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Numerical Methods and Software for Partial Differential

Equations

Lecturer: Dr Yvonne Fryer

Time: Mondays 10am-1pm

Location: QA210 / KW116 (Black)

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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

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Components of Course

Partial Differential Equations

Numerical Methods

Software

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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

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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

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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.

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Modelling in Industry: Automobiles

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Modelling in Industry: Aerospace

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Modelling in Industry: Electronics

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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.

Pdt

dP05.0

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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.

dt

df

y

u

x

u

Partial Differentials

Ordinary Differential

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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.

2

2

2

2

2

2

z

u

y

u

x

uK

t

u

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Partial Differential Equations• Example: Temperature on a

computer board.

02

2

2

2

k

Q

y

T

x

T

x

T=

T=

x

T

x

T

Heat Source Q

y

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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.

2

22

x

uu

yx

uu

t

uu xxxyt

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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:

y

x

2

2

2

22 .

yx

.

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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)

.))Grad((Div

.)(Div

)(Grad

uu

),(

),(),(

2

1

yxu

yxuyxu

),( yx

1u

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Classification of PDEs• The generalised 2nd 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 : b2 – 4ac > 0 – Parabolic if : b2 – 4ac = 0 – Elliptic if : b2 – 4ac < 0

02

22

2

2

gfuy

ue

x

ud

y

uc

yx

ub

x

ua

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Exercise• Classify the following PDEs

1)

2)

3)

4)

02

22

2

2

y

u

yx

u

x

u

0622

22

2

2

y

u

x

u

y

u

yx

u

x

u

0962

22

2

2

y

u

yx

u

x

u

yx

u

x

u

2

2

2

9

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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

02

2

2

2

y

u

x

u

02

2

2

2

gy

u

x

u

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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

02

2

2

2

k

Q

y

u

x

uHeat Source Q

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Further ExamplesPoisson’s equation can be used to model many different phenomena

02

2

2

2

gy

u

x

u

Flow in Porous Media

Current in extended bodies

Diffusion Torsion in a bar

constitutive law

g

Darcy HookeOhmFick

Du

V: Voltage

fluid supply

VDu cDu

D: conductivityD: permeability D: diffusion coeffs

rate of twistelectric chargeion supply

c: concentration : piezometric head

T

xyu

,

: Prandtl stress function

u

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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.

02

2

ax

ua

t

u

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Parabolic PDE - Example

• One-dimensional heat diffusion along a pipe. Pipe is heated from one end.

2

2

10x

u

t

u

Tim

eInitial Conditions

Steady state conditions

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Hyperbolic PDE - Example• A continuously-vibrating undamped Violin or

Guitar string.

• Example – wave equation

Tim

e

2

22

2

2

x

ua

t

u

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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.

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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):

cx

u

cux

u

)()0,( xfxu

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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

25o C

x

Insulated

100o C

(10,5)

50100),10(25),0(

1000)5,()0,(:(BCs)toSubject

50,100,0:Solve2

2

2

2

yyuyu

xxy

ux

y

u

yxy

u

x

u

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Boundary and Initial Conditions

02

2

2

2

y

u

x

uu(

10,y

) =

100

u(0,

y) =

25

0)0,(

xy

u

0)5,(

xy

u

MODEL RESULT

Insulated

Insulated

100o C

25o C

(0,0)

(10,5)

REAL WORLD

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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 (http://www.mathworks.com)

• Can solve PDEs

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Numerical Software

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Example• Simulate temperature between two rooms using

Laplace Equation• All external walls insulated

5 Meters3 Meters

Frid

ge (

-100

C)

Fire

Pla

ce (

40C

)

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Results from Example