1 Numerical Methods and Software for Partial Differential Equations Lecturer:Dr Yvonne Fryer Time:Mondays 10am-1pm Location:QA210 / KW116 (Black)

2 Aim of Lecture During this lecture we will discuss: –Course outline –Partial differential equations Elliptic Parabolic Hyperbolic –Numerical Methods Finite Difference Finite Volume Finite Element –Software : Excel, PDE-Toolbox

3 Components of Course Partial Differential Equations Numerical Methods Software

4 Course Outline 10 Week course + 2 other weeks 1 Coursework 1 Exam Assessment 50:50 (Exam:CW) Approximate times (lectures lengths vary): –Lecture 10am – ~11:30am Queen Anne A210 –Class work: ~11:30am – 12noon Queen Anne A210 –Lab work: 12noon – 1pm King William W116 (Black) –Homework: finish off Class & Lab work

5 Textbooks Morton & Mayers, Numerical Solution of Partial Differential Equations (Cambridge University Press, 2005) is recommended if you want to do some background reading Burden & Faires, Numerical Analysis (Brooks Cole, 2001) has a good chapter on Numerical Methods for PDEs However you do not require either of these books – the lecture notes should suffice

6 Partial Differential Equations PDEs are used extensively to represent real world phenomena and processes. –Heat transfer in nuclear reactors. –Airflow around an aircraft. –Structural dynamics of a bridge. –Movement of money in financial markets. –Etc. Modelling & simulation of such processes requires solution of these PDEs.

7 Modelling in Industry: Automobiles

8 Modelling in Industry: Aerospace

9 Modelling in Industry: Electronics

10 Partial Differential Equations Ordinary Differential Equation Describes rate of change in population (P) over time (t). Only one independent variable. Partial Differential Equations have more than one independent variable.

11 Partial Differential Equations Note the difference in the differential terms. The dependent variable f = f(t) is dependent on only time (t) for the ODE. For the PDE the dependent variable u is dependent on both directions x and y. Partial Differentials Ordinary Differential

12 Introduction Partial Differential Equations (PDEs) can be used to represent, mathematically, a large amount of real-world phenomena. For example the heat conduction across the earth: Where u(t,x,y,z) is temperature, K, , and  are material properties and x, y, z and t are space locations and time.

13 Partial Differential Equations Example: Temperature on a computer board. x T=  Heat Source Q y

14 Terminology For simplicity, we will deal only with only two independent variables: –two space variables: x and y, or –one space variable and one time variable denoted by x and t respectively. The unknown function is denoted by u and its partial derivatives: Mostly we shall use the longer form. However, watch out for the short form in past exam papers.

15 Terminology In two dimensions the gradient (Grad) operator is given by the vector: Divergence (Div) is given by the dot product The Laplacian is given by:

16 Classroom discussion Consider the scalar function and the vector function i.e. is the component of u in the x direction Calculate the following and state what the result represents (e.g. vector, scalar)

17 Classification of PDEs The generalised 2 nd order linear PDE can be written: where a, b, c, d, e, f, g are constants (some may be 0). A PDE in this form is said to be: –Hyperbolic if : b 2 – 4ac > 0 –Parabolic if: b 2 – 4ac = 0 –Elliptic if : b 2 – 4ac < 0

18 Exercise Classify the following PDEs 1) 2) 3) 4)

19 Elliptic PDEs Elliptic PDEs represent phenomena that have already reached a steady state and are, hence, time independent. Two classic Elliptic Equations are: –Laplace Equation –Poisson's Equation u(x,y) is independent variable and g is a constant

20 Elliptic PDE - Example Temperature, u(x,y) profile around two computer chips on a printed circuit board. Q is the power source and K is the thermal conductivity Heat Source Q

21 Further Examples Poisson’s equation can be used to model many different phenomena Flow in Porous Media Current in extended bodies DiffusionTorsion in a bar constitutive law g Darcy Hooke Ohm Fick V: Voltage fluid supply D: conductivityD: permeabilityD: diffusion coeffs rate of twistelectric chargeion supply c: concentration : piezometric head : Prandtl stress function

22 Parabolic PDEs Parabolic PDEs describe time dependent phenomena, such as conduction of heat, that are evolving towards steady state. Classical parabolic equation is the one dimensional heat or diffusion equation.

23 Parabolic PDE - Example One-dimensional heat diffusion along a pipe. Pipe is heated from one end. Time Initial Conditions Steady state conditions

24 Hyperbolic PDE - Example A continuously-vibrating undamped Violin or Guitar string. Example – wave equation Time

25 Boundary and Initial Conditions For a PDE based mathematical model of a physical system to have a solution then we must have: –The PDE –The physical domain of interest –The boundary and initial conditions. Elliptic problems require boundary conditions Parabolic & Hyperbolic equations require both initial and boundary conditions.

26 Boundary and Initial Conditions Two types of boundary condition may be given: –Dirichlet : u(x,y) = c –Neumann : (and mixed : ) where c is a constant. Initial condition for u(x,t):

27 Boundary and Initial Conditions For example consider the temperature, u(x,y), across the following plate. Mathematical model to represent temperature u(x,y) is: Insulated (0,0) y 25 o C x Insulated 100 o C (10,5)

28 Boundary and Initial Conditions u(10,y) =100 u(0,y) =25 MODEL RESULT Insulated 100 o C 25 o C (0,0) (10,5) REAL WORLD

29 Numerical Methods Usually we cannot solve the PDEs by analytical means. In this case numerical methods are used. Such methods are: –Finite Differences –Finite Volumes –Finite Elements These methods discretise the governing equations at discrete points in the domain. These discretised equations are then solved using computers.

Microsoft Excel (standard spreadsheet software) MATLAB ( Can solve PDEs 30 Numerical Software

31 Example Simulate temperature between two rooms using Laplace Equation All external walls insulated 5 Meters 3 Meters Fridge (-100C) Fire Place (40C)

32 Results from Example