Mat1062: Introductory Numerical Methods for PDE
|
|
- Lindsay Davis
- 6 years ago
- Views:
Transcription
1 Mat1062: Introductory Numerical Methods for PDE Mary Pugh January 13, Ownership These notes are the joint property of Rob Almgren and Mary Pugh 2 Boundary Conditions We now want to discuss in detail methods for solving the linear diusion equation for u(x, t) u t = D u xx (1) We shall suppose that the domain is X L x X R, and that initial data u 0 (x) is given at t = 0: u(x, 0) = u 0 (x), X L x X R and that suitable boundary conditions are given on x = X L and x = X R for t > 0 We will consider boundary conditions that are Dirichlet, Neumann, or Robin Dirichlet boundary conditions specify the value of u at the endpoints: u(x L, t) = u L (t), u(x R, t) = u R (t) where u L and u R are specied functions of time Neumann conditions specify the derivative u x at the endpoints: u x (X L, t) = u L (t), u x (X R, t) = u R (t) 1
2 2 Mary Pugh Mat1062 Jan 13, 2009 Robin boundary conditions specify a linear combination of u and u x at the endpoints: a u(x L, t) + b u x (X L, t) = u L (t), c u(x R, t) + d u x (X R, t) = u R (t) where a, b, c, and d are xed constants usually determined by material properties 21 Space Discretization For given X L and X R, we choose a number N Our grid spacing is and we place grid points at locations h = X R X L N x j = X L + j h, j = 0, 1,, N, so that x 0 = X L and x N = X R We represent our solution by a vector U(t) of length N + 1, whose components are U j (t) u ( x j, t), j = 0,, N That is, the jth component of the vector, U j (t) is supposed to approximate the PDE's solution at the x j grid point: u(x j, t) We take the initial data to be an exact sample of the true initial data u 0 (x): U j (0) = u 0 ( xj ) Given Du xx,, a domain [X L, X R ], and boundary conditions, these determine a linear operator on a function space We will approximate this linear operator by a matrix operating on the vector U This allows us to dene U(t) as satisfying a linear ODE U t = LU, which we can solve by standard methods As we will see below, even if the domain is unchanged and the operation Du xx is unchanged, dierent boundary conditions lead to dierent linear operators and hence dierent dynamics This holds in the continuous case (where the operator is acting on a function space) and in the discretized case (where the approximating operator is just a matrix) Boundary conditions matter!
3 Mat1062 Jan 13, 2009 Mary Pugh 3 22 Dirichlet Boundary Conditions Let us consider the case of Dirichlet boundary conditions: u(x L, t) = u L (t) and u(x R, t) = u R (t) There are N+1 meshpoints and U(t) has a component for each meshpoint However, the the components U 0 (t) and U N (t) are specied by the boundary conditions and so there are only N 1 unknowns: U 1 (t) U N 1 (t) From the January 8 notes, we need to approximate u xx (x j, t) for 1 j N 1 This is done with the nite-dierence approximation In matrix form this is U 0 u xx (x 1 ) U 1 u xx (x N 1 ) U N However, these expressions are not quite what we want The vector on the left is of size N 1 and will determine the interior grid values The vector on the right is of size N + 1; it involves both the interior grid values and the boundary values We need to express derivatives at the interior grid points in terms only of interior grid values and the Dirichlet boundary conditions This means that we need to eliminate U 0 and U N from the above Substituting the boundary values u 0 = u L (t), u N = u R (t), we approximate u xx (with Dirichlet boundary conditions on [X L, X R ]) via the inhomogeneous linear operator L 1 L 1 U = M 1 U + R 1, where M 1 is a (N 1) (N 1) tridiagonal matrix and R 1 is an (N 1)-vector containing the boundary data: 2 1 u L (t) M 1 = , R 1 (t) = 1 0 (2) u R (t) R 1 depends on time if the boundary conditions do In practice, it is often more convenient to store the boundary values in their natural locations at U N
