methods are based on an appropriate discretization of the Frobenius--Perron operator, and two essentially different mathematical concepts are used: our idea is to combine classical convergence results for finite dimensional approximations of compact operators with results from ergodic theory concerning the approximation of SRB-measures by invariant measures of stochastically perturbed systems. The efficiency of the methods is illustrated by several numerical examples.

Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients

Abstract: We propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients. The main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution. This is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions. As a consequence, the base functions are adapted to the local properties of the differential operator. In this paper, we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale. While such a simplifying assumption is not required by our method, it allows us to use homogenization theory to obtain a useful asymptotic solution structure. The issue of boundary conditions for the base functions will be discussed. Our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution, independently of the small scale in the homogenization limit. Application of our method to problems with continuous scales is also considered.

A critical assessment of the truly Meshless Local Petrov-Galerkin (MLPG), and Local Boundary Integral Equation (LBIE) methods

Abstract: The essential features of the Meshless Local Petrov-Galerkin (MLPG) method, and of the Local Boundary Integral Equation (LBIE) method, are critically examined from the points of view of a non-element interpolation of the field variables, and of the meshless numerical integration of the weak form to generate the stiffness matrix. As truly meshless methods, the MLPG and the LBIE methods hold a great promise in computational mechanics, because these methods do not require a mesh, either to construct the shape functions, or to integrate the Petrov-Galerkin weak form. The characteristics of various meshless interpolations, such as the moving least square, Shepard function, and partition of unity, as candidates for trial and test functions are investigated, and the advantages and disadvantages are pointed out. Emphasis is placed on the characteristics of the global forms of the nodal trial and test functions, which are non-zero only over local sub-domains ΩtrJ and ΩteI, respectively. These nodal trial and test functions are centered at the nodes J and I (which are the centers of the domains ΩtrJ and ΩteI), respectively, and, in general, vanish at the boundaries ∂ΩtrJ and ∂ΩteI of ΩtrJ and ΩteI, respectively. The local domains ΩtrJ and ΩteI can be of arbitrary shapes, such as spheres, rectangular parallelopipeds, and ellipsoids, in 3-Dimensional geometries. The sizes of ΩtrJ and ΩteI can be arbitrary, different from each other, and different for each J, and I, in general. It is shown that the LBIE is but a special form of the MLPG, if the nodal test functions are specifically chosen so as to be the modified fundamental solutions to the differential equations in ΩteI, and to vanish at the boundary ∂ΩteI. The difficulty in the numerical integration of the weak form, to generate the stiffness matrix, is discussed, and a new integration method is proposed. In this new method, the Ith row in the stiffness matrix is generated by integrating over the fixed sub-domain ΩteI (which is the support for the test function centered at node I); or, alternatively the entry KIJ in the global stiffness matrix is generated by integrating over the intersections of the sub-domain ΩtrJ (which is the sub-domain, with node J as its center, and over which the trial function is non-zero), with ΩteI (which is the sub-domain centered at node I over which the test function is non-zero). The generality of the MLPG method is emphasized, and it is pointed that the MLPG can also be the basis of a Galerkin method that leads to a symmetric stiffness matrix. This paper also points out a new but elementary method, to satisfy the essential boundary conditions exactly, in the MLPG method, while using meshless interpolations of the MLS type. This paper presents a critical appraisal of the basic frameworks of the truly meshless MLPG/LBIE methods, and the numerical examples show that the MLPG approach gives good results. It now apears that the MLPG method may replace the well-known Galerkin finite element method (GFEM) as a general tool for numerical modeling, in the not too distant a future.

