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I.Popescu:Numerical methods 1/16/2012

1

Hydroinformatics yModule 4: Numerical Methods I

Lecture 1 and 2: Models, ODE, Explicit and implicit methods

I.Popescu

Remarks:1. Schedule2. Prerequisites

• Basic mathematics, • Fluid dynamics and fluid mechanics,• Mathematical representation of fluid flow equations

3 Marking

1/16/2012 I.Popescu: Numerical Methods 2

3.Marking• Assignments (15%)• Written examination (30%)


2

Course

Main aimintroduce numerical techniques which can be used on computersused on computersoverview of

WHAT it can be doneHOW it can be done

At the end of lectures you will:apply standard methods to simple flow problems and


apply standard methods to simple flow problems and determine the optimal values for time steps and grid sizeassess simulation results produced by different methodsevaluate the relevance of their useidentify and analyse the reason for failure of particular methods and, possibly, provide solutions

1-Why Numerical Methods?


3

Computational Fluid Dynamics

Computational Fluid Dynamics (CFD) is the science of predicting fluid flow, heat transfer mass transfer chemical reactionstransfer, mass transfer, chemical reactions, and related phenomena by solving the mathematical equations which govern these processes using a numerical process (that is, on a computer).


Solution of Equations Phenomena in nature -described in terms of properties that prevail at each point of space and time separately

Formulating PDE’s governing the phenomena

SOLUTIONS

HOW ??



4



Analytical

SOLUTIONS

Too complex equations?


Separationof

variables

Integral Solution(Green)

Integraltransforms(Fourier

& Laplace)



Analytical

SOLUTIONS


Separationof

variables



& Laplace)


5

What is a model?

Is this a model?


What is a model?

Is this a model?



6

What is a model?

Is this a model?

– Naomi Campbel


What is a model?

Is this a model?



7

What is a model?

Is this a model?


What is a model?

Is this a model?



8

What is a model?


What is a model?A simplification of reality, but with essential features -- an example:

RealityReality ModelModel



9

What is a model?

Different formsEngineering problem solving

– A model is a simplified, schematic representations of the real world, a representations that retains enough aspects of the original system to make it useful to the modeller.

In fluid dynamics– physical models (scale models);


– physical models (scale models);– mathematical models

– conceptual– simulation models

Error and uncertainty in modellingDefinitions of model:

"A system which will convert a given input (geometry, boundary conditions, force, etc.) into an output (flow

t l l t ) t b d i i ilrates, levels, pressures, etc.) to be used in civil engineering design and operation" (Novak)"A simplified representation of the natural system" (Resfgaard)

It is not reasonable to expect that a "simplified


representation" convert input into exact output. Errors are inevitable. The actual values of the output variables are uncertain.


10

What’s the difference between a modeling program, and a model?

Modelling program = softwareSoftware with an array of equations that simulate river flowsimulate river flow

Model = software + dataA modelling program + data specific to a certain river


What is a model?

Mathematical models

Q

QQ


H


11



Analytical Numericalt h i

SOLUTIONS


techniquesSeparation

of variables



& Laplace)

FDM FVM FEMBEM

Ph i l

DifferentialEquations

AnalyticalMethods

SimpleDifferentialEquations

Why numerical methods?

Physicalphenomenon

Equations+

BoundaryConditions

ApproximateMethods

FDMor

FVMFEM,

BEM


BEMGeneral differential equations are too complicated to be solved analyticallyFDM, FEM, FVM, BEM are approximate numerical approaches for solving differential equations


12

Why Numerical Methods?

Analytical solutions– derived only for few engineering problems – limited number of closed form solutions– useful to provide insight to the behaviour of an

engineering system

Numerical solutions


– fill the gap between theory and experiment


Easy input - preprocessor.Solves many types of problemsy yp pModular design

– fluids, dynamics, heat, etc.

