The 26Th International Domain Decomposition Conference Dd Xxvi
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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 ............. -
A Parallel Preconditioner for a FETI-DP Method for the Crouzeix-Raviart finite Element
A parallel preconditioner for a FETI-DP method for the Crouzeix-Raviart finite element Leszek Marcinkowski∗1 Talal Rahman2 1 Introduction In this paper, we present a Neumann-Dirichlet type parallel preconditioner for a FETI-DP method for the nonconforming Crouzeix-Raviart (CR) finite element dis- cretization of a model second order elliptic problem. The proposed method is almost optimal, in fact, the condition number of the preconditioned problem grows poly- logarithmically with respect to the mesh parameters of the local triangulations. In many scientific applications, where partial differential equations are used to model, the Crouzeix-Raviart (CR) finite element has been one of the most com- monly used nonconforming finite element for the numerical solution. This includes applications like the Poisson equation (cf. [11, 23]), the Darcy-Stokes problem (cf. [8]), the elasticity problem (cf. [3]). We also would like to add that there is a close relationship between mixed finite elements and the nonconforming finite element for the second order elliptic problem; cf. [1, 2]. The CR element has also been used in the framework of finite volume element method; cf. [9]. There exists quite a number of works focusing on iterative methods for the CR finite element for second order problems; cf. [4, 5, 10, 13, 16, 18, 19, 20, 21, 22] and references therein. The purpose of this paper is to propose a parallel algorithm based on a Neumann-Dirichlet preconditioner for a FETI-DP formulation of the CR finite element method for the second order elliptic problem. To our knowledge, this is apparently the first work on such preconditioner for the FETI-DP method for the Crouzeix-Raviart (CR) finite element. -
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) . -
Coupling of the Finite Volume Element Method and the Boundary Element Method – an a Priori Convergence Result
COUPLING OF THE FINITE VOLUME ELEMENT METHOD AND THE BOUNDARY ELEMENT METHOD – AN A PRIORI CONVERGENCE RESULT CHRISTOPH ERATH∗ Abstract. The coupling of the finite volume element method and the boundary element method is an interesting approach to simulate a coupled system of a diffusion convection reaction process in an interior domain and a diffusion process in the corresponding unbounded exterior domain. This discrete system maintains naturally local conservation and a possible weighted upwind scheme guarantees the stability of the discrete system also for convection dominated problems. We show existence and uniqueness of the continuous system with appropriate transmission conditions on the coupling boundary, provide a convergence and an a priori analysis in an energy (semi-) norm, and an existence and an uniqueness result for the discrete system. All results are also valid for the upwind version. Numerical experiments show that our coupling is an efficient method for the numerical treatment of transmission problems, which can be also convection dominated. Key words. finite volume element method, boundary element method, coupling, existence and uniqueness, convergence, a priori estimate 1. Introduction. The transport of a concentration in a fluid or the heat propa- gation in an interior domain often follows a diffusion convection reaction process. The influence of such a system to the corresponding unbounded exterior domain can be described by a homogeneous diffusion process. Such a coupled system is usually solved by a numerical scheme and therefore, a method which ensures local conservation and stability with respect to the convection term is preferable. In the literature one can find various discretization schemes to solve similar prob- lems. -
Hp-Finite Element Methods for Hyperbolic Problems
¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ Eidgen¨ossische Ecole polytechnique f´ed´erale de Zurich ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ Technische Hochschule Politecnico federale di Zurigo ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ Z¨urich Swiss Federal Institute of Technology Zurich hp-Finite Element Methods for Hyperbolic Problems E. S¨uli†, P. Houston† and C. Schwab ∗ Research Report No. 99-14 July 1999 Seminar f¨urAngewandte Mathematik Eidgen¨ossische Technische Hochschule CH-8092 Z¨urich Switzerland †Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, United Kingdom ∗E. S¨uliand P. Houston acknowledge the financial support of the EPSRC (Grant GR/K76221) hp-Finite Element Methods for Hyperbolic Problems E. S¨uli†, P. Houston† and C. Schwab ∗ Seminar f¨urAngewandte Mathematik Eidgen¨ossische Technische Hochschule CH-8092 Z¨urich Switzerland Research Report No. 99-14 July 1999 Abstract This paper is devoted to the a priori and a posteriori error analysis of the hp-version of the discontinuous Galerkin finite element method for partial differential equations of hyperbolic and nearly-hyperbolic character. We consider second-order partial dif- ferential equations with nonnegative characteristic form, a large class of equations which includes convection-dominated diffusion problems, degenerate elliptic equa- tions and second-order problems of mixed elliptic-hyperbolic-parabolic type. An a priori error bound is derived for the method in the so-called DG-norm which is optimal in terms of the mesh size h; the error bound is either 1 degree or 1/2 de- gree below optimal in terms of the polynomial degree p, depending on whether the problem is convection-dominated, or diffusion-dominated, respectively. In the case of a first-order hyperbolic equation the error bound is hp-optimal in the DG-norm. -
