Basics of Discretization Methods
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Chapter 4 Finite Difference Gradient Information
Chapter 4 Finite Difference Gradient Information In the masters thesis Thoresson (2003), the feasibility of using gradient- based approximation methods for the optimisation of the spring and damper characteristics of an off-road vehicle, for both ride comfort and handling, was investigated. The Sequential Quadratic Programming (SQP) algorithm and the locally developed Dynamic-Q method were the two successive approximation methods used for the optimisation. The determination of the objective function value is performed using computationally expensive numerical simulations that exhibit severe inherent numerical noise. The use of forward finite differences and central finite differences for the determination of the gradients of the objective function within Dynamic-Q is also investigated. The results of this study, presented here, proved that the use of central finite differencing for gradient information improved the optimisation convergence history, and helped to reduce the difficulties associated with noise in the objective and constraint functions. This chapter presents the feasibility investigation of using gradient-based successive approximation methods to overcome the problems of poor gradient information due to severe numerical noise. The two approximation methods applied here are the locally developed Dynamic-Q optimisation algorithm of Snyman and Hay (2002) and the well known Sequential Quadratic 47 CHAPTER 4. FINITE DIFFERENCE GRADIENT INFORMATION 48 Programming (SQP) algorithm, the MATLAB implementation being used for this research (Mathworks 2000b). This chapter aims to provide the reader with more information regarding the Dynamic-Q successive approximation algorithm, that may be used as an alternative to the more established SQP method. Both algorithms are evaluated for effectiveness and efficiency in determining optimal spring and damper characteristics for both ride comfort and handling of a vehicle. -
Finite Difference Methods for Solving Differential
FINITE DIFFERENCE METHODS FOR SOLVING DIFFERENTIAL EQUATIONS I-Liang Chern Department of Mathematics National Taiwan University May 16, 2013 2 Contents 1 Introduction 3 1.1 Finite Difference Approximation . ........ 3 1.2 Basic Numerical Methods for Ordinary Differential Equations ........... 5 1.3 Runge-Kuttamethods..... ..... ...... ..... ...... ... ... 8 1.4 Multistepmethods ................................ 10 1.5 Linear difference equation . ...... 14 1.6 Stabilityanalysis ............................... 17 1.6.1 ZeroStability................................. 18 2 Finite Difference Methods for Linear Parabolic Equations 23 2.1 Finite Difference Methods for the Heat Equation . ........... 23 2.1.1 Some discretization methods . 23 2.1.2 Stability and Convergence for the Forward Euler method.......... 25 2.2 L2 Stability – von Neumann Analysis . 26 2.3 Energymethod .................................... 28 2.4 Stability Analysis for Montone Operators– Entropy Estimates ........... 29 2.5 Entropy estimate for backward Euler method . ......... 30 2.6 ExistenceTheory ................................. 32 2.6.1 Existence via forward Euler method . ..... 32 2.6.2 A Sharper Energy Estimate for backward Euler method . ......... 33 2.7 Relaxationoferrors.............................. 34 2.8 BoundaryConditions ..... ..... ...... ..... ...... ... 36 2.8.1 Dirichlet boundary condition . ..... 36 2.8.2 Neumann boundary condition . 37 2.9 The discrete Laplacian and its inversion . .......... 38 2.9.1 Dirichlet boundary condition . ..... 38 3 Finite Difference Methods for Linear elliptic Equations 41 3.1 Discrete Laplacian in two dimensions . ........ 41 3.1.1 Discretization methods . 41 3.1.2 The 9-point discrete Laplacian . ..... 42 3.2 Stability of the discrete Laplacian . ......... 43 3 4 CONTENTS 3.2.1 Fouriermethod ................................ 43 3.2.2 Energymethod ................................ 44 4 Finite Difference Theory For Linear Hyperbolic Equations 47 4.1 A review of smooth theory of linear hyperbolic equations ............. -
Exercises from Finite Difference Methods for Ordinary and Partial