Can run on PC’s now.Relatively low cost, even sometimes


cheaper


13


Models are no longer restricted to closed solution problemsRepresent accurately realistically complex system such as:

– non-linear systems– time dependent state variables

Sometimes physical experiments are


Sometimes physical experiments are impossible to be simulated

Simple Mathematical Models

( )var var , ,Dependent iable f independent iable parameters forcing function=

Dependent variableh h ll fl h b h– A characteristic that usually reflects the behavior or

state of the system

Independent variable– Are usually dimensions, such as time and space

Parameters


– Are reflective of system’s properties or compositions

Forcing functions– External influences acting on the system


14

Basic principle of Numerical Methods

DISCRETISATIONAn exact ‘continuous’ solution is represented by an approximate ‘discrete’ solution at a number of points.

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1


0 1 2 3 4 5 6 7-1

-0.8

0 1 2 3 4 5 6 7-1

-0.8

Exact solution Discretised approximation

Basic principle of Numerical Methods

DISCRETISATION of space and time

Continuous in time

Discrete in timeApproximation

of the derivative

t

( ) ( )


t4t3t2t1

Δt

t ( ) ( )2 1 2 1

2 1 2 1

U t U tU U Ut t t t t

−Δ −= =

Δ − −


15

Example - simple mathematical model

Newton Second LawFF ma or am

= =m

c

g

( ) ( ) ( ) ( )1 1

new value old value slope x ste

i i i i icv t v t g v t t tm

+ +

= +

⎡ ⎤= + − −⎢ ⎥⎣ ⎦↑ ↑ ↑ ↑

p size


/( ) 1 ct mgmv t ec

−⎡ ⎤= −⎣ ⎦Analytical solution -

Example-simple mathematical model

Newton Second Law- Falling objectM=68.1 kg

c=12 5 kg/sc=12.5 kg/s

t step= 2 secTime (s) Computed velocity (m/s)

Numerically Analytically0.0 0.000 0.0002.0 19.600 16.4044.0 32.005 27.769


6.0 39.856 35.6418.0 44.824 41.09510.0 47.969 44.87312.0 49.959 47.490

: 53.390 53.390


16

Example - simple mathematical model

Newton Second Law- Falling object60

0

10

20

30

40

50

Velo

city

M=68.1 kg

c=12.5 kg/s

t step= 2 sec


00 10 20 30 40 50

Time

numerically analitically

2- Differential equations

- Quick review -


17

Introduction and First Definitions1

Differential equations are equations that contain derivatives

In an algebraic equation, the unknown is a number. InIn an algebraic equation, the unknown is a number. In a differential equation, the unknown is a function.Most often the unknown function we seek, expresses the behavior of the state variable or variables as a function of time. This function it is called state equation

The order of the differential equation is the order

1/16/2012 Review of Mathematics 33

The order of the differential equation is the order of the derivative of the unknown function involved in the equation.

Introduction and First Definitions2

A linear differential equation of order n is a differential written in the following form:

)()()(...)()( 011

1

1 xfyxadxdyxa

dxydxa

dxydxa n

n

nn

n

n =++++ −

−

−

where an(x) is not the zero function.

Note: Some books may use the notation y’, ’’ ’’’ (4) f th d i ti


y’’, y’’’, y(4), …for the derivatives.


18

Linear Differential EquationsA linear equation obliges the unknown function y(x) to have some restrictions. Accepted operation on y are:Accepted operation on y are:

Differentiating yMultiplying y and its derivatives by a function of the variable x.Adding what you obtained in


Adding what you obtained in previous step and let it be equal to a function of x.

Solving differential equations (DE)A major field of study

Existence: Does a differential equation have a solution?Uniqueness: Does a differential equation have more than one solution? If yes, how can we find a solution which satisfies particular y , pconditions?

Differential equations require integration to be solvedIf the values of the unknown function y(x) and its derivatives at some point are known the solution to DE is called an initial value problem (in short IVP). If no initial conditions are given, the description of all solutions to the DE is called the general solution

Some kinds of D E ’s can be solved exactly; many not


Some kinds of D.E. s can be solved exactly; many not for those that cannot, there are various ways to approximate the solution


19

d

Single VariableWe know:

Differential Equations

)( yFdtdy

=

We want to know:)(tfy =


Several VariableWe know: We want to know:

Differential Equations

),...,,(1 qxyFdtdy

=

),...,,(2 qxyFdtdx

=

M M

)(2 tfx =

)(1 tfy =


),...,,( qxyFdtdq

n=

M M)(3 tfq =


20

Methods for solving DEs

Solving a differential equation can be done in three major ways:

l i ll- analytically, - qualitatively, - numerically.


