Lecture 1. Finite difference and finite element methods. Partial differential equations (PDEs) Solving the heat equation numerically
|
|
- Donald Hardy
- 6 years ago
- Views:
Transcription
1 Finite difference and finite element methods Lecture 1
2 Scope of the course Analysis and implementation of numerical methods for pricing options. Models: Black-Scholes, stochastic volatility, exponential Lévy. Options: European, American, Asian, barrier, compound... In this course: Focus on deterministic (PDE based) methods Finite difference methods (FDM) Finite element methods (FEM) This course will be complemented by the course Monte Carlo methods in autumn 2009.
3 Organization of the course 14 lectures (2 hours) + 13 exercise classes (1 hour). No lectures on April, 14. Testat: 70% of solved homework assignments (theoretical exercises + MATLAB programming). Examination: On Tuesday, May 26, Written, closed-book examination includes theoretical and MATLAB programming problems. Examination takes place on ETH-workstations running MATLAB under LINUX. Own computer will NOT be required.
4 Outline Partial differential equations (PDEs)
5 Definitions and notation Let α = (α 1,..., α d ) N d 0 be a multiindex. Set α = d i=1 α i. For u : R, x = (x 1,..., x d ) R d define Let k N 0. Then D α u(x) := α u(x) x α 1 1 xα d d = α 1 x 1 α d x d u. D k u(x) := {D α u(x) : α = k} is the set of all partial derivatives of order k. If k = 1, we regard the elements of D 1 u(x) =: Du(x) as being arranged in a vector Du = ( x1 u,..., xd u).
6 If k = 2, we regard the elements of D 2 u(x) as being arranged in a matrix x1 x1 u x1 xd u D 2 u =.... xd x1 u xd xd u In the following: write xi x j instead of xi xj. Hence, the Laplacian u of u can be written as u := d xi x i u = tr(d 2 u). i=1 A partial differential equation (PDE) is an equation involving an unknown function of two or more variables and certain of its derivatives.
7 Let R d be open, x = (x 1,..., x d ) and N k 1. Definition An expression of the form F (D k u(x), D k 1 u(x),..., Du(x), u(x), x) = 0, x is called a k-th order PDE, where F : R dk R dk 1 R d R R is given and is the unknown. u : R
8 Let a ij (x), b i (x), c(x) and f(x) be given functions. For a linear 2nd order PDE in d + 1 variables, F has the form F (D 2 u, Du, u, x) = d d a ij (x) xi x j u+ b i (x) xi u+c(x)u f(x). i,j=0 Assume that the matrix A(x) := {a ij (x)} d i,j=0 is symmetric with real eigenvalues λ 0 (x) λ 1 (x) λ d (x). i=0 Definition Let S = {0,..., d}. At x R d+1, the PDE is called (i) elliptic λ i (x) 0, i sign(λ 0 (x)) =... = sign(λ d (x)) (ii) parabolic!j S : λ j (x) = 0 rank(a(x), b(x)) = d + 1 (iii) hyperbolic λ i (x) 0, i!j S : signλ j (x) signλ k (x), k S \ {j} The PDE is called elliptic, parabolic, hyperbolic on, if it is elliptic, parabolic, hyperbolic x.
9 Examples t u u = f(t, x) is parabolic (set x 0 = t). The Poisson equation u = f(x) is elliptic. The wave equation tt u u = f(t, x) is hyperbolic (set x 0 = t) The Black Scholes equation for the value of a European option v(t, s) t v 1 2 σ2 s 2 ss v rs s v + rv = 0 with σ, r 0 is parabolic at (t, s) (0, T ) (0, R) and degenerates to a ordinary differential equation as s 0. Note: PDEs can have infinitely many solutions. To obtain a unique solution, we have to pose boundary conditions.
10 Outline Partial differential equations (PDEs) Partial differential equations (PDEs)
11 The PDE Partial differential equations (PDEs) Let = (a, b) R be a open interval and let J := (0, T ), T > 0. Find u : J R such that Remark t u xx u = f(t, x) in J u = 0 on J u(0, ) = u 0 in (i) The equation u(0, ) = u 0 in is the initial condition. (ii) The equation u = 0 on J is the boundary condition. Here it is of Dirichlet type and homogeneous. oal: approximate u(t, x).