Exercises from Finite Difference Methods for Ordinary and Partial Differential Equations by Randall J. LeVeque SIAM, Philadelphia, 2007 http://www.amath.washington.edu/ rjl/fdmbook ∼ Under construction | more to appear. Contents Chapter 1 4 Exercise 1.1 (derivation of finite difference formula) . 4 Exercise 1.2 (use of fdstencil) . 4 Chapter 2 5 Exercise 2.1 (inverse matrix and Green's functions) . 5 Exercise 2.2 (Green's function with Neumann boundary conditions) . 5 Exercise 2.3 (solvability condition for Neumann problem) . 5 Exercise 2.4 (boundary conditions in bvp codes) . 5 Exercise 2.5 (accuracy on nonuniform grids) . 6 Exercise 2.6 (ill-posed boundary value problem) . 6 Exercise 2.7 (nonlinear pendulum) . 7 Chapter 3 8 Exercise 3.1 (code for Poisson problem) . 8 Exercise 3.2 (9-point Laplacian) . 8 Chapter 4 9 Exercise 4.1 (Convergence of SOR) . 9 Exercise 4.2 (Forward vs. backward Gauss-Seidel) . 9 Chapter 5 11 Exercise 5.1 (Uniqueness for an ODE) . 11 Exercise 5.2 (Lipschitz constant for an ODE) . 11 Exercise 5.3 (Lipschitz constant for a system of ODEs) . 11 Exercise 5.4 (Duhamel's principle) . 11 Exercise 5.5 (matrix exponential form of solution) . 11 Exercise 5.6 (matrix exponential form of solution) . 12 Exercise 5.7 (matrix exponential for a defective matrix) . 12 Exercise 5.8 (Use of ode113 and ode45) . 12 Exercise 5.9 (truncation errors) . 13 Exercise 5.10 (Derivation of Adams-Moulton) . 13 Exercise 5.11 (Characteristic polynomials) . 14 Exercise 5.12 (predictor-corrector methods) . 14 Exercise 5.13 (Order of accuracy of Runge-Kutta methods) . -
On the Use of Discrete Fourier Transform for Solving Biperiodic Boundary Value Problem of Biharmonic Equation in the Unit Rectangle
On the Use of Discrete Fourier Transform for Solving Biperiodic Boundary Value Problem of Biharmonic Equation in the Unit Rectangle Agah D. Garnadi 31 December 2017 Department of Mathematics, Faculty of Mathematics and Natural Sciences, Bogor Agricultural University Jl. Meranti, Kampus IPB Darmaga, Bogor, 16680 Indonesia Abstract This note is addressed to solving biperiodic boundary value prob- lem of biharmonic equation in the unit rectangle. First, we describe the necessary tools, which is discrete Fourier transform for one di- mensional periodic sequence, and then extended the results to 2- dimensional biperiodic sequence. Next, we use the discrete Fourier transform 2-dimensional biperiodic sequence to solve discretization of the biperiodic boundary value problem of Biharmonic Equation. MSC: 15A09; 35K35; 41A15; 41A29; 94A12. Key words: Biharmonic equation, biperiodic boundary value problem, discrete Fourier transform, Partial differential equations, Interpola- tion. 1 Introduction This note is an extension of a work by Henrici [2] on fast solver for Poisson equation to biperiodic boundary value problem of biharmonic equation. This 1 note is organized as follows, in the first part we address the tools needed to solve the problem, i.e. introducing discrete Fourier transform (DFT) follows Henrici's work [2]. The second part, we apply the tools already described in the first part to solve discretization of biperiodic boundary value problem of biharmonic equation in the unit rectangle. The need to solve discretization of biharmonic equation is motivated by its occurence in some applications such as elasticity and fluid flows [1, 4], and recently in image inpainting [3]. Note that the work of Henrici [2] on fast solver for biperiodic Poisson equation can be seen as solving harmonic inpainting. -
TERMINOLOGY UNIT- I Finite Element Method (FEM)
TERMINOLOGY UNIT- I The Finite Element Method (FEM) is a numerical technique to find Finite Element approximate solutions of partial differential equations. It was originated from Method (FEM) the need of solving complex elasticity and structural analysis problems in Civil, Mechanical and Aerospace engineering. Any continuum/domain can be divided into a number of pieces with very Finite small dimensions. These small pieces of finite dimension are called Finite Elements Elements. Field The field variables, displacements (strains) & stresses or stress resultants must Conditions satisfy the governing condition which can be mathematically expressed. A set of independent functions satisfying the boundary conditions is chosen Functional and a linear combination of a finite number of them is taken to approximately Approximation specify the field variable at any point. A structure can have infinite number of displacements. Approximation with a Degrees reasonable level of accuracy can be achieved by assuming a limited number of of Freedom displacements. This finite number of displacements is the number of degrees of freedom of the structure. The formulation for structural analysis is generally based on the three Numerical fundamental relations: equilibrium, constitutive and compatibility. There are Methods two major approaches to the analysis: Analytical and Numerical. Analytical approach which leads to closed-form solutions is effective in case of simple geometry, boundary conditions, loadings and material properties. Analytical However, in reality, such simple cases may not arise. As a result, various approach numerical methods are evolved for solving such problems which are complex in nature. For numerical approach, the solutions will be approximate when any of these Numerical relations are only approximately satisfied. -