Partial derivatives

When a function changes with more than one variable at once – for example, both time and space – we use partial derivativesspace we use partial derivativesNotation:

= means the rate of change in y with tty∂∂


= means the rate of change in y with xxy∂∂


21

Total time derivative

The total time derivative is used to describe the change in a variable in elation to a mo ing (as opposed torelation to a moving (as opposed to

stationary) objectExample:

xyu

ty

DtDy

∂∂

+∂∂

=


u is velocity

xtDt ∂∂

Time & space derivatives: reviewNotation:

= ordinary derivative: y changes only with tdy ordinary derivative: y changes only with t

= partial derivative: change in y with t at a fixed point

= total time derivative: change in y as oneDy

ty∂∂dt


total time derivative: change in y as one moves about (e.g., in a fluid)Dt


22

Classification of DE ( Clasess)dimension of unknown:

ordinary differential equation (ODE) – unknown is a function of one variable, e.g. y(t)partial differential equation (PDE) – unknown is a function of multiple variables, e.g. u(t,x,y)p , g ( , ,y)

number of equations:single differential equation, e.g. y’=ysystem of differential equations (coupled), e.g. y1’=y2, y2’=-g

ordernth order DE has nth derivative, and no higher, e.g. y’’=-g

linear & nonlinear:linear differential equation: all terms linear in unknown and its


qderivativese.g x’’+ax’+bx+c=0 , x’=t2x – linear;x’’=1/x – nonlinear

Classification of PDEs2 2 2

2 2

u u u u ua b c d e fu gx x y y x y

∂ ∂ ∂ ∂ ∂+ + = + + +

∂ ∂ ∂ ∂ ∂ ∂2 4b acΔ = −

elliptic PDE - b2- 4ac < 0 – smooth solutions

hyperbolic PDE - b2- 4ac > 0 – solutions with discontinuities

4b acΔ


parabolic PDE - b2- 4ac =0


23

General features of PDEs

Elliptic PDEs – describe processes that have already reached

steady state– time-independent (e.g. steady-state aquifer flow or

heat diffusion).

Hyperbolic PDEs – describe time-dependent, conservative physical

processes, such as convection, that are not evolving towards steady state (e.g. free surface flow, pipe


y ( g , p pflow).

Parabolic PDEs – describe time-dependent, dissipative physical

processes, such as diffusion, that are evolving towards steady state. (e.g. transient aquifer flow).

What do we need to solve DE?

Bou

ndar

y va

lues

Bou

ndar

y va

lues

Problem domain


a b

BB

Initial values


24

Boundary and initial conditions

Initial value problems (IVP)derivative with respect to time (either dU/dt, or d2U/dt2 etc ) in the equation it is necessaryor d2U/dt2, etc.) in the equation, it is necessary to know the initial value of U in order to compute its value in the future, within the model domain. initial condition of U must be given. this problems are also called initial value


this problems are also called initial value problems

Boundary and initial conditions

Boundary value problems (BVP)– Initial conditions are not always sufficient to ensure

the existence of a unique solutions. Conditions onthe existence of a unique solutions. Conditions on the boundary of the domain of the problem must be given as well. These conditions are called boundary conditions.

C CComes from the boundary

time t time t + Δ t


x x


25

Reminder - Basic formulas

What is a derivative?– Rate of change of a function

slope of the tangent line to the graph of a function– slope of the tangent line to the graph of a function– best linear approximation to a function

f

Real derivative

( ) ( )( ) limx o

f x x f xf xxΔ →

+ Δ −′ =Δ


Approx.derivatives

xx2x2x1 ( ) ( )( ) f x x f xf xx

+Δ −′ ≈Δ

Δx is small


Taylor series expantion2 3 4 5

( ) ( ) ( ) ( ) ( ) ( ) ( )2! 3! 4! 5!x x x xf x x f x xf x f x f x f x f xΔ Δ Δ Δ′ ′′ ′′′ ′′′′ ′′′′′±Δ = ±Δ + ± + ± +K( ) ( ) ( ) ( ) ( ) ( ) ( )2! 3! 4! 5!

