cfd numerical error

8132019 CFD numerical Error

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1

AE3213 Computational Fluid Dynamics

Numerical Error and Stability1

Numerical Error and Stability

bull Contents

Define numerical error and understand how numerical schemes aresusceptible to error growth (instability)

bull Illustrate how different numerical systems display different instabilitycharacteristics

bull Describe the popular von Neumann technique for the analysis ofnumerical stability with examples

bull Discuss stability-imposed constraints and their mitigation

Analyse discretization errors and their impact upon the solution

bull Concept of modified PDE

bull Numerical dissipation and dispersion



bull Differential Equation

bull This equation describes simple one-dimensional problems ofconvection and diffusion

eg propagation of a paint spill in the Thames

bull Explicit Centred-Difference Equation is

bull We can (and will) solve this difference equation to obtain thesolution u at spatial point j and time step k

We have to choose the mesh spacing ∆ x and ∆t

Example ndash Linear Burgers Equation

2

2

2

x

u

x

uc

t

u

part

part=

part

part+

part

part micro

( )( )

( )k

j

k

j

k

j

k

j

k

j

k

j

k

j uuu x

t uu

x

t cuu 11211

12

2 minus+minus+

++minus

∆

∆+minus

∆

∆minus=

micro



2




bull We use the example of a paint spill in the Thames

bull We are interested in the concentration of the paint spill as theriver carries it downstream

bull We want to look at the paint concentration profile u at different

points along the river rather than at different times

bull We want to see how the mean flow rate c of the Thames affects

the concentration profiles

For example in the different seasons of the year

bull We want to use the same amount of compute time in each case

The lower the mean flow rate c the larger the time step ∆t

3



bull Numerical Solution for case where velocity c = 25 ∆t = 00002


4

983085983088983086983090983088

983088983086983088983088

983088983086983090983088

983088983086983092983088

983088983086983094983088

983088983086983096983088

983089983086983088983088

983089983086983090983088

983088983086983088983088 983088983086983093983088 983089983086983088983088 983089983086983093983088 983090983086983088983088

983125

983128

983116983145983150983141983137983154 983106983157983154983143983141983154983155 983109983153983157983137983156983145983151983150

983125 983137983156 983155983156983141983152 983088

983125 983137983156 983155983156983141983152 983093983088

983125 983137983156 983155983156983141983152 983089983088983088

983125 983137983156 983155983156983141983152 983089983093983088

983125 983137983156 983155983156983141983152 983090983088983088



3





5

983085983088983086983090983088

983088983086983088983088

983088983086983090983088

983088983086983092983088

983088983086983094983088

983088983086983096983088

983089983086983088983088

983089983086983090983088

983088983086983088983088 983088983086983093983088 983089983086983088983088 983089983086983093983088 983090983086983088983088

983125

983128

983116983145983150983141983137983154 983106983157983154983143983141983154983155 983109983153983157983137983156983145983151983150

983125 983137983156 983155983156983141983152 983088

983125 983137983156 983155983156983141983152 983093983088

983125 983137983156 983155983156983141983152 983089983088983088

983125 983137983156 983155983156983141983152 983089983093983088

983125 983137983156 983155983156983141983152 983090983088983088

983085983088983086983090983088

983088983086983088983088

983088983086983090983088

983088983086983092983088

983088983086983094983088

983088983086983096983088

983089983086983088983088

983089983086983090983088

983088983086983088983088 983088983086983093983088 983089983086983088983088 983089983086983093983088 983090983086983088983088

983125

983128

983116983145983150983141983137983154 983106983157983154983143983141983154983155 983109983153983157983137983156983145983151983150

983125 983137983156 983155983156983141983152 983088

983125 983137983156 983155983156983141983152 983093983088

983125 983137983156 983155983156983141983152 983089983088983088

983125 983137983156 983155983156983141983152 983089983093983088

983125 983137983156 983155983156983141983152 983090983088983088



983085983088983086983090983088

983088983086983088983088

983088983086983090983088

983088983086983092983088

983088983086983094983088

983088983086983096983088

983089983086983088983088

983089983086983090983088

983088983086983088983088 983088983086983093983088 983089983086983088983088 983089983086983093983088 983090983086983088983088

983125

983128

983116983145983150983141983137983154 983106983157983154983143983141983154983155 983109983153983157983137983156983145983151983150

983125 983137983156 983155983156983141983152 983088

983125 983137983156 983155983156983141983152 983093983088

983125 983137983156 983155983156983141983152 983089983088983088

983125 983137983156 983155983156983141983152 983089983093983088

983125 983137983156 983155983156983141983152 983090983088983088



6



4




983085983089983086983093983088

983085983089983086983088983088

983085983088983086983093983088

983088983086983088983088

983088983086983093983088

983089983086983088983088

983089983086983093983088

983090983086983088983088

983090983086983093983088

983088983086983088983088 983088983086983093983088 983089983086983088983088 983089983086983093983088 983090983086983088983088

