Finite Volume Discretization of the Heat Equation


 Laureen Mathews
 1 years ago
 Views:
Transcription
1 Lecture Notes 3 Finite Volume Discretization of te Heat Equation We consider finite volume discretizations of te onedimensional variable coefficient eat equation, wit Neumann boundary conditions u t x (k(x) x u) = S(t, x), 0 < x <, t > 0, () Te coefficient k(x) is strictly positive. u(0, x) = f(x), 0 < x <, u x (t, 0) = u x (t, ) = 0, t 0. Semidiscrete approximation By semidiscretization we mean discretization only in space, not in time. Tis approac is also called metod of lines. Discretization We discretize space into N equal size grid cells (bins) of size = /N, and define x j = /2 + j, so tat x j is te center of cell j, see figure. Te edges of cell j are ten and x j+/2. x 0 x x N Unknowns In a finite volume metod te unknowns approximate te average of te solution over a grid cell. More precisely, we let q j (t) be te approximation q j (t) u j (t) := u(t, x)dx. Note te contrast wit finite difference metods, were pointwise values are approximated, and finite element metods, were basis function coefficients are approximated. (0)
2 Exact update formula We derive an exact update formula for u j (t), te exact local averages. Integrating () over cell j and dividing by we get u t (t, x)dx = Upon defining te flux and te local average of te source we get te exact update formula du j (t) x (k(x) x u)dx + S(t, x)dx = k(x j+/2)u x (t, x j+/2 ) k( )u x (t, ) = F j+/2(t) F j /2 (t) F j (t) = F (t, x j ) = k(x j )u x (t, x j ), S j (t) = + S j (t). S(t, x)dx, Te fluxes F j+/2 and F j /2 ten represents ow muc eat flows out troug te left and rigt boundary of te cell. Note, tis is an instance te conservation law in integral form, d V udv + S F nds = V SdV, were we ave picked V as te interval [, x j+/2 ], and scaled by V =. Approximation of te flux + S(t, x)dx. To use te exact update formula as te basis for a numerical sceme we must approximate te fluxes F j±/2. Since te value in te midpoint of te cell is a second order approximation of te average, we ave for smoot u, F j /2 (t) = k( )u x (t, ) = k( ) u(t, x j) u(t, x j ) = k( ) u j(t) u j (t) + O( 2 ). + O( 2 ) We terefore use F j /2 (t) F j /2 (t) = k( ) q j(t) q j (t). as approximation. Tis leads to te numerical sceme for inner points j N 2, dq j (t) F j+/2 (t) F j /2 (t) = + S j (t) (2) = k(x j+/2)(q j+ (t) q j (t)) k( )(q j (t) q j (t)) 2 + S j (t). 2 (0)
3 Hence, wit k j := k(x j ), dq j = k j+/2q j+ (k j+/2 + k j /2 )q j + k j /2 q j 2 + S j, (3) for j =,..., N 2. Tis is a second order approximation. Boundary conditions To complete te sceme (3) we need update formulae also for te boundary points j = 0 and j = N. Tese must be derived by taking te boundary conditions into account. We introduce te gost cells j = and j = N wic are located just outside te domain. Te boundary conditions are used to fill tese cells wit values q and q N, based on te values q j in te interior cells. Te same update formula (3) as before can ten be used also for j = 0 and j = N. Let us consider our boundary condition u x = 0 at x = 0. (We can also tink of tis as a "no flux" condition, F = 0.) We formally extend te definition of te solution u for x < 0, i.e. outside te domain, and, as before, approximate 0 = u x (t, 0) = u(t, x 0) u(t, x ) + O( 2 ) = u 0(t) u (t) + O( 2 ) u (t) = u 0 (t) + O( 3 ). Replacing u j by our approximation q j and dropping te O( 2 ) term we get an expression for q in terms of q 0 as te boundary rule q (t) = q 0 (t). We now insert tis into te update formula (3) for j = 0, dq 0 = k /2q (k /2 + k /2 )q 0 + k /2 q 2 + S 0 = k /2 q q S 0. (4) In exactly te same way we obtain for j = N tat q N (t) = q N (t) and terefore dq N = k N 3/2 q N 2 q N 2 + S N. (5) Remark Diriclet boundary conditions u(t, 0) = u(t, ) = 0 can be approximated to second order in two ways. First, one can use a sifted grid, x j = j so tat x 0 and x N, te centers of cells 0 and N, are precisely on te boundary. Ten one does not need gost cells; one just sets q 0 = q N = 0. Note tat te number of unknows are now only N, so A R (N ) (N ) etc. Second, one can take te average of two cells to approximate te value in between, 0 = u(t, 0) = u(t, x 0) + u(t, x ) 2 leading to te approximations + O( 2 ) = u 0(t) + u (t) 2 q = q 0, q N = q N. + O( 2 ), 3 (0)
4 Matrix form We put all te formulae (3), (4), (5) togeter and write tem in matrix form. Introduce q = and k /2 k /2 A = 2 q 0. q N, S = S 0. S N k /2 (k /2 + k 3/2 ) k 3/ Ten we get te linear ODE system, q, S R k N 5/2 (k N 5/2 + k N 3/2 ) k N 3/2 k N 3/2 k N 3/2 dq(t) R N N. (6) = Aq(t) + S(t). (7) Hence, in tis semidiscretization te timedependent PDE as been approximated by a system of ODEs, were te matrix A is a discrete approximation of te second order differential operator x k(x) x, including its boundary conditions.. Brief outline of extensions to 2D Te same strategy can be used in 2D. We give a cursory description of te main steps ere. To simplify tings we consider te constant coefficient problem, u t u = S(t, x, y), 0 < x <, 0 < y <, t > 0, u(0, x, y) = f(x, y), 0 < x <, 0 < y <, u x (t, 0, y) = u x (t, 2π, y) = u y (t, x, 0) = u y (t, x, 2π) = 0, 0 < x <, 0 < y <, t 0. Discretization We discretize te domain [0, ] 2 into N N equal size grid cells of size, were = /N. We define x j = /2 + j and y k = /2 + k and denote te cell wit center (x j, y k ) by I jk. 4 (0)