f f f f f f f

2 3( ) ( ) 1 ( ) ( ) ( ) ( ) ( )2! 3!

f x x f x x xxf x f x f x f x O xx x

⎡ ⎤+ Δ − Δ Δ′ ′′ ′′′ ′= Δ + + + = + Δ⎢ ⎥Δ Δ ⎣ ⎦K

32( ) ( ) 1 2 ( ) 2 ( ) ( ) ( )f x x f x x xxf x f x f x O x

⎡ ⎤+ Δ − − Δ Δ′ ′′′ ′Δ + + + Δ⎢ ⎥

first order accurate in xΔ


2 ( ) 2 ( ) ( ) ( )2 2 3!

xf x f x f x O xx x

= Δ + + = + Δ⎢ ⎥Δ Δ ⎣ ⎦K

second order accurate in xΔ


26


( ) ( )f x f x x− − Δf(x )

fo rw a rd d i f fe r e n c eb a c k w a rd d i f fe r e n c e

c e n tra l d i f fe r e n c e

( ) ( )f x x f xx

+ Δ −Δ

( ) ( )2

f x x f x xx

+ Δ − −ΔΔ

xΔ


xx xx x xi-1 i i+ 1= x

2 xΔ

Why study Numerical Methods ?NO numerical method is completely trouble free in all situations!

– How should one choose/use an algorithm to get trouble free and accurate answers?trouble free and accurate answers?

No numerical method is error free!,– What level of error/accuracy do one take in the way

the problem is solved?

No numerical method is optimal for all types/forms of an equation!


yp qIMPORTANT : In order to solve a physical problem numerically, you must understand the behavior of the numerical methods used as well as the physics of the problem


27

Typical difficulties with NM

Instability



Inconsistency



28


Inaccurate


Errors

Errors due to computers: – Round-off error

Exponent underflow– Exponent underflow – Exponent overflow

Errors due to numerical methods – Truncation error – Propagated error

M f


Measures for errors– absolute error– relative error


29

Errors


Hydroinformaticianas a developer and user must understand how a numerical method performs for his/her given problemhis/her given problemunderstanding how various numerical algorithmsare derived,how the physics effects the numericshow the numerics effects the physics

h t th / t bilit ti


what are the accuracy/stability propertieswhat are the cost of a method for given level ofaccuracy (this varies substantially from method tomethod )


30

Properties of Numerical Methods

Convergence– numerical scheme solution is convergent if it comes

closer and closer to the analytical solution of the realcloser and closer to the analytical solution of the realODE/PDE when the time step decreases;

LaxTheorem:2conditions need it for convergence– Consistency; and– Stability



ConsistencyA scheme is consistent if it gives a correct

i ti f th ODE/PDE thapproximation of the ODE/PDE as thetime/space step is decreasedverified using Taylor Series expansion



31


StabilityA scheme is stable if any initially finite

t b ti i b d d tiperturbation remains bounded as time grows

Stability verificationMatrix methodFourier methodDomains of dependence


Domains of dependence

Notations

Un = U (tn)Uj = U (xj)Uj

n = U (tn, xj)t

n + 1

Δ t

Δ x


x

n

j -1 j j+ 1


32

4-Finite difference methods for ODE

The general Initial Value Problem1

Given

),( tufdu=

and initial condition given

If you know values at time n, find the

),(fdt


values at time n+1


33

The general IVP2

All initial value problems are solved by Integrating forward in tThere are two main types of integration yp gprocedure

– One-step methods– Multi-step methods

u u


t0 ti ti+1

t tt0

General classification

Numerical scheme– a particular discretization of a differential equation

Schemes:Schemes: – Explicit (Euler method) - the value of the variable at

time level n+1 can be computed directly (or explicitly) from the value at time level n.

– Implicit (Improved Euler) - involves values of the variable at time level n+1, there is an implicit


relationship between the derivative and the variable which has to be computed.

– Mixed (Crank-Nicholson ) - schemes in which the explicit and implicit schemes are combined.


34

Explicit schemes/Euler/One Step Method

u

Forward difference approximation

= ),( tufdtdu

u

Δt∑−

=

+

Δ+=

⋅Δ+≈

1

0

1

),(

),(

N

n

nninitialfinal

nnnn

tuftuu

tuftuu

dt


This is the same as saying:new value = old value + (slope) x (step size)t n t n+1

These schemes are also called one step methods or “marching in time”, because new value is calculated as one step forward from the old value.