The Original Euler's Calculus-Of-Variations Method: Key
Submitted to EJP 1 Jozef Hanc, [email protected] The original Euler’s calculus-of-variations method: Key to Lagrangian mechanics for beginners Jozef Hanca) Technical University, Vysokoskolska 4, 042 00 Kosice, Slovakia Leonhard Euler's original version of the calculus of variations (1744) used elementary mathematics and was intuitive, geometric, and easily visualized. In 1755 Euler (1707-1783) abandoned his version and adopted instead the more rigorous and formal algebraic method of Lagrange. Lagrange’s elegant technique of variations not only bypassed the need for Euler’s intuitive use of a limit-taking process leading to the Euler-Lagrange equation but also eliminated Euler’s geometrical insight. More recently Euler's method has been resurrected, shown to be rigorous, and applied as one of the direct variational methods important in analysis and in computer solutions of physical processes. In our classrooms, however, the study of advanced mechanics is still dominated by Lagrange's analytic method, which students often apply uncritically using "variational recipes" because they have difficulty understanding it intuitively. The present paper describes an adaptation of Euler's method that restores intuition and geometric visualization. This adaptation can be used as an introductory variational treatment in almost all of undergraduate physics and is especially powerful in modern physics. Finally, we present Euler's method as a natural introduction to computer-executed numerical analysis of boundary value problems and the finite element method. I. INTRODUCTION In his pioneering 1744 work The method of finding plane curves that show some property of maximum and minimum,1 Leonhard Euler introduced a general mathematical procedure or method for the systematic investigation of variational problems. -
Finite Difference and Discontinuous Galerkin Methods for Wave Equations
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1522 Finite Difference and Discontinuous Galerkin Methods for Wave Equations SIYANG WANG ACTA UNIVERSITATIS UPSALIENSIS ISSN 1651-6214 ISBN 978-91-554-9927-3 UPPSALA urn:nbn:se:uu:diva-320614 2017 Dissertation presented at Uppsala University to be publicly examined in Room 2446, Polacksbacken, Lägerhyddsvägen 2, Uppsala, Tuesday, 13 June 2017 at 10:15 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Professor Thomas Hagstrom (Department of Mathematics, Southern Methodist University). Abstract Wang, S. 2017. Finite Difference and Discontinuous Galerkin Methods for Wave Equations. Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1522. 53 pp. Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-554-9927-3. Wave propagation problems can be modeled by partial differential equations. In this thesis, we study wave propagation in fluids and in solids, modeled by the acoustic wave equation and the elastic wave equation, respectively. In real-world applications, waves often propagate in heterogeneous media with complex geometries, which makes it impossible to derive exact solutions to the governing equations. Alternatively, we seek approximated solutions by constructing numerical methods and implementing on modern computers. An efficient numerical method produces accurate approximations at low computational cost. There are many choices of numerical methods for solving partial differential equations. Which method is more efficient than the others depends on the particular problem we consider. In this thesis, we study two numerical methods: the finite difference method and the discontinuous Galerkin method. -
Upwind Summation by Parts Finite Difference Methods for Large Scale