4 4 Mary Pugh Mat1062 Jan 13, 2009 the endpoints of a linear grid, and then the derivatives can be computed by doing the (N 1) (N + 1) multiplication above Of course, the matrix M 1 never actually exists in the computer, nor do we ever actually do a matrix multiply as written above: since most entries are zero, storing the matrix and doing a matrix multiply would be a huge waste of memory and time \Multiplication" really means \apply the linear operator," which we do by taking dierences 23 Neumann Boundary Conditions For Neumann boundary conditions (u x (X L, t) = u L (t), u x (X R, t) = u R (t)) the values at the endponts, U 0 (t) and U N (t), are unknown Again, we approximate the dierentiation operation as U U 0 u xx (x 0 ) U 1 u xx (x N ) U N U N U N+1 Note that we've done something strange here We refer to U 1 and U N+1 which are the values at X L h and X R + h respectively These grid points are not in our domain They are artices introduced to help implement the nite dierence approximation of u xx at the endpoints Such points are often called \ghost points" The vector on the left is of size N + 1 and will determine the interior grid values and the boundary grid values The vector on the right is of size N + 3; it involves the interior grid values, the boundary grid values, and the values at the two ghost points We need to express the right-hand side in terms of interior grid values and the boundary grid values only This means that we need to use the Neumann boundary conditions to determine the values at the ghost points This is done as follows: u L (t) = u x (X L, t) U 1 U 1 as a result = U 1 = U 1 u L (t) u xx (X L, t) U 1 2U 0 + U 1 = 2U 1 2U 0 2u L(t) h
5 Mat1062 Jan 13, 2009 Mary Pugh 5 Similarly, at the right-hand endpoint hence 0 = u x (X R, t) U N+1 U N 1 u xx (X R, t) U N+1 2U N + U N 1 = U N+1 = U N 1 + u R (t) = 2U N 1 2U N + 2u R(t) h And so, we approximate u xx (with Neumann boundary conditions on [X L, X R ]) with the inhomogeneous linear operator M 2 U + R 2, where M 2 is a (N+1) (N+1) tridiagonal matrix and R 2 is an N+1-vector containing the boundary data: 2 2 u L (t) M 2 = , R 2 (t) = 2 0 h (3) u R (t) A natural question to ask is why we used centered nite dierences above to approximate u x (X L ) and u x (X R ) rather than using a left-dierence for u x (X L ) and a right-dierence for u x (X R ): 0 = u x (X L ) U 0 U 1 h 0 = u x (X R ) U N+1 U N h We will revisit this question when we discuss the convergence and accuracy of our scheme 24 Robin Boundary Conditions We handle Robin boundary conditions (a u(x L, t) + b u x (X L, t) = u L (t), c u(x R, t) + d u x (X R, t) = u R (t)) in the same manner that we handle Neumann boundary conditions: by introducing ghost grid points the values at which are then determined from the boundary conditions u L (t) = a u(x L, t) + b u x (X L, t) a U 0 + b U 1 U 1 = U 1 = U 1 + 2ah b U 0 u L(t) b
6 6 Mary Pugh Mat1062 Jan 13, 2009 as a result Similarly, hence u xx (X L, t) U 1 2U 0 + U 1 = 2U 1 (2 2ah/b)U 0 2u L(t) bh u R (t) = c u(x R, t) + d u x (X R, t) c U N + d U N+1 U N 1 = U N+1 = U N 1 2ch d u xx (X R, t) U 1 2U 0 + U 1 U N + u R(t) d = 2U N 1 (2 + 2ch/d)U N + 2u R(t) dh In this way, we can approximate u xx (with Robin boundary conditions on [X L, X R ]) as the inhomogeneous linear operator = M 3 U + R 3 where M 2 is a (N + 1) (N + 1) tridiagonal matrix: 2 + 2ah/b 2 M 3 = , R 3 (t) = 2 h ch/d (4) As a reality check, note that if a = c = 0 and if b = d = 1 then this matrix is the same matrix as for the Neumann boundary conditions u L (t)/b 0 0 u R (t)/d 25 Summary The spatial derivative operator Lu = Du xx on [X L, X R ] u(x L, t) = u L (t) u(x R, t) = u R (t)
7 Mat1062 Jan 13, 2009 Mary Pugh 7 has the discrete approximation LU = MU + R where M = DM 1 and R = DR 1 with M 1 and R 1 given in (2) Similarly, the spatial derivative operator Lu = Du xx on [X L, X R ] u x (X L, t) = u L (t) u x (X R, t) = u R (t) has the discrete approximation LU = MU + R where M = DM 2 and R = DR 2 with M 2 and R 2 given in (3) Finally, the spatial derivative operator Lu = Du xx on [X L, X R ] au(x L, t) + bu x (X L, t) = u L (t) cu(x R, t) + Du x (X R, t) = u R (t) has the discrete approximation LU = MU + R where M = DM 3 and R = DR 3 with M 3 and R 3 given in (4) In all three cases, we have L which is an ane linear operator But the three matrices M 1, M 2, and M 3 are all dierent from one another They will have dierent eigenvalues and eigenvectors All of the eigenvalues of M 1 and M 2 will be either zero or negative The matrix M 3 can have positive eigenvalues They correspond to modes that grow exponentially in time, rather than decaying this shows how powerful boundary eects can be!