5 y N I jk y y 0 x 0 x x N Unknowns Te unknowns are now q jk (t), wic are approximations of te average of te solution over te grid cell I jk, q jk (t) u jk (t) := 2 I jk u(t, x, y)dxdy. Exact update formula Te update formula is again an instance of te conservation law in integral form were we pick te volume V as I jk and scale by I jk = 2, d 2 udxdy + F nds = I jk 2 I jk 2 Sdxdy, I jk were F = u. Upon defining S jk (t) := 2 I jk S(t, x, y)dxdy, tis can be written as du jk (t) = 2 I jk F nds + Sjk (t). Let F = (f, f 2 ) and define te average flux troug eac side of te cell, F j,k±/2 (t) := we get te exact formula f 2 (t, x, y k±/2 )dx, F j±/2,k (t) := yk+/2 y k /2 f (t, x j±/2, y)dy, du jk = (F j+/2,k F j /2,k + F j,k+/2 F j,k /2 ) + S jk. (8) 5 (0)
6 Again, te fluxes F j,k±/2 and F j±/2,k represent te eat flux (upto sign) out troug te four sides of te cell. Approximation of te flux We use te same type of approximation as in D for F = u. We get for instance F j,k /2 = = 2 u y (t, x, y k /2 )dx = u(t, x, y k )dx + 2 = 3 I jk u(t, x, y)dxdy + 3 = u jk + u j,k + O( 2 ). u(t, x, y k ) u(t, x, y k ) u(t, x, y k )dx + O( 2 ) I j,k u(t, x, y)dxdy + O( 2 ) dx + O( 2 ) Replacing u jk by q jk, dropping O( 2 ) and inserting in te exact formula (8) we get te fivepoint formula, dq jk = 2 (q j+,k + q j,k + q j,k+ + q j,k 4q jk ) + S jk. Boundary conditions is done as in D. Te matrix form is more complicated and te subject of Homework. 2 Properties of te semidiscrete approximation 2. Conservation Wen S = 0 in () we ave seen tat te solution as te conservation property 0 u(t, x)dx = constant = 0 f(x)dx. In fact tis olds for all conservation laws wit "noflux" boundary conditions F = 0, wic is easily seen from te integral form of te PDE. An analogue of tis olds also for te semidiscrete approximation. More precisely, let us define Ten Q(t) = N q j (t). Q(t) = constant. (9) Moreover, if we start te approximation wit exact values, q j (0) = u j (0), ten it follows tat Q(t) is exactly te integral of te solution u for all time, Q(t) = constant = Q(0) = N u j (0) = N We prove te exact discrete conservation (9) in two ways. 6 (0) u(0, x)dx = 0 u(0, x)dx = 0 u(t, x)dx.
7 Recall from (2) tat wen S = 0, Ten N dq = dq j (t) N dq j = F j+/2 (t) F j /2 (t) =. (0) F j /2 (t) F j+/2 = F /2 F N /2 = 0, were we used te boundary conditions (4) and (5) wic implies tat F /2 = k /2 q 0 q = 0, FN /2 = k N /2 q N q N Note tat tis discrete conservation property is true for any discretization of te type (0) if F /2 = F N /2, regardless of ow te fluxes F j±/2 are computed. Recall from (7) tat wen S = 0, Moreover, dq(t) = Aq(t). Q(t) = T q, =. RN. Ten dq(t) = T dq(t) = T Aq(t) = 0, since T Aq(t) = (A) T q(t) by te symmetry of A and A are te row sums of A wic are zero, by (6). 2.2 Maximum principle We ave seen before tat in te continuous case tat wen S = 0 in (), te maximum value of te solution u(t, x) in [0, T ] [0, ] is eiter attained on te boundary x {0, } or for te initial data at t = 0 (regardless of boundary conditions). Te corresponding result in te semidiscrete case says tat q = max sup q j (t) j 0,...,N 0 t<t is attained eiter for j = 0, j = N or for te initial data t = 0. Tis is easily sown by contradiction. Suppose te maximum is attained at t = t > 0 and tat it is strictly larger tan q 0 (t ) and q N (t ). Ten, tere must be an interior index j [, N 2] suc tat Terefore dq j (t ) q j (t ) < q j (t ), q j (t ) q j +(t ). = k j +/2 (q j + q j ) }{{} 0 k j /2 (q j q j ) < 0. }{{} >0 Hence, tere is an ε suc tat q j (t) > q j (t ) for all t (t ε, t ), wic contradicts te assumption tat q j (t ) is a maximum. Furtermore, as in te continuous case local spatial maximum (minimum) of q j in te interior cannot increase (decrease) in time. = 0. 7 (0)
8 Forward Euler Im Backward Euler Im D D Re Re Figure. Stability regions D for te Forward and Backward Euler metods. 3 Fully discrete approximation Te semidiscrete approximation leads to a system of ODEs. dq(t) = Aq(t) + S(t). () Tis can be solved by standard numerical metods for ODEs wit a time step t, e.g. Forward Euler metod u n+ = u n + tf(t n, u n ), t n = n t. Applied to () tis would give te fully discrete sceme for (), q n+ = q n + t[aq n + S(t n )], q n q(t n ). As for any ODE metod we must verify its absolute stability: tat for our coice of step size t te tλ k D, k, were D is te stability region of te ODE solver and {λ k } are te eigenvalues of A. For parabolic problems te real part of te eigenvalues are negative, and te size of tem in general grow as / 2. Tis is a major difficulty. It means tat wen te stability region D is bounded, as in explicit metods, we get a time step restriction of te type t C 2. Tis is a severe restriction, wic is seldom warranted from an accuracy point of view. It leads to unnecessarily expensive metods. Example In te constant coefficient case k(x) te eigenvalues of A are precisely λ k = 4 ( ) kπ 2 sin2, k = 0,..., N. 2 Tis gives te stability condition t 2 /2 for Forward Euler. 8 (0)