Implicit schemesBackward difference approximation

U ( , )dU f U t=U

Δt( )1 1 1,n n n nU f U t t U+ + +− Δ =

( , )f U tdt

Iteration need it, ussualy


t n t n+1


35

Mixed schemesAveraging implicit and explicit schemes

U

Use this “average” slopeto predict Un+1


( ) ( )1 11

, ,

2

n n nnn n

f U t f U tU U t

+ ++

+= + Δ

{t n t n+1

Numerical schemes for ODE - Example

Ground water reservoir

Q

h

dh hdt

α= −


toh h e α−=

-exact solution


36

Multi step methods: Midpoint MethodsA better estimate of the function U would come from evaluating the derivative at the midpoint of the Δt interval.

U

Δuu n+1/2=?

The problem: we know t at the midpoint but we don’t know the u(t) at the midpoint (yet). The solution is to use Euler’s method to estimate Δu and then re-estimate Δu using the derivatives evaluated halfway t n t n+1/2 t n+1

Δt

u

t


along the line segment encompassing the original Δu.This method is called the second order Runge Kutta method, or the midpoint method.

Mixed schemes - Midpoint Methods

tkakauu nn Δ++=+ )( 22111

Where for two point schemes

U(analytical)

U(Midpoint)

U(t)

),(1nn tufk =

),( 11112 tpttkqufk nn ΔΔ= ++

;21

211

,aa

tscoefficienweightedaa−=−

Where for two point schemesU(Euler)

tn ttn+1


21

12 =pa

21

112 =qa

;21 1 aa


37

ExerciseApply the Euler and Mid-point methods to the following problem (with known solutions!):

fdu )(

utufutufttufttuf

fortuf

dtdu

−====

=

),(.4),(.3),(.2),(.1

),(

2


In each case, compare the errors in the two schemes after taking the same number (10, say) of identical timeteps (start with ). Is the mid-point method noticeably superior to Euler?

Mixed schemes - Multi Step Methods

The midpoint method can be extended by considering other intermediate estimates. The most frequently used variation is the fourth-order q yRunge-Kutta method which considers one estimate at the initial point, two estimates at the midpoint, and one estimate at a trial endpoint.This is popular for two reasons

It is easy to program


It is stable

HoweverIt requires more computing time


38

Runge-Kutta Method - 44thth--orderorder

f2uu

f1

f3

f4

f

u


ti ti + h/2 ti + h

( )4321 2261 fffff +++=

t

Rungge Kutta - 4 point scheme

tkakakakauu nn Δ++++=+ )( 443322111

WhereWhere

)()2/,2/()2/,2/(

),(

),(),(

23

12

1

kfktttkufktttkufk

tufk

tuatderivativethetuf

ΔΔΔ+Δ+=Δ+Δ+=

=

−U(analytical)

U(t)

2

3

4


),( 34 tttkufk Δ+Δ+=tn

ttn+1

12


39

Runge-Kutta 4th order - solution

(tn + k1Δt/2, un+ k1Δt/2)

Δt/2

slope = k1

t

u

)(

),(),(

tfk

tuatderivativethetuf −(tn ,un )

(k1Δt/2)


),()2/,2/()2/,2/(

),(

34

23

12

1

tttkufktttkufktttkufk

tufk

Δ+Δ+=Δ+Δ+=Δ+Δ+=

=

point to

Runge-Kutta versus Euler-example

2007.06 uudtdu

−=

Solve the equation below using both Euler and Runge Kutta methods

point to estimate

Problem: estimate the slope to

calculate Δu

Δuu


Δ t = 0.5


40

u0 = 10 analytical u

Euler solution

u

k1 = du/dt at u0

k1 = 6*10-.007*(10)2

Δu = k1*Δt

uest= u0 + Δu

Δu

estimated u


Δ t = 0.5

Step 1: Evaluate slope at current value of state variable.

u0 = 10

k1 = du/dt at u0

k1 = 6*10-.007*(10)2

k1 = 59.3k1=slope 1

u



41

Step 2: Calculate u1at t +Δt/2 using k1.