12 Discretization of the domain Computational domain J is replaced by discrete grid: {(t m, x i )}, i = 0,..., N + 1, m = 0,..., M, where x i are space grid points with space step size h and t m are the time levels with time step size k: x i = a + ih, h = b a N + 1, t m = mk, k = T M. We represent the exact solution u(t, x) by its values on the grid: u(t, x) {u m i = u(t m, x i )}, i = 0,..., N +1, m = 0,..., M. The goal is to approximate the values {u m i }. Values of the solution between grid points are then found by some interpolation.
13 Difference Quotients (= Finite Differences) We want to approximate the derivatives of u using only its values on the grid. First, let us consider a function f(x) of one variable. Assume that f C 2. Then, using Taylor s formula, f (x) = f(x + h) f(x) h + h 2 f (ξ), ξ [x, x + h]. If f i = f(x i ) are the values of f on the grid {x i }, we obtain f (x i ) = f i+1 f i h Similarly, for f C 4 + O(h) =: ( δ + x f) i + O(h). f (x i ) = f i+1 2f i + f i 1 h 2 + O(h 2 ) =: ( δ xx f ) i + O(h2 ).
14 FD scheme Partial differential equations (PDEs) Let θ [0, 1]. We replace the PDE t u xx u = f by the set of algebraic equations E m i = θf m+1 i + (1 θ)f m i i = 1,..., N, m = 0,..., M 1 u 0 i = u 0 (x i ) i = 1,..., N u m k = 0 k {0, N + 1}, m = 0,..., M, where E m i Ei m := k 1( u m+1 i u m i is the finite difference operator = um+1 i k u m i [θ um+1 i+1 2um+1 ) [ θ(δxx u) m+1 i + (1 θ)(δ xx u) m i i + u m+1 i 1 h 2 + (1 θ) um i+1 2um i + u m ] i 1 h 2. ]
15 FD scheme in matrix form Introduce the column vectors u m = (u m 1,..., um N ), E m = (E m 1,..., E m N ), f m = (f m 1,..., f m N ) and the tridiagonal N N matrix = h 2 tridiag( 1, 2, 1). Then the FD-scheme E m = θf m+1 + (1 θ)f m becomes, in matrix form: iven u 0 = (u 0 (x 1 ),..., u 0 (x N )) R N, for m = 0,..., M 1 find u m+1 R N such that ( I } + {{ θk } =:B or ) u m+1 + ( I + (1 θ)k }{{} =: C ) u m = k[θf m+1 + (1 θ)f m ], }{{} =:F m Bu m+1 = Cu m + kf m, m = 0,..., M 1.
16 Variational formulation We do not require the PDE to be hold pointwise. Take a smooth test function v C0 () satisfying v(a) = v(b) = 0. Multiply the PDE with v and integrate by parts: t uv dx xx uv dx = fv dx d dt uv dx [ x u(t, x)v(x)] x=b x=a + }{{} x u x v dx = fv dx =0 The variational or weak formulation of the heat equation reads: Find u such that u(0) = u 0 and such that v C0 () d u(t, x)v(x)dx + u (t, x)v (x)dx = f(t, x)v(x)dx. dt
17 alerkin discretization Let V N be a finite (N) dimensional subspace of H 1 0 (). The idea is to approximate u(t, x) by an element u N (t, x) V N, for each t J. Find u N (t, x) V N such that u N (0, x) = u 0,N (x) and such that v N V N d dt u N (t, x)v N (x)dx+ Let {b j } N j=1 be a basis of V N. Then u N (t, x) = N j=1 u N,j(t)b j (x), where u N(t, x)v N (x)dx = u N (t) = (u N,1 (t), u N,2 (t),..., u N,N (t)) is a vector of unknown functions. Similarly, v N V N v N (x) = N i=1 v N,ib i (x). f(t, x)v N (x)dx.
18 Hence (we skip the argument x in ) d u N (t)v N + u N dt (t)v N = f(t)v N, v N V N d ( )( ) u N,j (t)b j v N,i b i dt j i ( ) ( ) + u N,j (t)b j v N,i b i = f(t) v N,i b i j i i [ ] v N,i u N,j b j b i + u N,j b jb i f(t)b i = 0 i j [ ] v N M u N (t) + Au N (t) f N (t) = 0, v N R N M u N (t) + Au N (t) = f N (t).
19 Semi discrete scheme Thus, d dt (u N (t), v N ) + a ( ) u N (t), v N = (f(t), vn ), is equivalent to the ODE M u N (t) + Au N (t) = f N (t), v N V N where M (mass matrix) and A (stiffness matrix) are N N matrices with M ij = b j (x)b i (x)dx, A ij = b j (x)b i (x)dx. Similarly, f N (t) R N with entries f N,i (t) = f(t, x)b i (x)dx.