9 Te underlying reason for te severe timestep restriction is te fact tat parabolic problems include processes on all timescales: ig frequencies decay fast, low frequencies slowly. Teir semidiscretization ave timescales spread out all over te interval [ / 2, 0], wic means tat te ODEs are stiff. Te consequence is tat implicit metods sould be used for parabolic problems. Implicit metods typically ave unbounded stability domains D and ave no stability restriction on te timestep tey are unconditionally stable. Of course, implicit metods are more expensive per timestep tan explicit metods, since a system of equations must be solved, but tis is outweiged by te fact tat muc longer timesteps can be taken. Moreover, w en te coefficients do not vary wit time, matrices etc. can be constructed once, and refactored only as canges of timestep make it necessary. Te "θmetod" is a class of ODE metods defined as u n+ = u n + t [ θf(t n+, u n+ ) + ( θ)f(t n, u n ) ], 0 θ. Tis includes some common metods: Applied to () we ave or θ = 0 Forward Euler (explicit, st order), θ = /2 Crank Nicolson (implicit, 2nd order), θ = Backward Euler (implicit, st order), q n+ = q n + ta[θq n+ + ( θ)q n ] + t [θs(t n+ ) + ( θ)s(t n )], (2) }{{} S n θ ( θ ta)q n+ = ( + ( θ) ta)q n + ts n θ. For te constant coefficient problem k one can sow te timestep restriction t 2 { 2( 2θ), θ < /2,, /2 θ, (unconditionally stable). Remark 2 Te CrankNicolson sceme is second order accurate but gives slowly decaying oscillations for large eigenvalues. It is unsuitable for parabolic problems wit rapidly decaying transients. Te θ = sceme damps all components, and sould be used in te initial steps. Remark 3 Te most used family of timestepping scemes for parabolic problems are te Backward Differentiation Formulas (BDF), of order troug 5 wic are A(α)stable. Tey are multistep metods generalizing Backward Euler to iger order. For instance, te second order BDF metod is u n+ = 4 3 un 3 un tf(t n+, u n+ ). BDF metods are also known as Gear s metods and available in Matlab as ODE5S. 3. Fully discrete conservation For te θmetod we also ave discrete conservation wen S 0. Let Q n N q n j = T q n. 9 (0)
10 Ten upon multiplying by from te left in (2) we get Q n+ = T q n+ = T q n + t T A[θq n+ +( θ)q n ] = Q n + t(a) T [θq n+ +( θ)q n ] = Q n, since as before A = 0. In particular, if initial data is exact, q 0 j = u j(0), ten Q n = 0 u(t n, x)dx, for all n 0. Te same is true for te second order BDF metod if te initialization of te first step Q is conservative so tat Q = Q 0. Ten q n+ = 4 3 qn 3 qn taqn+ implies Q n+ = T q n+ = 4 3 T q n 3 T q n T taq n+ = 4 3 Qn 3 Qn. Wit te stipulated initial data tis difference equation as te solution Q n = Q 0 for all n 0. 4 Acknowledgement Part of tese notes are based on earlier notes by Prof. Jesper Oppelstrup. 0 (0)
Verifying Numerical Convergence Rates
1 Order of accuracy Verifying Numerical Convergence Rates We consider a numerical approximation of an exact value u. Te approximation depends on a small parameter, suc as te grid size or time step, and
More informationFINITE DIFFERENCE METHODS
FINITE DIFFERENCE METHODS LONG CHEN Te best known metods, finite difference, consists of replacing eac derivative by a difference quotient in te classic formulation. It is simple to code and economic to
More informationInstantaneous Rate of Change:
Instantaneous Rate of Cange: Last section we discovered tat te average rate of cange in F(x) can also be interpreted as te slope of a scant line. Te average rate of cange involves te cange in F(x) over
More informationFinite Difference Approximations
Capter Finite Difference Approximations Our goal is to approximate solutions to differential equations, i.e., to find a function (or some discrete approximation to tis function) tat satisfies a given relationsip
More informationTrapezoid Rule. y 2. y L
Trapezoid Rule and Simpson s Rule c 2002, 2008, 200 Donald Kreider and Dwigt Lar Trapezoid Rule Many applications of calculus involve definite integrals. If we can find an antiderivative for te integrand,
More informationUnderstanding the Derivative Backward and Forward by Dave Slomer
Understanding te Derivative Backward and Forward by Dave Slomer Slopes of lines are important, giving average rates of cange. Slopes of curves are even more important, giving instantaneous rates of cange.