PROPOSITION of PROJECTS necessary to get credit in the 'Introduction to Parallel Programing in 2014'
PROPOSITION of PROJECTS necessary to get credit in the 'Introduction to Parallel Programing in 2014' GENERALITIES The text may be written in Polish, or English. The text has to contain two parts: 1. Theoretical
More informationFinite Difference Methods for Boundary Value Problems
Finite Difference Methods for Boundary Value Problems October 2, 2013 () Finite Differences October 2, 2013 1 / 52 Goals Learn steps to approximate BVPs using the Finite Difference Method Start with two-point
More informationNumerical Analysis of Differential Equations Numerical Solution of Elliptic Boundary Value
Numerical Analysis of Differential Equations 188 5 Numerical Solution of Elliptic Boundary Value Problems 5 Numerical Solution of Elliptic Boundary Value Problems TU Bergakademie Freiberg, SS 2012 Numerical
More informationProgress in developing a 3D background error correlation model using an implicit diusion operator
Progress in developing a 3D background error correlation model using an implicit diusion operator Isabelle Mirouze 1 et Anthony T. Weaver 2 1 CERFACS / CNRS-UMR 5219 2 CERFACS / SUC URA 1875 VODA: intermediary
More informationHOMEWORK 4: Numerical solution of PDEs for Mathmatical Models, Analysis and Simulation, Fall 2011 Report due Mon Nov 12, Maximum score 6.0 pts.
HOMEWORK 4: Numerical solution of PDEs for Mathmatical Models, Analysis and Simulation, Fall 2011 Report due Mon Nov 12, 2012. Maximum score 6.0 pts. Read Strang s (old) book, Sections 3.1-3.4 and 6.4-6.5.
More information= main diagonal, in the order in which their corresponding eigenvectors appear as columns of E.
3.3 Diagonalization Let A = 4. Then and are eigenvectors of A, with corresponding eigenvalues 2 and 6 respectively (check). This means 4 = 2, 4 = 6. 2 2 2 2 Thus 4 = 2 2 6 2 = 2 6 4 2 We have 4 = 2 0 0
More informationFinite difference method for elliptic problems: I
Finite difference method for elliptic problems: I Praveen. C praveen@math.tifrbng.res.in Tata Institute of Fundamental Research Center for Applicable Mathematics Bangalore 560065 http://math.tifrbng.res.in/~praveen
More informationIntroduction to numerical schemes
236861 Numerical Geometry of Images Tutorial 2 Introduction to numerical schemes Heat equation The simple parabolic PDE with the initial values u t = K 2 u 2 x u(0, x) = u 0 (x) and some boundary conditions
More informationby Ron Buckmire Mathematics Department Occidental College Los Angeles, CA , U.S.A.
A new nite-dierence scheme for singular dierential equations in cylinical or spherical coordinates by Ron Buckmire Mathematics Department Occidental College Los Angeles, CA 90041-3338, U.S.A. December
More informationMath 1270 Honors ODE I Fall, 2008 Class notes # 14. x 0 = F (x; y) y 0 = G (x; y) u 0 = au + bv = cu + dv
Math 1270 Honors ODE I Fall, 2008 Class notes # 1 We have learned how to study nonlinear systems x 0 = F (x; y) y 0 = G (x; y) (1) by linearizing around equilibrium points. If (x 0 ; y 0 ) is an equilibrium
More informationLaboratory #3: Linear Algebra. Contents. Grace Yung
Laboratory #3: Linear Algebra Grace Yung Contents List of Problems. Introduction. Objectives.2 Prerequisites 2. Linear Systems 2. What is a Matrix 2.2 Quick Review 2.3 Gaussian Elimination 2.3. Decomposition
More information2.4 Eigenvalue problems
2.4 Eigenvalue problems Associated with the boundary problem (2.1) (Poisson eq.), we call λ an eigenvalue if Lu = λu (2.36) for a nonzero function u C 2 0 ((0, 1)). Recall Lu = u. Then u is called an eigenfunction.
More informationNumerical Schemes from the Perspective of Consensus
Numerical Schemes from the Perspective of Consensus Exploring Connections between Agreement Problems and PDEs Department of Electrical Engineering and Computer Sciences University of California, Berkeley
More informationExtreme Values and Positive/ Negative Definite Matrix Conditions
Extreme Values and Positive/ Negative Definite Matrix Conditions James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 8, 016 Outline 1
More information1e N
Spectral schemes on triangular elements by Wilhelm Heinrichs and Birgit I. Loch Abstract The Poisson problem with homogeneous Dirichlet boundary conditions is considered on a triangle. The mapping between
More informationMAT 22B - Lecture Notes
MAT 22B - Lecture Notes 4 September 205 Solving Systems of ODE Last time we talked a bit about how systems of ODE arise and why they are nice for visualization. Now we'll talk about the basics of how to
More informationAM 205: lecture 14. Last time: Boundary value problems Today: Numerical solution of PDEs
AM 205: lecture 14 Last time: Boundary value problems Today: Numerical solution of PDEs ODE BVPs A more general approach is to formulate a coupled system of equations for the BVP based on a finite difference
More informationAn Introduction to Numerical Methods for Differential Equations. Janet Peterson
An Introduction to Numerical Methods for Differential Equations Janet Peterson Fall 2015 2 Chapter 1 Introduction Differential equations arise in many disciplines such as engineering, mathematics, sciences
More informationAMSC/CMSC 466 Problem set 3
AMSC/CMSC 466 Problem set 3 1. Problem 1 of KC, p180, parts (a), (b) and (c). Do part (a) by hand, with and without pivoting. Use MATLAB to check your answer. Use the command A\b to get the solution, and
More informationFinite Difference Methods (FDMs) 1
Finite Difference Methods (FDMs) 1 1 st - order Approxima9on Recall Taylor series expansion: Forward difference: Backward difference: Central difference: 2 nd - order Approxima9on Forward difference: Backward
More informationAMS527: Numerical Analysis II
AMS527: Numerical Analysis II Supplementary Material on Finite Different Methods for Two-Point Boundary-Value Problems Xiangmin Jiao Stony Brook University Xiangmin Jiao Stony Brook University AMS527:
More informationLecture 1. Finite difference and finite element methods. Partial differential equations (PDEs) Solving the heat equation numerically
Finite difference and finite element methods Lecture 1 Scope of the course Analysis and implementation of numerical methods for pricing options. Models: Black-Scholes, stochastic volatility, exponential
More informationProblem set 3: Solutions Math 207B, Winter Suppose that u(x) is a non-zero solution of the eigenvalue problem. (u ) 2 dx, u 2 dx.