983125

983128

983116983145983150983141983137983154 983106983157983154983143983141983154983155 983109983153983157983137983156983145983151983150

983125 983137983156 983155983156983141983152 983088

983125 983137983156 983155983156983141983152 983093983088

983125 983137983156 983155983156983141983152 983089983088983088

983125 983137983156 983155983156983141983152 983089983093983088

983125 983137983156 983155983156983141983152 983090983088983088


7




(again)


8

983085983088983086983090983088

983088983086983088983088

983088983086983090983088

983088983086983092983088

983088983086983094983088

983088983086983096983088

983089983086983088983088

983089983086983090983088

983088983086983088983088 983088983086983093983088 983089983086983088983088 983089983086983093983088 983090983086983088983088

983125

983128

983116983145983150983141983137983154 983106983157983154983143983141983154983155 983109983153983157983137983156983145983151983150

983125 983137983156 983155983156983141983152 983088

983125 983137983156 983155983156983141983152 983093983088

983125 983137983156 983155983156983141983152 983089983088983088

983125 983137983156 983155983156983141983152 983089983093983088

983125 983137983156 983155983156983141983152 983090983088983088



5




983085983088983086983094983088

983085983088983086983092983088

983085983088983086983090983088

983088983086983088983088

983088983086983090983088

983088983086983092983088

983088983086983094983088

983088983086983096983088

983089983086983088983088

983089983086983090983088

983089983086983092983088

983088983086983088983088 983088983086983093983088 983089983086983088983088 983089983086983093983088 983090983086983088983088

983125

983128

983116983145983150983141983137983154 983106983157983154983143983141983154983155 983109983153983157983137983156983145983151983150

983125 983137983156 983155983156983141983152 983088

983125 983137983156 983155983156983141983152 983093983088

983125 983137983156 983155983156983141983152 983089983088983088

983125 983137983156 983155983156983141983152 983089983093983088

983125 983137983156 983155983156983141983152 983090983088983088


9




bull We see some strange results for apparently sensiblecombinations of mean flow rate c and time step ∆t

What is going on

10



6



Numerical Errors

bull Numerical solutions of partial differential equations contain twotypes of errors

bull Discretization error is the summation of truncation errorsassociated with the finite differences

and similar errors in the numerical boundary conditions

bull Round-off error is due to the finite accuracy (associated with theword length) of computers

This error can in certain circumstances become unstable

We will study the basic ideas of stability using simple one-dimensional equations first qualitatively then quantitatively



Round-off error and instability

12



7



Round-off error (1) computed solution

bull A numerical solution from a real computer with finite accuracy

13

-02

-01

0

01

02

0 02 04 06 08 1

x

U N

Numerical solution produced by a real computer with finite accuracy



Round-off error (2) exact solution

bull A numerical solution from a real computer with finite accuracyincludes the exact solution of the difference equation and

14

Exact solution of difference equation

-02

-01

0

01

02

0 02 04 06 08 1

x

U D



8



Round-off error (3) computed minus exact

bull A numerical solution from a real computer with finite accuracyincludes the exact solution of the difference equation and the

round-off error

15

Round-off error εεεε = UN - UD -0002

-0001

0

0001

0002

0 02 04 06 08 1

x

ε ε



Error and Instability

bull We are interested in whether the round-off error grows or decaysas the solution is updated

bull We can examine the stability of the error by considering the effectof the numerical algorithm on the error profilehellip

Different algorithms tend to promote different types of instability

Wersquoll look in turn at dynamic and static instability

These are associated with diffusion and convection respectively

16



9



Dynamic instability (1)

bull Consider the one-dimensional diffusion equation

bull A numerical scheme to solve the above equation might be

(forward-time centred-space uniform grid)

17

2

2

x

u

t

u

part

part=

part

part micro

( ) ( )k

j

k

j

k

j

k

j

k

juuu

xt

uu112

1

2minus+

+

+minus∆

=∆

minus micro




bull We now consider the error involved in solving

bull We substitute where

u is the computed solution

D is the exact solution of the difference equation

ε is the error

to obtain

18

( ) ( )k

j

k

j

k

j

k

j

k

juuu

xt

uu112

1

2minus+

+

+minus∆

=∆

minus micro

k

j

k

j

k

j Du ε +=

( ) ( )