More informationLecture 10: What is a Function, definition, piecewise defined functions, difference quotient, domain of a function
Lecture 10: Wat is a Function, definition, piecewise defined functions, difference quotient, domain of a function A function arises wen one quantity depends on anoter. Many everyday relationsips between
More information7.6 Complex Fractions
Section 7.6 Comple Fractions 695 7.6 Comple Fractions In tis section we learn ow to simplify wat are called comple fractions, an eample of wic follows. 2 + 3 Note tat bot te numerator and denominator are
More informationOptimal Pricing Strategy for Second Degree Price Discrimination
Optimal Pricing Strategy for Second Degree Price Discrimination Alex O Brien May 5, 2005 Abstract Second Degree price discrimination is a coupon strategy tat allows all consumers access to te coupon. Purcases
More informationDerivatives Math 120 Calculus I D Joyce, Fall 2013
Derivatives Mat 20 Calculus I D Joyce, Fall 203 Since we ave a good understanding of its, we can develop derivatives very quickly. Recall tat we defined te derivative f x of a function f at x to be te
More informationThis supplement is meant to be read after Venema s Section 9.2. Throughout this section, we assume all nine axioms of Euclidean geometry.
Mat 444/445 Geometry for Teacers Summer 2008 Supplement : Similar Triangles Tis supplement is meant to be read after Venema s Section 9.2. Trougout tis section, we assume all nine axioms of uclidean geometry.
More information2.0 5Minute Review: Polynomial Functions
mat 3 day 3: intro to limits 5Minute Review: Polynomial Functions You sould be familiar wit polynomials Tey are among te simplest of functions DEFINITION A polynomial is a function of te form y = p(x)
More informationThe EOQ Inventory Formula
Te EOQ Inventory Formula James M. Cargal Matematics Department Troy University Montgomery Campus A basic problem for businesses and manufacturers is, wen ordering supplies, to determine wat quantity of
More informationIn other words the graph of the polynomial should pass through the points
Capter 3 Interpolation Interpolation is te problem of fitting a smoot curve troug a given set of points, generally as te grap of a function. It is useful at least in data analysis (interpolation is a form
More informationA.4. Rational Expressions. Domain of an Algebraic Expression. What you should learn. Why you should learn it
A6 Appendi A Review of Fundamental Concepts of Algebra A.4 Rational Epressions Wat you sould learn Find domains of algebraic epressions. Simplify rational epressions. Add, subtract, multiply, and divide
More informationf(a + h) f(a) f (a) = lim
Lecture 7 : Derivative AS a Function In te previous section we defined te derivative of a function f at a number a (wen te function f is defined in an open interval containing a) to be f (a) 0 f(a + )
More information2 Limits and Derivatives
2 Limits and Derivatives 2.7 Tangent Lines, Velocity, and Derivatives A tangent line to a circle is a line tat intersects te circle at exactly one point. We would like to take tis idea of tangent line
More information1 Density functions, cummulative density functions, measures of central tendency, and measures of dispersion
Density functions, cummulative density functions, measures of central tendency, and measures of dispersion densityfunctionsintro.tex October, 9 Note tat tis section of notes is limitied to te consideration
More informationSolution Derivations for Capa #7
Solution Derivations for Capa #7 1) Consider te beavior of te circuit, wen various values increase or decrease. (Select Iincreases, Ddecreases, If te first is I and te rest D, enter IDDDD). A) If R1
More informationMATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1  BASIC DIFFERENTIATION
MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1  BASIC DIFFERENTIATION Tis tutorial is essential prerequisite material for anyone stuing mecanical engineering. Tis tutorial uses te principle of
More informationDifferentiable Functions
Capter 8 Differentiable Functions A differentiable function is a function tat can be approximated locally by a linear function. 8.. Te derivative Definition 8.. Suppose tat f : (a, b) R and a < c < b.