Problem set 3: Solutions Math 27B, Winter 216 1. Suppose that u(x) is a non-zero solution of the eigenvalue problem u = λu < x < 1, u() =, u(1) =. Show that λ = (u ) 2 dx u2 dx. Deduce that every eigenvalue
More informationDepartment of Mathematics California State University, Los Angeles Master s Degree Comprehensive Examination in. NUMERICAL ANALYSIS Spring 2015
Department of Mathematics California State University, Los Angeles Master s Degree Comprehensive Examination in NUMERICAL ANALYSIS Spring 2015 Instructions: Do exactly two problems from Part A AND two
More informationTutorial 2. Introduction to numerical schemes
236861 Numerical Geometry of Images Tutorial 2 Introduction to numerical schemes c 2012 Classifying PDEs Looking at the PDE Au xx + 2Bu xy + Cu yy + Du x + Eu y + Fu +.. = 0, and its discriminant, B 2
More informationMATH 333: Partial Differential Equations
MATH 333: Partial Differential Equations Problem Set 9, Final version Due Date: Tues., Nov. 29, 2011 Relevant sources: Farlow s book: Lessons 9, 37 39 MacCluer s book: Chapter 3 44 Show that the Poisson
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University The Implicit Schemes for the Model Problem The Crank-Nicolson scheme and θ-scheme
More informationGaussian Elimination and Back Substitution
Jim Lambers MAT 610 Summer Session 2009-10 Lecture 4 Notes These notes correspond to Sections 31 and 32 in the text Gaussian Elimination and Back Substitution The basic idea behind methods for solving
More informationMathematical Methods - Lecture 9
Mathematical Methods - Lecture 9 Yuliya Tarabalka Inria Sophia-Antipolis Méditerranée, Titane team, http://www-sop.inria.fr/members/yuliya.tarabalka/ Tel.: +33 (0)4 92 38 77 09 email: yuliya.tarabalka@inria.fr
More information2 Introduction T(,t) = = L Figure..: Geometry of the prismatical rod of Eample... T = u(,t) = L T Figure..2: Geometry of the taut string of Eample..2.
Chapter Introduction. Problems Leading to Partial Dierential Equations Many problems in science and engineering are modeled and analyzed using systems of partial dierential equations. Realistic models
More informationIntroduction - Motivation. Many phenomena (physical, chemical, biological, etc.) are model by differential equations. f f(x + h) f(x) (x) = lim
Introduction - Motivation Many phenomena (physical, chemical, biological, etc.) are model by differential equations. Recall the definition of the derivative of f(x) f f(x + h) f(x) (x) = lim. h 0 h Its
More informationA review of stability and dynamical behaviors of differential equations:
A review of stability and dynamical behaviors of differential equations: scalar ODE: u t = f(u), system of ODEs: u t = f(u, v), v t = g(u, v), reaction-diffusion equation: u t = D u + f(u), x Ω, with boundary
More informationu(0) = u 0, u(1) = u 1. To prove what we want we introduce a new function, where c = sup x [0,1] a(x) and ɛ 0:
6. Maximum Principles Goal: gives properties of a solution of a PDE without solving it. For the two-point boundary problem we shall show that the extreme values of the solution are attained on the boundary.
More informationMA 201: Method of Separation of Variables Finite Vibrating String Problem Lecture - 11 MA201(2016): PDE
MA 201: Method of Separation of Variables Finite Vibrating String Problem ecture - 11 IBVP for Vibrating string with no external forces We consider the problem in a computational domain (x,t) [0,] [0,
More informationLinear Systems of Equations. ChEn 2450
Linear Systems of Equations ChEn 450 LinearSystems-directkey - August 5, 04 Example Circuit analysis (also used in heat transfer) + v _ R R4 I I I3 R R5 R3 Kirchoff s Laws give the following equations
More informationTable 1: BEM as a solution method for a BVP dierential formulation FDM BVP integral formulation FEM boundary integral formulation BEM local view is ad
Chapter 1 Introduction to Boundary element Method - 1D Example For reference: Hong-Ki Hong and Jeng-Tzong Chen, Boundary Element Method, Chapter 1 Introduction to Boundary element Method - 1D Example,
More informationWeighted Residual Methods
Weighted Residual Methods Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Table of Contents Problem definition. oundary-value Problem..................