20 Fully discrete scheme We discretize in time. Write u m N := u N(t m ), f m N := f N(t m ), where the time levels t m, m = 0,..., M are as before. Proceeding exactly as in the FDM, the fully discrete scheme reads: iven u 0 N = (u 0(x i )) N i=1 RN, for m = 0,..., M 1 find R N such that u m+1 N Mk 1( u m+1 N um N ) ( ) +A θu m+1 N +(1 θ)um N = θf m+1 N +(1 θ)f m N.
21 Choice of V N Setting B := M + kθa, C := M k(1 θ)a and F m m+1 N := θf + (1 θ)f m this can be written as N N Bu m+1 N = Cu m N + kf m N, m = 0,..., M 1. It remains to chose a space V N. Probably the simplest choice: V N is the space of piecewise linear, continuous functions. Let T := {a = x 0 < x 1 < < x N+1 = b} = {K i } N+1 i=1 be an equidistant mesh on with K i := (x i 1, x i ). (x i as before). Set V N = S 1 0 := { u C 0 0 () : u K i is affin linear on K i T }. Note: dimv N = N.
22 g replacements A basis {b i } N i=1 of V N is given by the so-called hat-functions b i : [a, b] R 0, b i (x) = max{0, 1 h 1 x x i }, i = 1,..., N. b 1 (x) b 2 (x) b k (x) b N (x) a = x 0 x 1 x 2 x k 1 x k x k+1 x N 1 x N x N+1 = b With this basis, we find for the mass- and stiffness matrix M = h/6 tridiag(1, 4, 1), A = h 1 tridiag( 1, 2, 1).
23 For both FDM and FEM, we have to solve M systems of N linear equations of the form Bu m+1 = Cu m + kf m, m = 0,..., M 1. where F m = θf m+1 + (1 θ)f m and FDM FEM u m vector of u m i u(t m, x i ) coeff. vector of u N (t m, x) B I + kθ M + kθa C I k(1 θ) M k(1 θ)a f m = h 2 tridiag( 1, 2, 1) A = h 1 tridiag( 1, 2, 1) f(t m, x i ) f m = i f(t m, x)b i (x)dx
24 Matlab coding (FDM) function error = heateq_fdm(a,b,t,n,theta) h = (b-a)/(n+1); k = h; e = ones(n,1); I = speye(n); = h^(-2)*spdiags([-e 2*e -e],-1:1,n,n); B = I+k*theta*; C = I-k*(1-theta)* x = [a+h:h:b-h] ; u0 = x.*sin(pi*x); f = -(1-pi^2)*x.*sin(pi*x)-2*pi*cos(pi*x); u = zeros(n,t/k+1); u(:,1) = u0; for j = 1:T/k F = k*f*(theta*exp(-j*k)+(1-theta)*exp(-(j-1)*k)); u(:,j+1) = B\(C*u(:,j)+F); err(j) = norm(u(:,j+1)-exp(-k*j)*u0); end error = sqrt(h)*max(err)
25 Example Partial differential equations (PDEs) Let = (0, 1), T = 1, and u(x, t) = e t x sin(πx). We measure the discrete L (0, T ; L 2 ())-error defined by e := sup h 1 2 ɛ m l2, m ɛ m 2 l 2 := i u(t m, x i ) u m i 2. For θ = 0.5 and k = h, we obtain, in terms of the mesh width h, convergence both of FDM and FEM of second order, i.e., e = O(h s ), s = 2.