More informationMath 113 HW #5 Solutions
Mat 3 HW #5 Solutions. Exercise.5.6. Suppose f is continuous on [, 5] and te only solutions of te equation f(x) = 6 are x = and x =. If f() = 8, explain wy f(3) > 6. Answer: Suppose we ad tat f(3) 6. Ten
More informationSAT Subject Math Level 1 Facts & Formulas
Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences: PEMDAS (Parenteses
More informationPart II: Finite Difference/Volume Discretisation for CFD
Part II: Finite Difference/Volume Discretisation for CFD Finite Volume Metod of te AdvectionDiffusion Equation A Finite Difference/Volume Metod for te Incompressible NavierStokes Equations MarkerandCell
More informationProof of the Power Rule for Positive Integer Powers
Te Power Rule A function of te form f (x) = x r, were r is any real number, is a power function. From our previous work we know tat x x 2 x x x x 3 3 x x In te first two cases, te power r is a positive
More informationTangent Lines and Rates of Change
Tangent Lines and Rates of Cange 922005 Given a function y = f(x), ow do you find te slope of te tangent line to te grap at te point P(a, f(a))? (I m tinking of te tangent line as a line tat just skims
More information6. Differentiating the exponential and logarithm functions
1 6. Differentiating te exponential and logaritm functions We wis to find and use derivatives for functions of te form f(x) = a x, were a is a constant. By far te most convenient suc function for tis purpose
More informationACT Math Facts & Formulas
Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Rationals: fractions, tat is, anyting expressable as a ratio of integers Reals: integers plus rationals plus special numbers suc as
More informationThe differential amplifier
DiffAmp.doc 1 Te differential amplifier Te emitter coupled differential amplifier output is V o = A d V d + A c V C Were V d = V 1 V 2 and V C = (V 1 + V 2 ) / 2 In te ideal differential amplifier A c
More informationGeometric Stratification of Accounting Data
Stratification of Accounting Data Patricia Gunning * Jane Mary Horgan ** William Yancey *** Abstract: We suggest a new procedure for defining te boundaries of te strata in igly skewed populations, usual
More informationWeek #15  Word Problems & Differential Equations Section 8.2
Week #1  Word Problems & Differential Equations Section 8. From Calculus, Single Variable by HugesHallett, Gleason, McCallum et. al. Copyrigt 00 by Jon Wiley & Sons, Inc. Tis material is used by permission
More informationP.4 Rational Expressions
7_0P04.qp /7/06 9:4 AM Page 7 Section P.4 Rational Epressions 7 P.4 Rational Epressions Domain of an Algebraic Epression Te set of real numbers for wic an algebraic epression is defined is te domain of
More informationFourier series. Jan Philip Solovej. English summary of notes for Analysis 1. May 8, 2012
Fourier series Jan Philip Solovej English summary of notes for Analysis 1 May 8, 2012 1 JPS, Fourier series 2 Contents 1 Introduction 2 2 Fourier series 3 2.1 Periodic functions, trigonometric polynomials
More informationMath WarmUp for Exam 1 Name: Solutions
Disclaimer: Tese review problems do not represent te exact questions tat will appear te exam. Tis is just a warmup to elp you begin studying. It is your responsibility to review te omework problems, webwork
More informationA STUDY ON NUMERICAL COMPUTATION OF POTENTIAL DISTRIBUTION AROUND A GROUNDING ROD DRIVEN IN SOIL
A STUDY ON NUMERICAL COMPUTATION OF POTENTIAL DISTRIBUTION AROUND A GROUNDING ROD DRIVEN IN SOIL Ersan Şentürk 1 Nurettin Umurkan Özcan Kalenderli 3 email: ersan.senturk@turkcell.com.tr email: umurkan@yildiz.edu.tr
More informationChapter 7 Numerical Differentiation and Integration
45 We ave a abit in writing articles publised in scientiþc journals to make te work as Þnised as possible, to cover up all te tracks, to not worry about te blind alleys or describe ow you ad te wrong idea
More informationSections 3.1/3.2: Introducing the Derivative/Rules of Differentiation
Sections 3.1/3.2: Introucing te Derivative/Rules of Differentiation 1 Tangent Line Before looking at te erivative, refer back to Section 2.1, looking at average velocity an instantaneous velocity. Here
More informationME422 Mechanical Control Systems Modeling Fluid Systems
Cal Poly San Luis Obispo Mecanical Engineering ME422 Mecanical Control Systems Modeling Fluid Systems Owen/Ridgely, last update Mar 2003 Te dynamic euations for fluid flow are very similar to te dynamic
More informationModule 1: Introduction to Finite Element Analysis Lecture 1: Introduction
Module : Introduction to Finite Element Analysis Lecture : Introduction.. Introduction Te Finite Element Metod (FEM) is a numerical tecnique to find approximate solutions of partial differential equations.
More information11.2 Instantaneous Rate of Change
11. Instantaneous Rate of Cange Question 1: How do you estimate te instantaneous rate of cange? Question : How do you compute te instantaneous rate of cange using a limit? Te average rate of cange is useful
More informationSolving partial differential equations (PDEs)
Solving partial differential equations (PDEs) Hans Fangor Engineering and te Environment University of Soutampton United Kingdom fangor@soton.ac.uk May 3, 2012 1 / 47 Outline I 1 Introduction: wat are
More informationOPTIMAL DISCONTINUOUS GALERKIN METHODS FOR THE ACOUSTIC WAVE EQUATION IN HIGHER DIMENSIONS
OPTIMAL DISCONTINUOUS GALERKIN METHODS FOR THE ACOUSTIC WAVE EQUATION IN HIGHER DIMENSIONS ERIC T. CHUNG AND BJÖRN ENGQUIST Abstract. In tis paper, we developed and analyzed a new class of discontinuous
More informationACTIVITY: Deriving the Area Formula of a Trapezoid
4.3 Areas of Trapezoids a trapezoid? How can you derive a formula for te area of ACTIVITY: Deriving te Area Formula of a Trapezoid Work wit a partner. Use a piece of centimeter grid paper. a. Draw any
More informationTheoretical calculation of the heat capacity
eoretical calculation of te eat capacity Principle of equipartition of energy Heat capacity of ideal and real gases Heat capacity of solids: DulongPetit, Einstein, Debye models Heat capacity of metals
More information1 Derivatives of Piecewise Defined Functions
MATH 1010E University Matematics Lecture Notes (week 4) Martin Li 1 Derivatives of Piecewise Define Functions For piecewise efine functions, we often ave to be very careful in computing te erivatives.