More informationERIK STERNER. Discretize the equations in space utilizing a nite volume or nite dierence scheme. 3. Integrate the corresponding time-dependent problem
Convergence Acceleration for the Navier{Stokes Equations Using Optimal Semicirculant Approximations Sverker Holmgren y, Henrik Branden y and Erik Sterner y Abstract. The iterative solution of systems of
More informationIntroduction to PDEs and Numerical Methods Tutorial 1: Overview of essential linear algebra and analysis
Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Introduction to PDEs and Numerical Methods Tutorial 1: Overview of essential linear algebra and analysis Dr. Noemi Friedman, 25.10.201.
More informationLecture 1 INF-MAT3350/ : Some Tridiagonal Matrix Problems
Lecture 1 INF-MAT3350/4350 2007: Some Tridiagonal Matrix Problems Tom Lyche University of Oslo Norway Lecture 1 INF-MAT3350/4350 2007: Some Tridiagonal Matrix Problems p.1/33 Plan for the day 1. Notation
More information9. Iterative Methods for Large Linear Systems
EE507 - Computational Techniques for EE Jitkomut Songsiri 9. Iterative Methods for Large Linear Systems introduction splitting method Jacobi method Gauss-Seidel method successive overrelaxation (SOR) 9-1
More informationNumerical Analysis and Methods for PDE I
Numerical Analysis and Methods for PDE I A. J. Meir Department of Mathematics and Statistics Auburn University US-Africa Advanced Study Institute on Analysis, Dynamical Systems, and Mathematical Modeling
More informationPIECEWISE LINEAR FINITE ELEMENT METHODS ARE NOT LOCALIZED
PIECEWISE LINEAR FINITE ELEMENT METHODS ARE NOT LOCALIZED ALAN DEMLOW Abstract. Recent results of Schatz show that standard Galerkin finite element methods employing piecewise polynomial elements of degree
More informationPhysics 202 Laboratory 5. Linear Algebra 1. Laboratory 5. Physics 202 Laboratory
Physics 202 Laboratory 5 Linear Algebra Laboratory 5 Physics 202 Laboratory We close our whirlwind tour of numerical methods by advertising some elements of (numerical) linear algebra. There are three
More informationDepartment of Applied Mathematics Preliminary Examination in Numerical Analysis August, 2013
Department of Applied Mathematics Preliminary Examination in Numerical Analysis August, 013 August 8, 013 Solutions: 1 Root Finding (a) Let the root be x = α We subtract α from both sides of x n+1 = x
More information3.5 Finite Differences and Fast Poisson Solvers
3.5 Finite Differences and Fast Poisson Solvers It is extremely unusual to use eigenvectors to solve a linear system KU = F. You need to know all the eigenvectors of K, and (much more than that) the eigenvector
More informationOn the Arithmetic and Shift Complexities of Inverting of Dierence Operator Matrices
On the Arithmetic and Shift Complexities of Inverting of Dierence Operator Matrices S. Abramov Moscow, Comp. Centre of RAS, MSU S. Abramov the arithmetic and shift complexities 1/12 Let K be a dierence
More informationINTRODUCTION TO PDEs
INTRODUCTION TO PDEs In this course we are interested in the numerical approximation of PDEs using finite difference methods (FDM). We will use some simple prototype boundary value problems (BVP) and initial
More informationBefore we look at numerical methods, it is important to understand the types of equations we will be dealing with.
Chapter 1. Partial Differential Equations (PDEs) Required Readings: Chapter of Tannehill et al (text book) Chapter 1 of Lapidus and Pinder (Numerical Solution of Partial Differential Equations in Science
More informationWeighted Residual Methods
Weighted Residual Methods Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research of the Polish Academy of Sciences
More information1 Separation of Variables
Jim ambers ENERGY 281 Spring Quarter 27-8 ecture 2 Notes 1 Separation of Variables In the previous lecture, we learned how to derive a PDE that describes fluid flow. Now, we will learn a number of analytical
More information19.2 Mathematical description of the problem. = f(p; _p; q; _q) G(p; q) T ; (II.19.1) g(p; q) + r(t) _p _q. f(p; v. a p ; q; v q ) + G(p; q) T ; a q
II-9-9 Slider rank 9. General Information This problem was contributed by Bernd Simeon, March 998. The slider crank shows some typical properties of simulation problems in exible multibody systems, i.e.,
More informationADDITIVE SCHWARZ FOR SCHUR COMPLEMENT 305 the parallel implementation of both preconditioners on distributed memory platforms, and compare their perfo
35 Additive Schwarz for the Schur Complement Method Luiz M. Carvalho and Luc Giraud 1 Introduction Domain decomposition methods for solving elliptic boundary problems have been receiving increasing attention
More informationK. BLACK To avoid these diculties, Boyd has proposed a method that proceeds by mapping a semi-innite interval to a nite interval [2]. The method is co
Journal of Mathematical Systems, Estimation, and Control Vol. 8, No. 2, 1998, pp. 1{2 c 1998 Birkhauser-Boston Spectral Element Approximations and Innite Domains Kelly Black Abstract A spectral-element
More informationA Block Red-Black SOR Method. for a Two-Dimensional Parabolic. Equation Using Hermite. Collocation. Stephen H. Brill 1 and George F.