26 FDM FEM 10 5 e 10 6 eplacements s = h
Discretization of PDEs and Tools for the Parallel Solution of the Resulting Systems
Discretization of PDEs and Tools for the Parallel Solution of the Resulting Systems Stan Tomov Innovative Computing Laboratory Computer Science Department The University of Tennessee Wednesday April 4,
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 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 informationChapter Two: Numerical Methods for Elliptic PDEs. 1 Finite Difference Methods for Elliptic PDEs
Chapter Two: Numerical Methods for Elliptic PDEs Finite Difference Methods for Elliptic PDEs.. Finite difference scheme. We consider a simple example u := subject to Dirichlet boundary conditions ( ) u
More informationLECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES)
LECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES) RAYTCHO LAZAROV 1 Notations and Basic Functional Spaces Scalar function in R d, d 1 will be denoted by u,
More informationNumerical Methods for PDEs
Numerical Methods for PDEs Partial Differential Equations (Lecture 1, Week 1) Markus Schmuck Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt University, Edinburgh
More informationFrom Completing the Squares and Orthogonal Projection to Finite Element Methods
From Completing the Squares and Orthogonal Projection to Finite Element Methods Mo MU Background In scientific computing, it is important to start with an appropriate model in order to design effective
More informationFinite difference method for solving Advection-Diffusion Problem in 1D
Finite difference method for solving Advection-Diffusion Problem in 1D Author : Osei K. Tweneboah MATH 5370: Final Project Outline 1 Advection-Diffusion Problem Stationary Advection-Diffusion Problem in
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 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 informationFinite Difference and Finite Element Methods
Finite Difference and Finite Element Methods Georgy Gimel farb COMPSCI 369 Computational Science 1 / 39 1 Finite Differences Difference Equations 3 Finite Difference Methods: Euler FDMs 4 Finite Element
More information1 Discretizing BVP with Finite Element Methods.
1 Discretizing BVP with Finite Element Methods In this section, we will discuss a process for solving boundary value problems numerically, the Finite Element Method (FEM) We note that such method is a
More informationSecond Order Elliptic PDE
Second Order Elliptic PDE T. Muthukumar tmk@iitk.ac.in December 16, 2014 Contents 1 A Quick Introduction to PDE 1 2 Classification of Second Order PDE 3 3 Linear Second Order Elliptic Operators 4 4 Periodic
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 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 informationThe method of lines (MOL) for the diffusion equation
Chapter 1 The method of lines (MOL) for the diffusion equation The method of lines refers to an approximation of one or more partial differential equations with ordinary differential equations in just
More informationPDEs, part 1: Introduction and elliptic PDEs
PDEs, part 1: Introduction and elliptic PDEs Anna-Karin Tornberg Mathematical Models, Analysis and Simulation Fall semester, 2013 Partial di erential equations The solution depends on several variables,
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 information[2] (a) Develop and describe the piecewise linear Galerkin finite element approximation of,
269 C, Vese Practice problems [1] Write the differential equation u + u = f(x, y), (x, y) Ω u = 1 (x, y) Ω 1 n + u = x (x, y) Ω 2, Ω = {(x, y) x 2 + y 2 < 1}, Ω 1 = {(x, y) x 2 + y 2 = 1, x 0}, Ω 2 = {(x,
More informationMulti-Factor Finite Differences
February 17, 2017 Aims and outline Finite differences for more than one direction The θ-method, explicit, implicit, Crank-Nicolson Iterative solution of discretised equations Alternating directions implicit
More informationApplied Mathematics 205. Unit III: Numerical Calculus. Lecturer: Dr. David Knezevic
Applied Mathematics 205 Unit III: Numerical Calculus Lecturer: Dr. David Knezevic Unit III: Numerical Calculus Chapter III.3: Boundary Value Problems and PDEs 2 / 96 ODE Boundary Value Problems 3 / 96
More informationINTRODUCTION TO FINITE ELEMENT METHODS
INTRODUCTION TO FINITE ELEMENT METHODS LONG CHEN Finite element methods are based on the variational formulation of partial differential equations which only need to compute the gradient of a function.