More informationThe Heat Equation. Lectures INF2320 p. 1/88
The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)
More informationCan a LumpSum Transfer Make Everyone Enjoy the Gains. from Free Trade?
Can a LumpSum Transfer Make Everyone Enjoy te Gains from Free Trade? Yasukazu Icino Department of Economics, Konan University June 30, 2010 Abstract I examine lumpsum transfer rules to redistribute te
More information5.4 The Heat Equation and ConvectionDiffusion
5.4. THE HEAT EQUATION AND CONVECTIONDIFFUSION c 6 Gilbert Strang 5.4 The Heat Equation and ConvectionDiffusion The wave equation conserves energy. The heat equation u t = u xx dissipates energy. The
More informationAn Interest Rate Model
An Interest Rate Model Concepts and Buzzwords Building Price Tree from Rate Tree Lognormal Interest Rate Model Nonnegativity Volatility and te Level Effect Readings Tuckman, capters 11 and 12. Lognormal
More informationExercises for numerical integration. Øyvind Ryan
Exercises for numerical integration Øyvind Ryan February 25, 21 1. Vi ar r(t) = (t cos t, t sin t, t) Solution: Koden blir % Oppgave.1.11 b) quad(@(x)sqrt(2+t.^2),,2*pi) a. Finn astigeten, farten og akselerasjonen.
More information13 PERIMETER AND AREA OF 2D SHAPES
13 PERIMETER AND AREA OF D SHAPES 13.1 You can find te perimeter of sapes Key Points Te perimeter of a twodimensional (D) sape is te total distance around te edge of te sape. l To work out te perimeter
More informationPipe Flow Analysis. Pipes in Series. Pipes in Parallel
Pipe Flow Analysis Pipeline system used in water distribution, industrial application and in many engineering systems may range from simple arrangement to extremely complex one. Problems regarding pipelines
More informationFourier Series. Chapter Some Properties of Functions Goal Preliminary Remarks
Chapter 3 Fourier Series 3.1 Some Properties of Functions 3.1.1 Goal We review some results about functions which play an important role in the development of the theory of Fourier series. These results
More information1. Periodic Fourier series. The Fourier expansion of a 2πperiodic function f is:
CONVERGENCE OF FOURIER SERIES 1. Periodic Fourier series. The Fourier expansion of a 2πperiodic function f is: with coefficients given by: a n = 1 π f(x) a 0 2 + a n cos(nx) + b n sin(nx), n 1 f(x) cos(nx)dx
More informationLecture 10. Limits (cont d) Onesided limits. (Relevant section from Stewart, Seventh Edition: Section 2.4, pp. 113.)
Lecture 10 Limits (cont d) Onesided its (Relevant section from Stewart, Sevent Edition: Section 2.4, pp. 113.) As you may recall from your earlier course in Calculus, we may define onesided its, were
More informationDetermine the perimeter of a triangle using algebra Find the area of a triangle using the formula
Student Name: Date: Contact Person Name: Pone Number: Lesson 0 Perimeter, Area, and Similarity of Triangles Objectives Determine te perimeter of a triangle using algebra Find te area of a triangle using
More informationNote nine: Linear programming CSE 101. 1 Linear constraints and objective functions. 1.1 Introductory example. Copyright c Sanjoy Dasgupta 1
Copyrigt c Sanjoy Dasgupta Figure. (a) Te feasible region for a linear program wit two variables (see tet for details). (b) Contour lines of te objective function: for different values of (profit). Te
More informationArea of a Parallelogram
Area of a Parallelogram Focus on After tis lesson, you will be able to... φ develop te φ formula for te area of a parallelogram calculate te area of a parallelogram One of te sapes a marcing band can make
More informationSimilar interpretations can be made for total revenue and total profit functions.