Efficient Spectral-Galerkin Method I. Direct Solvers for the Second And
SIAM J. SCI. COMPUT. °c 1994 Society for Industrial and Applied Mathematics Vol. 15, No. 6, pp. 1489{1505, November 1994 013 E±cient Spectral-Galerkin Method I. Direct Solvers for the Second and Fourth Order Equations Using Legendre Polynomials¤ Jie Sheny Abstract. We present some e±cient algorithms based on the Legendre-Galerkin approxima- tions for the direct solution of the second and fourth order elliptic equations. The key to the e±ciency of our algorithms is to construct appropriate base functions, which lead to systems with sparse matri- ces for the discrete variational formulations. The complexities of the algorithms are a small multiple of N d+1 operations for a d dimensional domain with (N ¡ 1)d unknowns, while the convergence rates of the algorithms are exponential for problems with smooth solutions. In addition, the algorithms can be e®ectively parallelized since the bottlenecks of the algorithms are matrix-matrix multiplications. Key words. spectral-Galerkin method, Legendre polynomial, Helmholtz equation, biharmonic equation, direct solver AMS subject classi¯cations. 65N35, 65N22, 65F05, 35J05 1. Introduction. This article is the ¯rst in a series for developing e±cient spectral Galerkin algorithms for elliptic problems. The spectral method employs global polynomials as the trial functions for the discretization of partial di®erential equations. It provides very accurate approximations with a relatively small number of unknowns. Consequently it has gained increasing popularity in the last two decades, especially in the ¯eld of computational fluid dynamics (see [11], [8] and the references therein). The use of di®erent test functions in a variational formulation leads to three most commonly used spectral schemes, namely, the Galerkin, tau and collocation versions. -
Family Name Given Name Presentation Title Session Code
Family Name Given Name Presentation Title Session Code Abdoulaev Gassan Solving Optical Tomography Problem Using PDE-Constrained Optimization Method Poster P Acebron Juan Domain Decomposition Solution of Elliptic Boundary Value Problems via Monte Carlo and Quasi-Monte Carlo Methods Formulations2 C10 Adams Mark Ultrascalable Algebraic Multigrid Methods with Applications to Whole Bone Micro-Mechanics Problems Multigrid C7 Aitbayev Rakhim Convergence Analysis and Multilevel Preconditioners for a Quadrature Galerkin Approximation of a Biharmonic Problem Fourth-order & ElasticityC8 Anthonissen Martijn Convergence Analysis of the Local Defect Correction Method for 2D Convection-diffusion Equations Flows C3 Bacuta Constantin Partition of Unity Method on Nonmatching Grids for the Stokes Equations Applications1 C9 Bal Guillaume Some Convergence Results for the Parareal Algorithm Space-Time ParallelM5 Bank Randolph A Domain Decomposition Solver for a Parallel Adaptive Meshing Paradigm Plenary I6 Barbateu Mikael Construction of the Balancing Domain Decomposition Preconditioner for Nonlinear Elastodynamic Problems Balancing & FETIC4 Bavestrello Henri On Two Extensions of the FETI-DP Method to Constrained Linear Problems FETI & Neumann-NeumannM7 Berninger Heiko On Nonlinear Domain Decomposition Methods for Jumping Nonlinearities Heterogeneities C2 Bertoluzza Silvia The Fully Discrete Fat Boundary Method: Optimal Error Estimates Formulations2 C10 Biros George A Survey of Multilevel and Domain Decomposition Preconditioners for Inverse Problems in Time-dependent -
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. -
A Face-Centred Finite Volume Method for Second-Order Elliptic Problems