1 A lock ed-lack SO Method for a Two-Dimensional Parabolic Equation Using Hermite Collocation Stephen H. rill 1 and George F. Pinder 1 Department of Mathematics and Statistics University ofvermont urlington,
More information7 Hyperbolic Differential Equations
Numerical Analysis of Differential Equations 243 7 Hyperbolic Differential Equations While parabolic equations model diffusion processes, hyperbolic equations model wave propagation and transport phenomena.
More informationVector Space Basics. 1 Abstract Vector Spaces. 1. (commutativity of vector addition) u + v = v + u. 2. (associativity of vector addition)
Vector Space Basics (Remark: these notes are highly formal and may be a useful reference to some students however I am also posting Ray Heitmann's notes to Canvas for students interested in a direct computational
More informationMAT 22B - Lecture Notes
MAT 22B - Lecture Notes 2 September 2015 Systems of ODEs This is some really cool stu. Systems of ODEs are a natural and powerful way to model how multiple dierent interrelated quantities change through
More informationLecture 9 Approximations of Laplace s Equation, Finite Element Method. Mathématiques appliquées (MATH0504-1) B. Dewals, C.
Lecture 9 Approximations of Laplace s Equation, Finite Element Method Mathématiques appliquées (MATH54-1) B. Dewals, C. Geuzaine V1.2 23/11/218 1 Learning objectives of this lecture Apply the finite difference
More informationHigh order FEM vs. IgA by J.A.Evans, T.J.Hughes and A.Reali, 2014
High order FEM vs. IgA by J.A.Evans, T.J.Hughes and A.Reali, 2014 Rainer Schneckenleitner JKU Linz rainer.schneckenleitner@gmx.at Jan 20, 2017 Rainer Schneckenleitner (SS16) Isogeometric Analysis Jan 20,
More informationCovariance and Correlation Matrix
Covariance and Correlation Matrix Given sample {x n } N 1, where x Rd, x n = x 1n x 2n. x dn sample mean x = 1 N N n=1 x n, and entries of sample mean are x i = 1 N N n=1 x in sample covariance matrix
More informationIntroduction to Boundary Value Problems
Chapter 5 Introduction to Boundary Value Problems When we studied IVPs we saw that we were given the initial value of a function and a differential equation which governed its behavior for subsequent times.
More informationSolution of the Two-Dimensional Steady State Heat Conduction using the Finite Volume Method
Ninth International Conference on Computational Fluid Dynamics (ICCFD9), Istanbul, Turkey, July 11-15, 2016 ICCFD9-0113 Solution of the Two-Dimensional Steady State Heat Conduction using the Finite Volume
More informationTechnical University Hamburg { Harburg, Section of Mathematics, to reduce the number of degrees of freedom to manageable size.