EXERCISE 37 Tings to remember: 1. MARGINAL COST, REVENUE, AND PROFIT If is te number of units of a product produced in some time interval, ten: Total Cost C() Marginal Cost C'() Total Revenue R() Marginal
More informationMarkov Chains, part I
Markov Chains, part I December 8, 2010 1 Introduction A Markov Chain is a sequence of random variables X 0, X 1,, where each X i S, such that P(X i+1 = s i+1 X i = s i, X i 1 = s i 1,, X 0 = s 0 ) = P(X
More information1.6. Analyse Optimum Volume and Surface Area. Maximum Volume for a Given Surface Area. Example 1. Solution
1.6 Analyse Optimum Volume and Surface Area Estimation and oter informal metods of optimizing measures suc as surface area and volume often lead to reasonable solutions suc as te design of te tent in tis
More informationM(0) = 1 M(1) = 2 M(h) = M(h 1) + M(h 2) + 1 (h > 1)
Insertion and Deletion in VL Trees Submitted in Partial Fulfillment of te Requirements for Dr. Eric Kaltofen s 66621: nalysis of lgoritms by Robert McCloskey December 14, 1984 1 ackground ccording to Knut
More informationSection 3.3. Differentiation of Polynomials and Rational Functions. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section 3.3 Differentiation of Polynomials an Rational Functions In tis section we begin te task of iscovering rules for ifferentiating various classes of
More informationThe Derivative as a Function
Section 2.2 Te Derivative as a Function 200 Kiryl Tsiscanka Te Derivative as a Function DEFINITION: Te derivative of a function f at a number a, denoted by f (a), is if tis limit exists. f (a) f(a+) f(a)
More informationTHE DIFFUSION EQUATION
THE DIFFUSION EQUATION R. E. SHOWALTER 1. Heat Conduction in an Interval We shall describe the diffusion of heat energy through a long thin rod G with uniform cross section S. As before, we identify G
More informationAverage and Instantaneous Rates of Change: The Derivative
9.3 verage and Instantaneous Rates of Cange: Te Derivative 609 OBJECTIVES 9.3 To define and find average rates of cange To define te derivative as a rate of cange To use te definition of derivative to
More informationTRADING AWAY WIDE BRANDS FOR CHEAP BRANDS. Swati Dhingra London School of Economics and CEP. Online Appendix
TRADING AWAY WIDE BRANDS FOR CHEAP BRANDS Swati Dingra London Scool of Economics and CEP Online Appendix APPENDIX A. THEORETICAL & EMPIRICAL RESULTS A.1. CES and Logit Preferences: Invariance of Innovation
More information3.7 Nonautonomous linear systems of ODE. General theory
3.7 Nonautonomous linear systems of ODE. General theory Now I will study the ODE in the form ẋ = A(t)x + g(t), x(t) R k, A, g C(I), (3.1) where now the matrix A is time dependent and continuous on some
More informationA Multigrid Tutorial part two
A Multigrid Tutorial part two William L. Briggs Department of Matematics University of Colorado at Denver Van Emden Henson Center for Applied Scientific Computing Lawrence Livermore National Laboratory
More informationModule 2. The Science of Surface and Ground Water. Version 2 CE IIT, Kharagpur
Module Te Science of Surface and Ground Water Version CE IIT, Karagpur Lesson 6 Principles of Ground Water Flow Version CE IIT, Karagpur Instructional Objectives On completion of te lesson, te student
More informationResearch on the Antiperspective Correction Algorithm of QR Barcode
Researc on te Antiperspective Correction Algoritm of QR Barcode Jianua Li, YiWen Wang, YiJun Wang,Yi Cen, Guoceng Wang Key Laboratory of Electronic Tin Films and Integrated Devices University of Electronic
More informationDifferential Calculus: Differentiation (First Principles, Rules) and Sketching Graphs (Grade 12)
OpenStaxCNX moule: m39313 1 Differential Calculus: Differentiation (First Principles, Rules) an Sketcing Graps (Grae 12) Free Hig Scool Science Texts Project Tis work is prouce by OpenStaxCNX an license
More information1 The Collocation Method
CS410 Assignment 7 Due: 1/5/14 (Fri) at 6pm You must wor eiter on your own or wit one partner. You may discuss bacground issues and general solution strategies wit oters, but te solutions you submit must
More informationMATH 461: Fourier Series and Boundary Value Problems
MATH 461: Fourier Series and Boundary Value Problems Chapter III: Fourier Series Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2015 fasshauer@iit.edu MATH 461 Chapter
More informationEC201 Intermediate Macroeconomics. EC201 Intermediate Macroeconomics Problem set 8 Solution
EC201 Intermediate Macroeconomics EC201 Intermediate Macroeconomics Prolem set 8 Solution 1) Suppose tat te stock of mone in a given econom is given te sum of currenc and demand for current accounts tat
More informationThe finite element immersed boundary method: model, stability, and numerical results
Te finite element immersed boundary metod: model, stability, and numerical results Lucia Gastaldi Università di Brescia ttp://dm.ing.unibs.it/gastaldi/ INdAM Worksop, Cortona, September 18, 2006 Joint
More informationVOL. 6, NO. 9, SEPTEMBER 2011 ISSN ARPN Journal of Engineering and Applied Sciences
VOL. 6, NO. 9, SEPTEMBER 0 ISSN 896608 0060 Asian Researc Publising Network (ARPN). All rigts reserved. BIT ERROR RATE, PERFORMANCE ANALYSIS AND COMPARISION OF M x N EQUALIZER BASED MAXIMUM LIKELIHOOD
More informationChapter 10 Finite Difference Methods
Chapter 10 Finite Difference Methods Finite difference methods were used centuries ago, long before computers were available. As we have seen in Chap. 2, these methods arise quite naturally by going back
More informationCHAPTER 8: DIFFERENTIAL CALCULUS
CHAPTER 8: DIFFERENTIAL CALCULUS 1. Rules of Differentiation As we ave seen, calculating erivatives from first principles can be laborious an ifficult even for some relatively simple functions. It is clearly
More informationProblem 1 (10 pts) Find the radius of convergence and interval of convergence of the series
1 Problem 1 (10 pts) Find the radius of convergence and interval of convergence of the series a n n=1 n(x + 2) n 5 n 1. n(x + 2)n Solution: Do the ratio test for the absolute convergence. Let a n =. Then,
More informationf(x + h) f(x) h as representing the slope of a secant line. As h goes to 0, the slope of the secant line approaches the slope of the tangent line.