Evaluate slope at u1.

u1 = u0 + k1* Δt /2

u = 24 82u1 = 24.82

k2 = du/dt at u1

k2 = 6*24.8-.007*(24.8)2

k2 = 144.63k2=slope 2

u1

u


Δ t = 0.5/2

1

Step 3: Calculate u2 at t +Δt/2 using k2.


u2 = u0 + k2* Δt /2

u2 = 46.2 k3 = slope 3

k3 = du/dt at u2

k3 = 6*46.2-.007*(46.2)2

k3 = 263.0

u2

u


Δ t = 0.5/2


42

Step 4: Calculate u3 at t +Δt using k3.


u3 = u0 + k3* Δt

u3 =141.0 k4 = slope 4u3

k4 = du/dt at u2

k4 = 6*141.0-.007*(141.0)2

k4 = 706.9

u2

u


Δ t = 0.5

Step 5: Calculate weighted slope.

Use weighted slope to estimate u at t +Δt

1 2 3 41 ( 2 2 )6

k k k k+ + +weighted slope =

)22(14321

1 kkkktuu nn +++Δ+=+

true value

estimated valueweighted slope

)22(6 4321 kkkktuu +++Δ+

u


Δ t = 0.5


43

4th order Runge-Kutta offers substantial improvement over Eulers.Both techniques provide estimates, not “true” values.The accuracy of the estimate depends on the size of the step used in the algorithm.

Conclusions

Runge-Kutta

Analytical

Euleru



Ground water reservoir

Q

h

dh hdt

α= −


toh h e α−=

-exact solution


44

Numerical schemes for ODE - ExampleSolution methods :1. Explicit(Euler)

The right hand side is determined at current time (hn)

1 hh nn+

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

nh1+nh

)(th

)1(

1

1

thh

ht

hh

nn

nnn

Δ−=

−=Δ−

+

+

α

α


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50 t

)1(1

th

hn

n

Δ−=+

α

The right hand side is determined at future time (hn+1)

Numerical schemes for ODE - ExampleSolution methods :2. Implicit

11

++

−=Δ− n

nn

ht

hh α

11un+0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

nh1+nh

)(th


)1(

1tu

un Δ+=

α0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0 t


45


toh h e α−=

h

dh hdt

α= −1 (1 )n nh t hα+ = − Δ

Q


1

1

nn hh

tα+ =

+ Δ

The BIG question

≈≠=1/16/2012 I.Popescu: Numerical Methods 90


46

The BIG question

If equations are approximation then itshould:

behave in the same way as the function to which it• behave in the same way as the function to which itis an approximation, and

• become a better approximation as the time-step isreduced.


0.9

1Comparison of different solution methods

exactexplicit

Explicit vs Implicit

1nh +

)exp()exp

)1(exp 1

t-αtαn(

t)n(-αh

hn

n

Δ=Δ−Δ+

=+

Exact solution

0.3

0.4

0.5

0.6

0.7

0.8

pimplicit nh


0 0.5 1 1.5 2 2.5 30

0.1

0.2

Exact solution lies between explicit

and implicit solutions

tΔα


47

0.9

1Comparison of different solution methods

exact

1

n

n

hh +

Three different solution methods -comparison

0.3

0.4

0.5

0.6

0.7

0.8

exactexplicitimplicitcombined (Crank-Nicolson)

Crank-Nicolson is closest to

exact solution


0 0.5 1 1.5 2 2.5 30

0.1

0.2

tΔα

Stability, Consistency, Convergence

Lax Theorem:

Consistency + stability ⇔ convergence

This is why you can trust your models !This is why you can trust your models !

Consistency : The scheme approximates the ODE more correctlyas the discretisation is refined.

Stability: Any initial perturbation remains bounded.Convergence: The numerical solution becomes closer to the real

one as the discretisation is refined



48

The BIG question

If equations are approximation then itshould:

behave in the same way as the function to which it• behave in the same way as the function to which itis an approximation, and

• become a better approximation as the time-step isreduced.