Interior and modal masters in condensation methods for eigenvalue problems Heinrich Voss Technical University Hamburg { Harburg, Section of Mathematics, D { 21071 Hamburg, Germany EMail: voss @ tu-harburg.d400.de
More information1 The relation between a second order linear ode and a system of two rst order linear odes
Math 1280 Spring, 2010 1 The relation between a second order linear ode and a system of two rst order linear odes In Chapter 3 of the text you learn to solve some second order linear ode's, such as x 00
More informationBoundary-Layer Transition. and. NASA Langley Research Center, Hampton, VA Abstract
Eect of Far-Field Boundary Conditions on Boundary-Layer Transition Fabio P. Bertolotti y Institut fur Stromungsmechanik, DLR, 37073 Gottingen, Germany and Ronald D. Joslin Fluid Mechanics and Acoustics
More informationNumerical Solutions to PDE s
Introduction Numerical Solutions to PDE s Mathematical Modelling Week 5 Kurt Bryan Let s start by recalling a simple numerical scheme for solving ODE s. Suppose we have an ODE u (t) = f(t, u(t)) for some
More informationCHAPTER 4. Introduction to the. Heat Conduction Model
A SERIES OF CLASS NOTES FOR 005-006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 4 A COLLECTION OF HANDOUTS ON PARTIAL DIFFERENTIAL EQUATIONS
More informationSection 4.4 Reduction to Symmetric Tridiagonal Form
Section 4.4 Reduction to Symmetric Tridiagonal Form Key terms Symmetric matrix conditioning Tridiagonal matrix Similarity transformation Orthogonal matrix Orthogonal similarity transformation properties
More informationMA 201, Mathematics III, July-November 2016, Partial Differential Equations: 1D wave equation (contd.) and 1D heat conduction equation
MA 201, Mathematics III, July-November 2016, Partial Differential Equations: 1D wave equation (contd.) and 1D heat conduction equation Lecture 12 Lecture 12 MA 201, PDE (2016) 1 / 24 Formal Solution of
More informationPhysics 6303 Lecture 8 September 25, 2017
Physics 6303 Lecture 8 September 25, 2017 LAST TIME: Finished tensors, vectors, and matrices At the beginning of the course, I wrote several partial differential equations (PDEs) that are used in many
More informationn i,j+1/2 q i,j * qi+1,j * S i+1/2,j
Helsinki University of Technology CFD-group/ The Laboratory of Applied Thermodynamics MEMO No CFD/TERMO-5-97 DATE: December 9,997 TITLE A comparison of complete vs. simplied viscous terms in boundary layer
More information= c c + g (n; c) (.b) on the interior of the unit square (; ) (; ) in space and for t (; t final ] in time for the unknown functions n(x; y; t)
Splitting Methods for Mixed Hyperbolic-Parabolic Systems Alf Gerisch, David F. Griths y, Rudiger Weiner, and Mark A. J. Chaplain y Institut fur Numerische Mathematik, Fachbereich Mathematik und Informatik
More informationSplitting methods with boundary corrections
Splitting methods with boundary corrections Alexander Ostermann University of Innsbruck, Austria Joint work with Lukas Einkemmer Verona, April/May 2017 Strang s paper, SIAM J. Numer. Anal., 1968 S (5)
More informationMA2501 Numerical Methods Spring 2015
Norwegian University of Science and Technology Department of Mathematics MA2501 Numerical Methods Spring 2015 Solutions to exercise set 3 1 Attempt to verify experimentally the calculation from class that
More informationMy signature below certifies that I have complied with the University of Pennsylvania s Code of Academic Integrity in completing this exam.
My signature below certifies that I have complied with the University of Pennsylvania s Code of Academic Integrity in completing this exam. Signature Printed Name Math 241 Exam 1 Jerry Kazdan Feb. 17,
More informationTake Home Exam I Key
Take Home Exam I Key MA 336 1. (5 points) Read sections 2.1 to 2.6 in the text (which we ve already talked about in class), and the handout on Solving the Nonhomogeneous Heat Equation. Solution: You did
More informationStrauss PDEs 2e: Section Exercise 4 Page 1 of 6
Strauss PDEs 2e: Section 5.3 - Exercise 4 Page of 6 Exercise 4 Consider the problem u t = ku xx for < x < l, with the boundary conditions u(, t) = U, u x (l, t) =, and the initial condition u(x, ) =, where
More informationApplied Mathematics 505b January 22, Today Denitions, survey of applications. Denition A PDE is an equation of the form F x 1 ;x 2 ::::;x n ;~u
Applied Mathematics 505b January 22, 1998 1 Applied Mathematics 505b Partial Dierential Equations January 22, 1998 Text: Sobolev, Partial Dierentail Equations of Mathematical Physics available at bookstore
More informationFinite-Elements Method 2
Finite-Elements Method 2 January 29, 2014 2 From Applied Numerical Analysis Gerald-Wheatley (2004), Chapter 9. Finite-Elements Method 3 Introduction Finite-element methods (FEM) are based on some mathematical
More informationBOUNDARY-VALUE PROBLEMS IN RECTANGULAR COORDINATES
1 BOUNDARY-VALUE PROBLEMS IN RECTANGULAR COORDINATES 1.1 Separable Partial Differential Equations 1. Classical PDEs and Boundary-Value Problems 1.3 Heat Equation 1.4 Wave Equation 1.5 Laplace s Equation
More informationAIMS Exercise Set # 1
AIMS Exercise Set #. Determine the form of the single precision floating point arithmetic used in the computers at AIMS. What is the largest number that can be accurately represented? What is the smallest
More informationFDM for parabolic equations
FDM for parabolic equations Consider the heat equation where Well-posed problem Existence & Uniqueness Mass & Energy decreasing FDM for parabolic equations CNFD Crank-Nicolson + 2 nd order finite difference
More informationIntroduction to PDEs and Numerical Methods: Exam 1
Prof Dr Thomas Sonar, Institute of Analysis Winter Semester 2003/4 17122003 Introduction to PDEs and Numerical Methods: Exam 1 To obtain full points explain your solutions thoroughly and self-consistently
More informationFluctuationlessness Theorem and its Application to Boundary Value Problems of ODEs
Fluctuationlessness Theorem and its Application to Boundary Value Problems o ODEs NEJLA ALTAY İstanbul Technical University Inormatics Institute Maslak, 34469, İstanbul TÜRKİYE TURKEY) nejla@be.itu.edu.tr
More informationFinite element approximation on quadrilateral meshes
COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING Commun. Numer. Meth. Engng 2001; 17:805 812 (DOI: 10.1002/cnm.450) Finite element approximation on quadrilateral meshes Douglas N. Arnold 1;, Daniele
More information3. Replace any row by the sum of that row and a constant multiple of any other row.