Derivative of f(z) Dr. E. Jacobs Te erivative of a function is efine as a limit: f (x) 0 f(x + ) f(x) We can visualize te expression f(x+) f(x) as representing te slope of a secant line. As goes to 0,
More informationCHAPTER 7. Di erentiation
CHAPTER 7 Di erentiation 1. Te Derivative at a Point Definition 7.1. Let f be a function defined on a neigborood of x 0. f is di erentiable at x 0, if te following it exists: f 0 fx 0 + ) fx 0 ) x 0 )=.
More informationSurface Areas of Prisms and Cylinders
12.2 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS G.10.B G.11.C Surface Areas of Prisms and Cylinders Essential Question How can you find te surface area of a prism or a cylinder? Recall tat te surface area of
More informationFinite Difference Approach to Option Pricing
Finite Difference Approach to Option Pricing February 998 CS5 Lab Note. Ordinary differential equation An ordinary differential equation, or ODE, is an equation of the form du = fut ( (), t) (.) dt where
More informationConservation of Energy 1 of 8
Conservation of Energy 1 of 8 Conservation of Energy Te important conclusions of tis capter are: If a system is isolated and tere is no friction (no nonconservative forces), ten KE + PE = constant (Our
More informationMATH 425, PRACTICE FINAL EXAM SOLUTIONS.
MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator
More informationWarm medium, T H T T H T L. s Cold medium, T L
Refrigeration Cycle Heat flows in direction of decreasing temperature, i.e., from igtemperature to low temperature regions. Te transfer of eat from a lowtemperature to igtemperature requires a refrigerator
More informationA FLOW NETWORK ANALYSIS OF A LIQUID COOLING SYSTEM THAT INCORPORATES MICROCHANNEL HEAT SINKS
A FLOW NETWORK ANALYSIS OF A LIQUID COOLING SYSTEM THAT INCORPORATES MICROCHANNEL HEAT SINKS Amir Radmer and Suas V. Patankar Innovative Researc, Inc. 3025 Harbor Lane Nort, Suite 300 Plymout, MN 55447
More informationOptimized Data Indexing Algorithms for OLAP Systems
Database Systems Journal vol. I, no. 2/200 7 Optimized Data Indexing Algoritms for OLAP Systems Lucian BORNAZ Faculty of Cybernetics, Statistics and Economic Informatics Academy of Economic Studies, Bucarest
More information2.1: The Derivative and the Tangent Line Problem
.1.1.1: Te Derivative and te Tangent Line Problem Wat is te deinition o a tangent line to a curve? To answer te diiculty in writing a clear deinition o a tangent line, we can deine it as te iting position
More informationSolving the heat equation, wave equation, Poisson equation using separation of variables and eigenfunctions
Solving the heat equation, wave equation, Poisson equation using separation of variables and eigenfunctions 1 Review: Interval in one space dimension Our domain G = (0,L) is an interval of length L. The
More informationNumerically integrating equations of motion
Numerically integrating equations of motion 1 Introduction to numerical ODE integration algorithms Many models of physical processes involve differential equations: the rate at which some thing varies
More informationExamination paper for Solutions to Matematikk 4M and 4N
Department of Mathematical Sciences Examination paper for Solutions to Matematikk 4M and 4N Academic contact during examination: Trygve K. Karper Phone: 99 63 9 5 Examination date:. mai 04 Examination
More informationPyramidization of Polygonal Prisms and Frustums
European International Journal of cience and ecnology IN: 0499 www.eijst.org.uk Pyramidization of Polygonal Prisms and Frustums Javad Hamadani Zade Department of Matematics Dalton tate College Dalton,
More informationThe modelling of business rules for dashboard reporting using mutual information
8 t World IMACS / MODSIM Congress, Cairns, Australia 37 July 2009 ttp://mssanz.org.au/modsim09 Te modelling of business rules for dasboard reporting using mutual information Gregory Calbert Command, Control,
More informationExam 2 Review. . You need to be able to interpret what you get to answer various questions.
Exam Review Exam covers 1.6,.1.3, 1.5, 4.14., and 5.15.3. You sould know ow to do all te omework problems from tese sections and you sould practice your understanding on several old exams in te exam
More informationMath Test Sections. The College Board: Expanding College Opportunity
Taking te SAT I: Reasoning Test Mat Test Sections Te materials in tese files are intended for individual use by students getting ready to take an SAT Program test; permission for any oter use must be sougt
More informationDiscovering Area Formulas of Quadrilaterals by Using Composite Figures
Activity: Format: Ojectives: Related 009 SOL(s): Materials: Time Required: Directions: Discovering Area Formulas of Quadrilaterals y Using Composite Figures Small group or Large Group Participants will
More informationSIXTY STUDY QUESTIONS TO THE COURSE NUMERISK BEHANDLING AV DIFFERENTIALEKVATIONER I
Lennart Edsberg, Nada, KTH Autumn 2008 SIXTY STUDY QUESTIONS TO THE COURSE NUMERISK BEHANDLING AV DIFFERENTIALEKVATIONER I Parameter values and functions occurring in the questions belowwill be exchanged
More information