More informationFirst order Partial Differential equations
First order Partial Differential equations 0.1 Introduction Definition 0.1.1 A Partial Deferential equation is called linear if the dependent variable and all its derivatives have degree one and not multiple
More information1. Let a(x) > 0, and assume that u and u h are the solutions of the Dirichlet problem:
Mathematics Chalmers & GU TMA37/MMG800: Partial Differential Equations, 011 08 4; kl 8.30-13.30. Telephone: Ida Säfström: 0703-088304 Calculators, formula notes and other subject related material are not
More informationLecture Notes on PDEs
Lecture Notes on PDEs Alberto Bressan February 26, 2012 1 Elliptic equations Let IR n be a bounded open set Given measurable functions a ij, b i, c : IR, consider the linear, second order differential
More informationLecture Introduction
Lecture 1 1.1 Introduction The theory of Partial Differential Equations (PDEs) is central to mathematics, both pure and applied. The main difference between the theory of PDEs and the theory of Ordinary
More informationPartial Differential Equations
M3M3 Partial Differential Equations Solutions to problem sheet 3/4 1* (i) Show that the second order linear differential operators L and M, defined in some domain Ω R n, and given by Mφ = Lφ = j=1 j=1
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 informationLecture Notes: African Institute of Mathematics Senegal, January Topic Title: A short introduction to numerical methods for elliptic PDEs
Lecture Notes: African Institute of Mathematics Senegal, January 26 opic itle: A short introduction to numerical methods for elliptic PDEs Authors and Lecturers: Gerard Awanou (University of Illinois-Chicago)
More informationPartial Differential Equations
Part II Partial Differential Equations Year 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2015 Paper 4, Section II 29E Partial Differential Equations 72 (a) Show that the Cauchy problem for u(x,
More informationn 1 f n 1 c 1 n+1 = c 1 n $ c 1 n 1. After taking logs, this becomes
Root finding: 1 a The points {x n+1, }, {x n, f n }, {x n 1, f n 1 } should be co-linear Say they lie on the line x + y = This gives the relations x n+1 + = x n +f n = x n 1 +f n 1 = Eliminating α and
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Numerical Methods for Partial Differential Equations Finite Difference Methods
More informationPartial Differential Equations
Partial Differential Equations Introduction Deng Li Discretization Methods Chunfang Chen, Danny Thorne, Adam Zornes CS521 Feb.,7, 2006 What do You Stand For? A PDE is a Partial Differential Equation This
More informationUNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH
UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH Faculty of Science and Engineering Department of Mathematics and Statistics END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MA4006 SEMESTER: Spring 2011 MODULE TITLE:
More informationCLASSIFICATION AND PRINCIPLE OF SUPERPOSITION FOR SECOND ORDER LINEAR PDE
CLASSIFICATION AND PRINCIPLE OF SUPERPOSITION FOR SECOND ORDER LINEAR PDE 1. Linear Partial Differential Equations A partial differential equation (PDE) is an equation, for an unknown function u, that
More informationNumerical Analysis of Differential Equations Numerical Solution of Parabolic Equations
Numerical Analysis of Differential Equations 215 6 Numerical Solution of Parabolic Equations 6 Numerical Solution of Parabolic Equations TU Bergakademie Freiberg, SS 2012 Numerical Analysis of Differential
More informationExponential integrators for semilinear parabolic problems
Exponential integrators for semilinear parabolic problems Marlis Hochbruck Heinrich-Heine University Düsseldorf Germany Innsbruck, October 2004 p. Outline Exponential integrators general class of methods
More informationChapter 3 Second Order Linear Equations
Partial Differential Equations (Math 3303) A Ë@ Õæ Aë áöß @. X. @ 2015-2014 ú GA JË@ É Ë@ Chapter 3 Second Order Linear Equations Second-order partial differential equations for an known function u(x,
More informationAssignment on iterative solution methods and preconditioning
Division of Scientific Computing, Department of Information Technology, Uppsala University Numerical Linear Algebra October-November, 2018 Assignment on iterative solution methods and preconditioning 1.
More informationImplementation of Sparse Wavelet-Galerkin FEM for Stochastic PDEs
Implementation of Sparse Wavelet-Galerkin FEM for Stochastic PDEs Roman Andreev ETH ZÜRICH / 29 JAN 29 TOC of the Talk Motivation & Set-Up Model Problem Stochastic Galerkin FEM Conclusions & Outlook Motivation
More informationOutline. 1 Boundary Value Problems. 2 Numerical Methods for BVPs. Boundary Value Problems Numerical Methods for BVPs