Convergence - Explicit method

tn ehh α−→ 0ehh →

ththhthh

)1()1()1(

2012

01

Δ−=Δ−=Δ−=

ααα

K


non thh )1( Δ−= α

on

n hnth ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−+=

α1


49


Convergence - Explicit method (cont)

⎥⎤

⎢⎡

⎟⎞

⎜⎛

⎟⎞

⎜⎛ ⋅⎟

⎞⎜⎛⎟⎞

⎜⎛ ⋅ 2

210 ntnt αα

⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−+= −− K

11

111 210 n

ntn

nthh nnnn αα

where

!)!(!

rrnn

rn

−=⎟

⎠⎞⎜

⎝⎛

r⎞⎛


!)!( rrn⎠⎝

rnrnnnrrn

n→−−=

−))...(1(

!)!(! !r

nrn r

=⎟⎠⎞⎜

⎝⎛

Stability, Consistency, Convergence Convergence - Explicit method (cont)

nn ntnthh αα⎥⎤

⎢⎡

+⎟⎞

⎜⎛ ⋅

+⎟⎞

⎜⎛ ⋅

+→22

0 1

Back to the equation

tn ehtth

nnhh

ααα −→⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟

⎠⎞

⎜⎝⎛ ⋅−+⎟

⎠⎞

⎜⎝⎛ ⋅−+→

⎥⎥⎦⎢

⎢⎣

+⎟⎠

⎜⎝−+⎟

⎠⎜⎝−+→

02

0

!21

111

!2!11

K

K


⎥⎦⎢⎣ ⎠⎝⎠⎝ !211

Taylor series for te ⋅−α


50


1

1nh +

≤

Stability condition

111

≤≤+nh

Stability - Explicit schemes

1nh≤

αα 20

111 =Δ⇒⎨⎧ ≤Δ−

⇒≤Δ−≤− tt

t

11 ≤≤− nh


ααα

2111 maxΔ⇒

⎩⎨ ≤Δ

⇒≤Δ≤ tt

t


Example

1

2210

max

=

==Δ⇒=

hand

tTakeα

α

) d

dt

dt=0.125

a)0<dt<1


dt=0.5

dt=0.25

dt 0 5


51

b)1<=dt<2


dt=1.25

d 1


=>Stable But Oscillatory Approximations

dt=1.5dt=1.5

c)2<dtStability, Consistency, Convergence

dt=4.5

dt=3


=>Unstable (i.e. Bad) Approximations

dt=2.5


52



H

0 1 1 1 or 0 t tαα

≤ − Δ ≤ ≤ Δ ≤

Δt


t n t n+1



H

ΔtΔt0 1/α 2/α

oscillatory unstablestable


t n t n+1


53


Consistency - explicit schemes

( )2

2dh t d hh O tΔ+ Δ( )2

22h O t

dt dtα+ = − − Δ

Truncation error - first order accurate



Implicit schemes– convergence - FOR YOU TO TEST

stability unconditionally stable– stability - unconditionally stable

Mixed schemes– unconditionally stable



54

Amplification factor

U n+1/U n is called the numerical amplification factor


Amplitude portrait

Method EulerImproved Euler

Region of instability

1

0

1

0 1 2 3 4AN



-2

-1

a Δt



55

4-Finite difference methods for ODE

Quick summary

What you should remember

ODEs can be solved numerically by replacing the d/dt operator by a difference of the considered function at two time levels.function at two time levels.When the derivative is estimated using the values of the function at time step level n, the scheme is said to be explicit. When the derivative is estimated using the values at time n+1, the scheme is said to be implicit.


pThe discretized ODE is consistent to the real one if it comes closer to the real ODE as dt (or dx) comes closer to zero.


56

What you should rememberThe difference between the discretised ODE and the real one is called the truncation error. It is obtained by substituting Taylor series expansions into the discretisation.The solution is said to be stable if initial perturbations in the numerical solution remain bounded.


The stability of the solution is investigated by calculating the value of the amplification factor. If |AN| is smaller than unity, the solution is stable.

What you should remember

In general, explicit schemes are not stable for all values of ∆t (or ∆x). I li i h l blImplicit schemes are always stable.The numerical solution is said to converge to the real one if it comes closer to it as ∆t (or ∆x) comes closer to 0.Consistency and stability of a scheme are


Consistency and stability of a scheme are necessary and sufficient conditions to the convergence of the solution

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