Upwind summation by parts finite difference methods for large scale elastic wave simulations in 3D complex geometries ? Kenneth Durua,∗, Frederick Funga, Christopher Williamsa aMathematical Sciences Institute, Australian National University, Australia. Abstract High-order accurate summation-by-parts (SBP) finite difference (FD) methods constitute efficient numerical methods for simulating large-scale hyperbolic wave propagation problems. Traditional SBP FD operators that approximate first-order spatial derivatives with central-difference stencils often have spurious unresolved numerical wave-modes in their computed solutions. Recently derived high order accurate upwind SBP operators based non-central (upwind) FD stencils have the potential to suppress these poisonous spurious wave-modes on marginally resolved computational grids. In this paper, we demonstrate that not all high order upwind SBP FD operators are applicable. Numerical dispersion relation analysis shows that odd-order upwind SBP FD operators also support spurious unresolved high-frequencies on marginally resolved meshes. Meanwhile, even-order upwind SBP FD operators (of order 2; 4; 6) do not support spurious unresolved high frequency wave modes and also have better numerical dispersion properties. For all the upwind SBP FD operators we discretise the three space dimensional (3D) elastic wave equation on boundary-conforming curvilinear meshes. Using the energy method we prove that the semi-discrete ap- proximation is stable and energy-conserving. We derive a priori error estimate and prove the convergence of the numerical error. Numerical experiments for the 3D elastic wave equation in complex geometries corrobo- rate the theoretical analysis. Numerical simulations of the 3D elastic wave equation in heterogeneous media with complex non-planar free surface topography are given, including numerical simulations of community developed seismological benchmark problems. -
Lecture 7: the Complex Fourier Transform and the Discrete Fourier Transform (DFT)
Lecture 7: The Complex Fourier Transform and the Discrete Fourier Transform (DFT) c Christopher S. Bretherton Winter 2014 7.1 Fourier analysis and filtering Many data analysis problems involve characterizing data sampled on a regular grid of points, e. g. a time series sampled at some rate, a 2D image made of regularly spaced pixels, or a 3D velocity field from a numerical simulation of fluid turbulence on a regular grid. Often, such problems involve characterizing, detecting, separating or ma- nipulating variability on different scales, e. g. finding a weak systematic signal amidst noise in a time series, edge sharpening in an image, or quantifying the energy of motion across the different sizes of turbulent eddies. Fourier analysis using the Discrete Fourier Transform (DFT) is a fun- damental tool for such problems. It transforms the gridded data into a linear combination of oscillations of different wavelengths. This partitions it into scales which can be separately analyzed and manipulated. The computational utility of Fourier methods rests on the Fast Fourier Transform (FFT) algorithm, developed in the 1960s by Cooley and Tukey, which allows efficient calculation of discrete Fourier coefficients of a periodic function sampled on a regular grid of 2p points (or 2p3q5r with slightly reduced efficiency). 7.2 Example of the FFT Using Matlab, take the FFT of the HW1 wave height time series (length 24 × 60=25325) and plot the result (Fig. 1): load hw1 dat; zhat = fft(z); plot(abs(zhat),’x’) A few elements (the second and last, and to a lesser extent the third and the second from last) have magnitudes that stand above the noise. -
High Order Gradient, Curl and Divergence Conforming Spaces, with an Application to NURBS-Based Isogeometric Analysis
High order gradient, curl and divergence conforming spaces, with an application to compatible NURBS-based IsoGeometric Analysis R.R. Hiemstraa, R.H.M. Huijsmansa, M.I.Gerritsmab aDepartment of Marine Technology, Mekelweg 2, 2628CD Delft bDepartment of Aerospace Technology, Kluyverweg 2, 2629HT Delft Abstract Conservation laws, in for example, electromagnetism, solid and fluid mechanics, allow an exact discrete representation in terms of line, surface and volume integrals. We develop high order interpolants, from any basis that is a partition of unity, that satisfy these integral relations exactly, at cell level. The resulting gradient, curl and divergence conforming spaces have the propertythat the conservationlaws become completely independent of the basis functions. This means that the conservation laws are exactly satisfied even on curved meshes. As an example, we develop high ordergradient, curl and divergence conforming spaces from NURBS - non uniform rational B-splines - and thereby generalize the compatible spaces of B-splines developed