Boundary Value Problems Numerical Methods for BVPs Outline Boundary Value Problems 2 Numerical Methods for BVPs Michael T. Heath Scientific Computing 2 / 45 Boundary Value Problems Numerical Methods for
More informationIntroduction to multiscale modeling and simulation. Explicit methods for ODEs : forward Euler. y n+1 = y n + tf(y n ) dy dt = f(y), y(0) = y 0
Introduction to multiscale modeling and simulation Lecture 5 Numerical methods for ODEs, SDEs and PDEs The need for multiscale methods Two generic frameworks for multiscale computation Explicit methods
More informationFinite Element Methods
Solving Operator Equations Via Minimization We start with several definitions. Definition. Let V be an inner product space. A linear operator L: D V V is said to be positive definite if v, Lv > for every
More informationTime-dependent variational forms
Time-dependent variational forms Hans Petter Langtangen 1,2 1 Center for Biomedical Computing, Simula Research Laboratory 2 Department of Informatics, University of Oslo Oct 30, 2015 PRELIMINARY VERSION
More informationA very short introduction to the Finite Element Method
A very short introduction to the Finite Element Method Till Mathis Wagner Technical University of Munich JASS 2004, St Petersburg May 4, 2004 1 Introduction This is a short introduction to the finite element
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 informationOne-dimensional and nonlinear problems
Solving PDE s with FEniCS One-dimensional and nonlinear problems L. Ridgway Scott The Institute for Biophysical Dynamics, The Computation Institute, and the Departments of Computer Science and Mathematics,
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University A Model Problem in a 2D Box Region Let us consider a model problem of parabolic
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 informationDiscretization of SDEs: Euler Methods and Beyond
Discretization of SDEs: Euler Methods and Beyond 09-26-2006 / PRisMa 2006 Workshop Outline Introduction 1 Introduction Motivation Stochastic Differential Equations 2 The Time Discretization of SDEs Monte-Carlo
More informationSimple Examples on Rectangular Domains
84 Chapter 5 Simple Examples on Rectangular Domains In this chapter we consider simple elliptic boundary value problems in rectangular domains in R 2 or R 3 ; our prototype example is the Poisson equation
More informationAMS 529: Finite Element Methods: Fundamentals, Applications, and New Trends
AMS 529: Finite Element Methods: Fundamentals, Applications, and New Trends Lecture 3: Finite Elements in 2-D Xiangmin Jiao SUNY Stony Brook Xiangmin Jiao Finite Element Methods 1 / 18 Outline 1 Boundary
More informationScientific Computing I
Scientific Computing I Module 8: An Introduction to Finite Element Methods Tobias Neckel Winter 2013/2014 Module 8: An Introduction to Finite Element Methods, Winter 2013/2014 1 Part I: Introduction to
More informationNumerical Analysis and Simulation of Partial Differential Equations
Numerical Analysis and Simulation of Partial Differential Equations Roland Pulch Institut für Mathematik und Informatik Ernst-Moritz-Arndt-Universität Greifswald Contents: 1. Examples and Classification
More informationLehrstuhl Informatik V. Lehrstuhl Informatik V. 1. solve weak form of PDE to reduce regularity properties. Lehrstuhl Informatik V
Part I: Introduction to Finite Element Methods Scientific Computing I Module 8: An Introduction to Finite Element Methods Tobias Necel Winter 4/5 The Model Problem FEM Main Ingredients Wea Forms and Wea
More informationTheory of PDE Homework 2
Theory of PDE Homework 2 Adrienne Sands April 18, 2017 In the following exercises we assume the coefficients of the various PDE are smooth and satisfy the uniform ellipticity condition. R n is always an
More information1. Differential Equations (ODE and PDE)
1. Differential Equations (ODE and PDE) 1.1. Ordinary Differential Equations (ODE) So far we have dealt with Ordinary Differential Equations (ODE): involve derivatives with respect to only one variable
More informationPoisson Solvers. William McLean. April 21, Return to Math3301/Math5315 Common Material.
Poisson Solvers William McLean April 21, 2004 Return to Math3301/Math5315 Common Material 1 Introduction Many problems in applied mathematics lead to a partial differential equation of the form a 2 u +
More informationIntroduction of Partial Differential Equations and Boundary Value Problems
Introduction of Partial Differential Equations and Boundary Value Problems 2009 Outline Definition Classification Where PDEs come from? Well-posed problem, solutions Initial Conditions and Boundary Conditions
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 information(1) u (t) = f(t, u(t)), 0 t a.