A Simple Method for Compressible Multifluid Flows

Abstract: A simple second order accurate and fully Eulerian numerical method is presented for the simulation of multifluid compressible flows, governed by the stiffened gas equation of state, in hydrodynamic regime. Our numerical method relies on a second order Godunov-type scheme, with approximate Riemann solver for the resolution of conservation equations, and a set of nonconservative equations. It is valid for all mesh points and allows the resolution of interfaces. This method works for an arbitrary number of interfaces, for breakup and coalescence. It allows very high density ratios (up to 1000). It is able to compute very strong shock waves (pressure ratio up to 10 5). Contrary to all existing schemes (which consider the interface as a discontinuity) the method considers the interface as a numerical diffusion zone as contact discontinuities are computed in compressible single phase flows, but the variables describing the mixture zone are computed consistently with the density, momentum and energy. Several test problems are presented in one, two, and three dimensions. This method allows, for example, the computation of the interaction of a shock wave propagating in a liquid with a gas cylinder, as well as Richtmeyer--Meshkov instabilities, or hypervelocity impact, with realistic initial conditions. We illustrate our method with the Rusanov flux. However, the same principle can be applied to a more general class of schemes.

Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions

Abstract: For high wave numbers, the Helmholtz equation suffers the so-called 'pollution effect'. This effect is directly related to the dispersion. A method to measure the dispersion on any numerical method related to the classical Galerkin FEM is presented. This method does not require to compute the numerical solution of the problem and is extremely fast. Numerical results on the classical Galerkin FEM (p-method) is compared to modified methods presented in the literature. A study of the influence of the topology triangles is also carried out. The efficiency of the different methods is compared. The numerical results in two of the mesh and for square elements show that the high order elements control the dispersion well. The most effective modified method is the QSFEM [1, 2] but it is also very complicated in the general setting. The residual-free bubble [3, 4] is effective in one dimension but not in higher dimensions. The least-square method [1, 5] approach lowers the dispersion but relatively little. The results for triangular meshes show that the best topology is the 'criss-cross' pattern. Copyright © 1999 John Wiley & Sons, Ltd.

Numerical analysis of hygro-thermal behaviour and damage of concrete at high temperature

Abstract: A computational analysis of hygro-thermal and mechanical behaviour of concrete structures at high temperature is presented. The evaluation of thermal, hygral and mechanical performance of this material, including damage effects, needs the knowledge of the heat and mass transfer processes. These are simulated within the framework of a coupled model where non-linearities due to high temperatures are accounted for. The constitutive equations are discussed in some detail. The discretization of the governing equations is carried out by Finite Elements in space and Finite Differences in time. Copyright © 1999 John Wiley & Sons, Ltd.

Fluid Flow Phenomena: A Numerical Toolkit

Abstract: This book deals with the simulation of the incompressible Navier-Stokes equations for laminar and turbulent flows. The book is limited to explaining and employing the finite difference method. It furnishes a large number of source codes which permit to play with the Navier-Stokes equations and to understand the complex physics related to fluid mechanics. Numerical simulations are useful tools to understand the complexity of the flows, which often is difficult to derive from laboratory experiments. This book, then, can be very useful to scholars doing laboratory experiments, since they often do not have extra time to study the large variety of numerical methods; furthermore they cannot spend more time in transferring one of the methods into a computer language. By means of numerical simulations, for example, insights into the vorticity field can be obtained which are difficult to obtain by measurements. This book can be used by graduate as well as undergraduate students while reading books on theoretical fluid mechanics; it teaches how to simulate the dynamics of flow fields on personal computers. This will provide a better way of understanding the theory. Two chapters on Large Eddy Simulations have been included, since this is a methodology that in the near future will allow more universal turbulence models for practical applications. The direct simulation of the Navier-Stokes equations (DNS) is simple by finite-differences, that are satisfactory to reproduce the dynamics of turbulent flows. A large part of the book is devoted to the study of homogeneous and wall turbulent flows. In the second chapter the elementary concept of finite difference is given to solve parabolic and elliptical partial differential equations. In successive chapters the 1D, 2D, and 3D Navier-Stokes equations are solved in Cartesian and cylindrical coordinates. Finally, Large Eddy Simulations are performed to check the importance of the subgrid scale models. Results for turbulent and laminar flows are discussed, with particular emphasis on vortex dynamics. This volume will be of interest to graduate students and researchers wanting to compare experiments and numerical simulations, and to workers in the mechanical and aeronautic industries.