in [1]. We give several examples of 2D Stokes flow calculations which result, amongst others, in a point wise divergence free velocity field. Keywords: Compatible numerical methods, Mixed methods, NURBS, IsoGeometric Analyis Be careful of the naive view that a physical law is a mathematical relation between previously defined quantities. The situation is, rather, that a certain mathematical structure represents a given physical structure. Burke [2] 1. Introduction In deriving mathematical models for physical theories, we frequently start with analysis on finite dimensional geometric objects, like a control volume and its bounding surfaces. We assign global, ’measurable’, quantities to these different geometric objects and set up balance statements. -
A Quasi-Static Particle-In-Cell Algorithm Based on an Azimuthal Fourier Decomposition for Highly Efficient Simulations of Plasma-Based Acceleration
A quasi-static particle-in-cell algorithm based on an azimuthal Fourier decomposition for highly efficient simulations of plasma-based acceleration: QPAD Fei Lia,c, Weiming And,∗, Viktor K. Decykb, Xinlu Xuc, Mark J. Hoganc, Warren B. Moria,b aDepartment of Electrical Engineering, University of California Los Angeles, Los Angeles, CA 90095, USA bDepartment of Physics and Astronomy, University of California Los Angeles, Los Angeles, CA 90095, USA cSLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA dDepartment of Astronomy, Beijing Normal University, Beijing 100875, China Abstract The three-dimensional (3D) quasi-static particle-in-cell (PIC) algorithm is a very effi- cient method for modeling short-pulse laser or relativistic charged particle beam-plasma interactions. In this algorithm, the plasma response, i.e., plasma wave wake, to a non- evolving laser or particle beam is calculated using a set of Maxwell's equations based on the quasi-static approximate equations that exclude radiation. The plasma fields are then used to advance the laser or beam forward using a large time step. The algorithm is many orders of magnitude faster than a 3D fully explicit relativistic electromagnetic PIC algorithm. It has been shown to be capable to accurately model the evolution of lasers and particle beams in a variety of scenarios. At the same time, an algorithm in which the fields, currents and Maxwell equations are decomposed into azimuthal harmonics has been shown to reduce the complexity of a 3D explicit PIC algorithm to that of a 2D algorithm when the expansion is truncated while maintaining accuracy for problems with near azimuthal symmetry. -
Discretization of Integro-Differential Equations
Discretization of Integro-Differential Equations Modeling Dynamic Fractional Order Viscoelasticity K. Adolfsson1, M. Enelund1,S.Larsson2,andM.Racheva3 1 Dept. of Appl. Mech., Chalmers Univ. of Technology, SE–412 96 G¨oteborg, Sweden 2 Dept. of Mathematics, Chalmers Univ. of Technology, SE–412 96 G¨oteborg, Sweden 3 Dept. of Mathematics, Technical University of Gabrovo, 5300 Gabrovo, Bulgaria Abstract. We study a dynamic model for viscoelastic materials based on a constitutive equation of fractional order. This results in an integro- differential equation with a weakly singular convolution kernel. We dis- cretize in the spatial variable by a standard Galerkin finite element method. We prove stability and regularity estimates which show how the convolution term introduces dissipation into the equation of motion. These are then used to prove a priori error estimates. A numerical ex- periment is included. 1 Introduction Fractional order operators (integrals and derivatives) have proved to be very suit- able for modeling memory effects of various materials and systems of technical interest. In particular, they are very useful when modeling viscoelastic materials, see, e.g., [3]. Numerical methods for quasistatic viscoelasticity problems have been studied, e.g., in [2] and [8]. The drawback of the fractional order viscoelastic models is that the whole strain history must be saved and included in each time step. The most commonly used algorithms for this integration are based on the Lubich convolution quadrature [5] for fractional order operators. In [1, 2], we develop an efficient numerical algorithm based on sparse numerical quadrature earlier studied in [6]. While our earlier work focused on discretization in time for the quasistatic case, we now study space discretization for the fully dynamic equations of mo- tion, which take the form of an integro-differential equation with a weakly singu- lar convolution kernel.