I. Introduction 1. Ordinary Differential Equations. In most introductions to ordinary differential equations one learns a variety of methods for certain classes of equations, but the issues of existence
More informationFinite difference methods for the diffusion equation
Finite difference methods for the diffusion equation D150, Tillämpade numeriska metoder II Olof Runborg May 0, 003 These notes summarize a part of the material in Chapter 13 of Iserles. They are based
More informationLecture Note 3: Interpolation and Polynomial Approximation. Xiaoqun Zhang Shanghai Jiao Tong University
Lecture Note 3: Interpolation and Polynomial Approximation Xiaoqun Zhang Shanghai Jiao Tong University Last updated: October 10, 2015 2 Contents 1.1 Introduction................................ 3 1.1.1
More informationKrylov methods for the computation of matrix functions
Krylov methods for the computation of matrix functions Jitse Niesen (University of Leeds) in collaboration with Will Wright (Melbourne University) Heriot-Watt University, March 2010 Outline Definition
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Introduction to Hyperbolic Equations The Hyperbolic Equations n-d 1st Order Linear
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 informationPartial Differential Equations
Partial Differential Equations Xu Chen Assistant Professor United Technologies Engineering Build, Rm. 382 Department of Mechanical Engineering University of Connecticut xchen@engr.uconn.edu Contents 1
More informationMath 104: l Hospital s rule, Differential Equations and Integration
Math 104: l Hospital s rule, and Integration Ryan Blair University of Pennsylvania Tuesday January 22, 2013 Math 104:l Hospital s rule, andtuesday Integration January 22, 2013 1 / 8 Outline 1 l Hospital
More informationBrownian Motion. 1 Definition Brownian Motion Wiener measure... 3
Brownian Motion Contents 1 Definition 2 1.1 Brownian Motion................................. 2 1.2 Wiener measure.................................. 3 2 Construction 4 2.1 Gaussian process.................................
More information10 The Finite Element Method for a Parabolic Problem
1 The Finite Element Method for a Parabolic Problem In this chapter we consider the approximation of solutions of the model heat equation in two space dimensions by means of Galerkin s method, using piecewise
More informationThe continuity method
The continuity method The method of continuity is used in conjunction with a priori estimates to prove the existence of suitably regular solutions to elliptic partial differential equations. One crucial
More informationA note on accurate and efficient higher order Galerkin time stepping schemes for the nonstationary Stokes equations
A note on accurate and efficient higher order Galerkin time stepping schemes for the nonstationary Stokes equations S. Hussain, F. Schieweck, S. Turek Abstract In this note, we extend our recent work for
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 informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 11 Partial Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002.
More informationSolving PDEs with freefem++
Solving PDEs with freefem++ Tutorials at Basque Center BCA Olivier Pironneau 1 with Frederic Hecht, LJLL-University of Paris VI 1 March 13, 2011 Do not forget That everything about freefem++ is at www.freefem.org
More informationA Note on Integral Transforms and Partial Differential Equations
Applied Mathematical Sciences, Vol 4, 200, no 3, 09-8 A Note on Integral Transforms and Partial Differential Equations Adem Kılıçman and Hassan Eltayeb Department of Mathematics and Institute for Mathematical
More informationSolution to Homework 1
Solution to Homework Sec 2 (a) Yes It is condition (VS 3) (b) No If x, y are both zero vectors Then by condition (VS 3) x = x + y = y (c) No Let e be the zero vector We have e = 2e (d) No It will be false
More informationPDEs, part 3: Hyperbolic PDEs
PDEs, part 3: Hyperbolic PDEs Anna-Karin Tornberg Mathematical Models, Analysis and Simulation Fall semester, 2011 Hyperbolic equations (Sections 6.4 and 6.5 of Strang). Consider the model problem (the
More informationSolving Boundary Value Problems (with Gaussians)
What is a boundary value problem? Solving Boundary Value Problems (with Gaussians) Definition A differential equation with constraints on the boundary Michael McCourt Division Argonne National Laboratory
More informationUNIVERSITY OF MANITOBA
Question Points Score INSTRUCTIONS TO STUDENTS: This is a 6 hour examination. No extra time will be given. No texts, notes, or other aids are permitted. There are no calculators, cellphones or electronic
More informationPARTIAL DIFFERENTIAL EQUATIONS. Lecturer: D.M.A. Stuart MT 2007
PARTIAL DIFFERENTIAL EQUATIONS Lecturer: D.M.A. Stuart MT 2007 In addition to the sets of lecture notes written by previous lecturers ([1, 2]) the books [4, 7] are very good for the PDE topics in the course.