A face-centred finite volume method for second-order elliptic problems Ruben Sevilla Zienkiewicz Centre for Computational Engineering, College of Engineering, Swansea University, Wales, UK Matteo Giacomini, and Antonio Huerta Laboratori de C`alculNum`eric(LaC`aN), ETS de Ingenieros de Caminos, Canales y Puertos, Universitat Polit`ecnicade Catalunya, Barcelona, Spain December 19, 2017 Abstract This work proposes a novel finite volume paradigm, the face-centred finite volume (FCFV) method. Contrary to the popular vertex (VCFV) and cell (CCFV) centred finite volume methods, the novel FCFV defines the solution on the mesh faces (edges in 2D) to construct locally-conservative numerical schemes. The idea of the FCFV method stems from a hybridisable discontinuous Galerkin (HDG) formulation with constant degree of approximation, thus inheriting the convergence properties of the classical HDG. The resulting FCFV features a global problem in terms of a piecewise constant function defined on the faces of the mesh. The solution and its gradient in each element are then recovered by solving a set of independent element-by-element problems. The mathematical formulation of FCFV for Poisson and Stokes equation is derived and numerical evidence of optimal convergence in 2D and 3D is provided. Numerical examples are presented to illustrate the accuracy, efficiency and robustness of the proposed methodology. The results show that, contrary to other FV methods, arXiv:1712.06173v1 [math.NA] 17 Dec 2017 the accuracy of the FCFV method is not sensitive to mesh distortion and stretching. In addition, the FCFV method shows its better performance, accuracy and robustness using simplicial elements, facilitating its application to problems involving complex geometries in 3D. -
A Space-Time Discontinuous Galerkin Spectral Element Method for the Stefan Problem
Noname manuscript No. (will be inserted by the editor) A Space-Time Discontinuous Galerkin Spectral Element Method for the Stefan problem Chaoxu Pei · Mark Sussman · M. Yousuff Hussaini Received: / Accepted: Abstract A space-time Discontinuous Galerkin (DG) spectral element method is presented to solve the Stefan problem in an Eulerian coordinate system. In this scheme, the level set method describes the time evolving interface. To deal with the prior unknown interface, two global transformations, a backward transformation and a forward transformation, are introduced in the space-time mesh. The idea of the transformations is to combine an Eulerian description, i.e. a fixed frame of reference, with a Lagrangian description, i.e. a moving frame of reference. The backward transformation maps the unknown time- varying interface in the fixed frame of reference to a known stationary inter- face in the moving frame of reference. In the moving frame of reference, the transformed governing equations, written in the space-time framework, are discretized by a DG spectral element method in each space-time slab. The forward transformation is used to update the level set function and then to project the solution in each phase onto the new corresponding time-dependent domain. Two options for calculating the interface velocity are presented, and both methods exhibit spectral accuracy. Benchmark tests in one spatial di- mension indicate that the method converges with spectral accuracy in both space and time for both the temperature distribution and the interface ve- locity. The interrelation between the interface position and the temperature makes the Stefan problem a nonlinear problem; a Picard iteration algorithm is introduced in order to solve the nonlinear algebraic system of equations and it is found that only a few iterations lead to convergence. -
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. -
Spectral Element Simulations of Laminar and Turbulent Flows in Complex Geometries
Applied Numerical Mathematics 6 (1989/90) 85-105 85 North-Holland SPECTRAL ELEMENT SIMULATIONS OF LAMINAR AND TURBULENT FLOWS IN COMPLEX GEOMETRIES George Em KARNIADAKIS* Center for Turbulence Research, Stanford University/NASA Ames Research Center, Stanford, CA 94305, USA Spectral element methods are high-order weighted residual techniques based on spectral expansions of variables and geometry for the Navier-Stokes and transport equations . Their success in the recent past in simulating flows of industrial complexity derives from the flexibility of the method to accurately represent nontrivial geometries while preserving the good resolution properties of the spectral methods . In this paper, we review some of the main ideas of the method with emphasis placed on implementation and data management . These issues need special attention in order to make the method efficient in practice, especially in view of the fact that high computing cost as well as strenuous storage requirements have been a major drawback of high-order methods in the past . Several unsteady, laminar complex flows are simulated, and a direct simulation of turbulent channel flow is presented, for the first time, using spectral element techniques . 1 . Introduction Spectral methods have proven in recent years a very powerful tool for analyzing fluid flows and have been used almost exclusively in direct simulations of transitional and turbulent flows [8]. To date, however, only simple geometry turbulent flows have been studied accurately [14], chiefly because of the poor performance of global spectral methods in representing more complex geometries. The development of the spectral element method [26] and its success to accurately simulate highly unsteady, two-dimensional laminar flows [6,13] suggests that a three-dimensional implementation may be the right approach to direct simulations of turbulent flows in truly complex geometries.