Advanced topics in computational partial differential equations : numerical methods and Diffpack programming

Abstract: X. Cai, E. Acklam, H. P. Langtangen, A. Tveito: Parallel Computing.- X. Cai: Overlapping Domain Decomposition Methods.- K.-A. Mardal, G. W. Zumbusch, H. P. Langtangen: Software Tools for Multigrid Methods.- K.-A. Mardal, H. P. Langtangen: Mixed Finite Elements.- K.-A. Mardal, J. Sundnes, H. P. Langtangen, A. Tveito: Systems of PDEs and Block Preconditioning.- A. Odegard, H. P. Langtangen, A. Tveito: Object-Oriented Implementation of Fully Implicit Methods for Systems of PDEs.- H. P. Langtangen, H. Osnes: Stochastic Partial Differential Equations.- H. P. Langtangen and K.-A. Mardal: Using Diffpack from Python Scripts.- X. Cai, A. M. Bruaset, H. P. Langtangen, G. T. Lines, K. Samuelsson, W. Shen, A. Tveito, G. Zumbusch: Performance Modeling of PDE Solvers.- J. Sundnes, G.T. Lines, P. Grottum, A. Tveito: Numerical Methods and Software for Modeling the Electrical Activity in the Human Heart.- O. Skavhaug, B. F. Nielsen, A. Tveito: Mathematical Models of Financial Derivatives.- O. Skavhaug, B. F. Nielsen, A. Tveito: Numerical Methods for Financial Derivatives.- T. Thorvaldsen, H. P. Langtangen, H. Osnes: Finite Element Modeling of Elastic Structures.- K. M. Okstad, T. Kvamsdal: Simulation of Aluminum Extrusion.- A. Kjeldstad, H. P. Langtangen, J. Skogseid, K. Bjorlykke: Simulation of Deformations, Fluid Flow and HeatTransfer in Sedimentary Basins

Numerical Methods for Chemical Engineers With Matlab Applications

Abstract: General Algorithm for the Software Developed in This Book. Programs on the website. 1. Numerical Solution of Nonlinear Equations. 2. Numerical Solution of Simultaneous Linear Algebraic Equations. 3. Finite Difference Methods and Interpolation. 4. Numerical Differentiation and Integration. 5. Numerical Solution of Ordinary Differential Equations. 6. Numerical Solution of Partial Differential Equations. 7. Linear and Nonlinear Regression Analysis. Appendix: Introduction to MATLAB. Index.

A fast numerical scheme for computing the response of composites using grid refinement

Abstract: A numerical scheme is presented to compute the response of a composite material to an applied field. The scheme is based on an iteration that uses a Fourier transform based projection method to compute the response of a reference material to the applied field, and a contraction mapping to relate the response of the reference material to the response of the composite material. The scheme also utilizes nested uniform grids to improve performance, and to evaluate the refinement of the grid.

Backward Error Analysis for Numerical Integrators

Abstract: Backward error analysis has become an important tool for understanding the long time behavior of numerical integration methods. This is true in particular for the integration of Hamiltonian systems where backward error analysis can be used to show that a symplectic method will conserve energy over exponentially long periods of time. Such results are typically based on two aspects of backward error analysis: (i) It can be shown that the modified vector fields have some qualitative properties which they share with the given problem and (ii) an estimate is given for the difference between the best interpolating vector field and the numerical method. These aspects have been investigated recently, for example, by Benettin and Giorgilli in [ J. Statist. Phys., 74 (1994), pp. 1117--1143], by Hairer in [Ann. Numer. Math., 1 (1994), pp. 107--132], and by Hairer and Lubich in [ Numer. Math., 76 (1997), pp. 441--462]. In this paper we aim at providing a unifying framework and a simplification of the existing results and corresponding proofs. Our approach to backward error analysis is based on a simple recursive definition of the modified vector fields that does not require explicit Taylor series expansion of the numerical method and the corresponding flow maps as in the above-cited works. As an application we discuss the long time integration of chaotic Hamiltonian systems and the approximation of time averages along numerically computed trajectories.