More informationMATH 2331 Linear Algebra. Section 1.1 Systems of Linear Equations. Finding the solution to a set of two equations in two variables: Example 1: Solve:
MATH 2331 Linear Algebra Section 1.1 Systems of Linear Equations Finding the solution to a set of two equations in two variables: Example 1: Solve: x x = 3 1 2 2x + 4x = 12 1 2 Geometric meaning: Do these
More informationTMA4220 Numerical Solution of Partial Differential Equations Using Element Methods Høst 2014
Norwegian University of Science and Technology Department of Mathematical Sciences TMA422 Numerical Solution of Partial Differential Equations Using Element Methods Høst 214 Exercise set 2 1 Consider the
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 informationFinite Differences for Differential Equations 28 PART II. Finite Difference Methods for Differential Equations
Finite Differences for Differential Equations 28 PART II Finite Difference Methods for Differential Equations Finite Differences for Differential Equations 29 BOUNDARY VALUE PROBLEMS (I) Solving a TWO
More informationKINGS COLLEGE OF ENGINEERING DEPARTMENT OF MATHEMATICS ACADEMIC YEAR / EVEN SEMESTER QUESTION BANK
KINGS COLLEGE OF ENGINEERING MA5-NUMERICAL METHODS DEPARTMENT OF MATHEMATICS ACADEMIC YEAR 00-0 / EVEN SEMESTER QUESTION BANK SUBJECT NAME: NUMERICAL METHODS YEAR/SEM: II / IV UNIT - I SOLUTION OF EQUATIONS
More informationLecture 3 Partial Differential Equations
Lecture 3 Partial Differential Equations Prof. Massimo Guidolin Prep Course in Investments August-September 2016 Plan of the lecture Motivation and generalities The heat equation and its applications in
More informationApplied Math Qualifying Exam 11 October Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems.
Printed Name: Signature: Applied Math Qualifying Exam 11 October 2014 Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems. 2 Part 1 (1) Let Ω be an open subset of R
More informationExistence Theory: Green s Functions
Chapter 5 Existence Theory: Green s Functions In this chapter we describe a method for constructing a Green s Function The method outlined is formal (not rigorous) When we find a solution to a PDE by constructing
More informationBlock-Structured Adaptive Mesh Refinement
Block-Structured Adaptive Mesh Refinement Lecture 2 Incompressible Navier-Stokes Equations Fractional Step Scheme 1-D AMR for classical PDE s hyperbolic elliptic parabolic Accuracy considerations Bell
More informationIntroduction to Partial Differential Equations
Introduction to Partial Differential Equations Philippe B. Laval KSU Current Semester Philippe B. Laval (KSU) Key Concepts Current Semester 1 / 25 Introduction The purpose of this section is to define
More informationScientific Computing WS 2018/2019. Lecture 15. Jürgen Fuhrmann Lecture 15 Slide 1
Scientific Computing WS 2018/2019 Lecture 15 Jürgen Fuhrmann juergen.fuhrmann@wias-berlin.de Lecture 15 Slide 1 Lecture 15 Slide 2 Problems with strong formulation Writing the PDE with divergence and gradient
More informationFinite Volume Schemes: an introduction
Finite Volume Schemes: an introduction First lecture Annamaria Mazzia Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate Università di Padova mazzia@dmsa.unipd.it Scuola di dottorato
More informationi=1 α i. Given an m-times continuously
1 Fundamentals 1.1 Classification and characteristics Let Ω R d, d N, d 2, be an open set and α = (α 1,, α d ) T N d 0, N 0 := N {0}, a multiindex with α := d i=1 α i. Given an m-times continuously differentiable
More informationDEPARTMENT OF MATHEMATICS. The University of Queensland. MATH3203 Scientific Computation: Algorithmic Design and Implementation
DEPARTMENT OF MATHEMATICS The University of Queensland MATH303 Scientific Computation: Algorithmic Design and Implementation Section A Boundary Value Problem in dimension Semester, 00 A simplified problem
More informationChapter 1: The Finite Element Method
Chapter 1: The Finite Element Method Michael Hanke Read: Strang, p 428 436 A Model Problem Mathematical Models, Analysis and Simulation, Part Applications: u = fx), < x < 1 u) = u1) = D) axial deformation
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 informationSINC PACK, and Separation of Variables
SINC PACK, and Separation of Variables Frank Stenger Abstract This talk consists of a proof of part of Stenger s SINC-PACK computer package (an approx. 400-page tutorial + about 250 Matlab programs) that
More informationMonte Carlo Method for Finding the Solution of Dirichlet Partial Differential Equations
Applied Mathematical Sciences, Vol. 1, 2007, no. 10, 453-462 Monte Carlo Method for Finding the Solution of Dirichlet Partial Differential Equations Behrouz Fathi Vajargah Department of Mathematics Guilan
More informationAPM526 CLASSROOM NOTES PART IB. APM526, Spring 2018 Last update: MAR 12
APM526 CLASSROOM NOTES PART IB APM526, Spring 2018 Last update: MAR 12 1 PARTICLE METHODS 1. Particle based methods: Large systems of ODEs for individual particles. 2. Partial differential equations: Equations
More information