Section. Solution of Linear Systems by Gauss-Jordan Method A matrix is an ordered rectangular array of numbers, letters, symbols or algebraic expressions. A matrix with m rows and n columns has size or
More information1111: Linear Algebra I
1111: Linear Algebra I Dr. Vladimir Dotsenko (Vlad) Lecture 5 Dr. Vladimir Dotsenko (Vlad) 1111: Linear Algebra I Lecture 5 1 / 12 Systems of linear equations Geometrically, we are quite used to the fact
More information18.06 Problem Set 8 - Solutions Due Wednesday, 14 November 2007 at 4 pm in
806 Problem Set 8 - Solutions Due Wednesday, 4 November 2007 at 4 pm in 2-06 08 03 Problem : 205+5+5+5 Consider the matrix A 02 07 a Check that A is a positive Markov matrix, and find its steady state
More informationINF-MAT Mandatory Exercise 1
INF-MAT336 - Mandatory Exercise 1 Vegard Kjelseth University of Oslo vokjelse@ifi.uio.no March 8, 213 Exercise 1 a) The solution of the ODE u (t) = au (t), t >, u () = 1 is u (t) = Ce at. The value of
More information3.3. SYSTEMS OF ODES 1. y 0 " 2y" y 0 + 2y = x1. x2 x3. x = y(t) = c 1 e t + c 2 e t + c 3 e 2t. _x = A x + f; x(0) = x 0.
.. SYSTEMS OF ODES. Systems of ODEs MATH 94 FALL 98 PRELIM # 94FA8PQ.tex.. a) Convert the third order dierential equation into a rst oder system _x = A x, with y " y" y + y = x = @ x x x b) The equation
More informationMAT292 - Calculus III - Fall Solution for Term Test 2 - November 6, 2014 DO NOT WRITE ON THE QR CODE AT THE TOP OF THE PAGES.
MAT9 - Calculus III - Fall 4 Solution for Term Test - November 6, 4 Time allotted: 9 minutes. Aids permitted: None. Full Name: Last First Student ID: Email: @mail.utoronto.ca Instructions DO NOT WRITE
More informationLinear Systems of n equations for n unknowns
Linear Systems of n equations for n unknowns In many application problems we want to find n unknowns, and we have n linear equations Example: Find x,x,x such that the following three equations hold: x
More informationSolution of Linear Equations
Solution of Linear Equations (Com S 477/577 Notes) Yan-Bin Jia Sep 7, 07 We have discussed general methods for solving arbitrary equations, and looked at the special class of polynomial equations A subclass
More informationLinear Algebra Section 2.6 : LU Decomposition Section 2.7 : Permutations and transposes Wednesday, February 13th Math 301 Week #4
Linear Algebra Section. : LU Decomposition Section. : Permutations and transposes Wednesday, February 1th Math 01 Week # 1 The LU Decomposition We learned last time that we can factor a invertible matrix
More information1 Solutions to selected problems
Solutions to selected problems Section., #a,c,d. a. p x = n for i = n : 0 p x = xp x + i end b. z = x, y = x for i = : n y = y + x i z = zy end c. y = (t x ), p t = a for i = : n y = y(t x i ) p t = p
More informationEssentials of Intermediate Algebra
Essentials of Intermediate Algebra BY Tom K. Kim, Ph.D. Peninsula College, WA Randy Anderson, M.S. Peninsula College, WA 9/24/2012 Contents 1 Review 1 2 Rules of Exponents 2 2.1 Multiplying Two Exponentials
More informationA MULTIGRID ALGORITHM FOR. Richard E. Ewing and Jian Shen. Institute for Scientic Computation. Texas A&M University. College Station, Texas SUMMARY
A MULTIGRID ALGORITHM FOR THE CELL-CENTERED FINITE DIFFERENCE SCHEME Richard E. Ewing and Jian Shen Institute for Scientic Computation Texas A&M University College Station, Texas SUMMARY In this article,
More informationMultigrid absolute value preconditioning
Multigrid absolute value preconditioning Eugene Vecharynski 1 Andrew Knyazev 2 (speaker) 1 Department of Computer Science and Engineering University of Minnesota 2 Department of Mathematical and Statistical
More informationSolving Symmetric Indefinite Systems with Symmetric Positive Definite Preconditioners
Solving Symmetric Indefinite Systems with Symmetric Positive Definite Preconditioners Eugene Vecharynski 1 Andrew Knyazev 2 1 Department of Computer Science and Engineering University of Minnesota 2 Department
More information