Numerical Methods for Gasdynamic Systems on Unstructured Meshes

Abstract: This article considers stabilized finite element and finite volume discretization techniques for systems of conservation laws. Using newly developed techniques in entropy symmetrization theory, simplified forms of the Galerkin least-squares (GLS) and the discontinuous Galerkin (DG) finite element method are developed and analyzed. The use of symmetrization variables yields numerical schemes which inherit global entropy stability properties of the PDE system. Detailed consideration is given to symmetrization of the Euler, Navier-Stokes, and magneto-hydrodynamic (MHD) equations. Numerous calculations are presented to evaluate the spatial accuracy and feature resolution capability of the simplified DG and GLS discretizations. Next, upwind finite volume methods are reviewed. Specifically considered are generalizations of Godunov’s method to high order accuracy and unstructured meshes. An important component of high order accurate Godunov methods is the spatial reconstruction operator. A number of reconstruction operators are reviewed based on Green-Gauss formulas as well as least-squares approximation. Several theoretical results using maximum principle analysis are presented for the upwind finite volume method. To assess the performance of the upwind finite volume technique, various numerical calculations in computational fluid dynamics are provided.

A transient two-dimensional finite volume model for the simulation of vertical U-tube ground heat exchangers

Abstract: The ability to predict both the long-term and short-term behavior of ground loop heat exchangers is critical to the design and energy analysis of ground source heat pump systems. A numerical model for the simulation of transient heat transfer in vertical ground loop heat exchangers is presented. The model is based on a two-dimensional fully implicit finite volume formulation. Numerical grids have been generated for different pipe sizes, shank spacing and borehole sizes using an automated parametric grid generation algorithm. The numerical method and grid generation techniques have been validated against an analytical model. The model has been developed with two main purposes in mind. The first application is used in a parameter estimation technique used to find the borehole thermal properties from short time scale test data. The second application is the calculation of nondimensional temperature response factors for short time scales that can be used in annual energy simulation.

Computational Partial Differential Equations: Numerical Methods and Diffpack Programming

Abstract: From the Publisher: The target audience of this book is students and researchers in computational sciences who need to develop computer codes for solving partial differential equations. The exposition is focused on numerics and software related to mathematical models in solid and fluid mechanics. The book teaches finite element methods and basic finite difference methods from a computational point of view. The main emphasis regards development of flexible computer programs, using the numerical library Diffpack. The application of Diffpack is explained in detail for problems including model equations in applied mathematics, heat transfer, elasticity, and viscous fluid flow. Diffpack is a modern software development environment based on C++ and object-oriented programming.

A Gautschi-type method for oscillatory second-order differential equations

Abstract: We study a numerical method for second-order differential equations in which high-frequency oscillations are generated by a linear part. For example, semilinear wave equations are of this type. The numerical scheme is based on the requirement that it solves linear problems with constant inhomogeneity exactly. We prove that the method admits second-order error bounds which are independent of the product of the step size with the frequencies. Our analysis also provides new insight into the m ollified impulse method of Garcia-Archilla, Sanz-Serna, and Skeel. We include results of numerical experiments with the sine-Gordon equation.

A Second-Order Rosenbrock Method Applied to Photochemical Dispersion Problems

Abstract: A second-order, L-stable Rosenbrock method from the field of stiff ordinary differential equations is studied for application to atmospheric dispersion problems describing photochemistry, advective, and turbulent diffusive transport. Partial differential equation problems of this type occur in the field of air pollution modeling. The focal point of the paper is to examine the Rosenbrock method for reliable and efficient use as an atmospheric chemical kinetics box-model solver within Strang-type operator splitting. In addition, two W-method versions of the Rosenbrock method are discussed. These versions use an inexact Jacobian matrix and are meant to provide alternatives for Strang-splitting. Another alternative for Strang-splitting is a technique based on so-called source-splitting. This technique is briefly discussed.