( ) ( )k

j

k

j

k

j

k

j

k

j

k

j

k

j

k

j

k

j

k

j

x D D D

xt t

D D112112

11

22minus+minus+

++

+minus∆

++minus∆

=∆

minus+

∆

minusε ε ε

micro micro ε ε



10




bull By definition the exact solution D satisfies the difference

equation and can be eliminated

bull The remaining terms define the development of the error

Has same form as original equation but is there another solution

bull We can write where

19

ε ε ε j

k

j

k

j

k += +

1∆




bull Correction tends to bring back towards zero

20

Error at time k

-0002

-0001

0

0001

0002

0 02 04 06 08 1

x

ε εε ε

j minus 1 + 1

( ) ( )k

j

k

j

k

j

k

j x

t 112

2minus+

+minus∆

∆=∆ ε ε ε micro

ε



11




bull However if is too large the correction will overshoot and

may be greater than

bull This is dynamic instability and is related to the time step size

21

( )

micro ∆

∆

t

x2

k

jε 2



Static instability (1)

bull Now consider the one-dimensional advection equation

bull Again using forward differences in time and central differences in

space on an equi-spaced grid the above can be approximated to

bull The corresponding error equation can be written

22

x

uc

t

u

part

partminus=

part

part

( )k

j

k

j

k

j

k

juu

x

c

t

uu11

1

2 minus+

+

minus∆

minus=∆

minus



12




bull Correction tends to move further way from zero

23

Error at time k

-0002

-0001

0

0001

0002

0 02 04 06 08 1

x

ε εε ε

j minus 1 + 1

( )k

j

k

j

k

j x

t c11

2 minus+

minus∆

∆minus=∆ ε ε ε




bull will therefore increase irrespective of the time step

bull This is static instability and can only be overcome by changing thedifference scheme used to solve the equation

24

k

jε



13



Qualitative view of numerical stability

bull Round-off errors may or may not grow with time into ldquowigglesrdquo inthe solution

In extreme cases the solution may ldquoblow uprdquo at the first iteration

bull We have used greatly simplified examples

Bad news is that it is not possible to analyse qualitatively the stabilityof a numerical approximation of the full Navier Stokes equations

Good news is that different instability types are closely related to theindividual terms in the equations representing different physicalprocesses

25



Quantitative view of numerical stability

bull Nor is it possible to analyse quantitatively the stability of anumerical approximation of the full Navier Stokes equations

bull However the stability of simpler linear models of theseequations can be analysed

We can define numerical stability boundaries for mesh spacing timestep size etc

bull Can also predict the form which the instability will take

bull The results of these studies can provide guidance for the stability

limits of the more complex non-linear Navier Stokes equations

26



14



Quantitative stability analysisthe von Neumann method

27



Quantitative view of numerical stability (1)

bull The von Neumann stability method can be used to quantify thestability of linear difference equations

bull The essence of the method is to assume that the round-off error ε can be given by a Fourier series in space and time

where

N = number of mesh intervals L = domain length

a = temporal growth rate k m= wavenumber 2π m L

28

( ) sum=

+=

2

1

N

m

xik at met xε

L



15




bull Because the problem is linear we can consider any one term ofthe series ie forget the summation Σ

But we do need to capture the lsquoworst casersquo scenario

bull In discrete notation we write

NB donrsquot confuse time step k with spatial wavenumber k m

bull An algorithm is stable if ie for all k m

We now apply this method to the two equations we have just considered qualitatively

29

k

j

k

j ε ε le+1

11

le=+

k

j

k j

Gε

ε

( ) xik at met x +=ε

jmk xik at k

j e +

=ε



vN Analysis of the Diffusion Eqn (1)

bull We substitute into

remembering that

bull After some algebra we obtain

bull Noting that and we can write

30

2

11

12

xt

k

j

k

j

k

j

k

j

k

j

∆

+minus=

∆

minusminus+

+ ε ε ε micro

ε ε jmk

xik at k

j e +

=ε

2

2

x

u

t

u

part

part=

part

part micro

( ) ( ) ( ) x xik at k

j

x xik at k

j

xik t t ak

j jm

k jm

k jm

k

eee ∆minus+

minus

∆++

+

+∆++=== 11

1ε ε ε



16




bull For stability we therefore require for all k m

A) Consider for all k m

This is satisfied if (always true)

B) Consider for all k m

This is satisfied if (true if )

31

( )1

2sin41

2

2 le

∆

∆

∆minus

xk

x

t m micro

( )1

2sin41

2

2 le

∆

∆

∆minus

xk

x

t m micro

( ) 12sin41

2

2 minusge

∆

∆

∆minus

xk

x

t m micro

2

2

x

u

t

u

part

part=

part

part micro



vN Analysis of the Advection Eqn (1)


(note centred-differences in x) remembering that


bull Noting that we can write

32

jmk xik at k

j e +

=ε x

ct

k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus+

+

2

11

1 ε ε ε ε

x

uc

t

u

part

partminus=

part

part

( ) ( ) ( ) x xik at k

j

x xik at k

j

xik t t ak

j jm

k jm

k jm

k

eee ∆minus+

minus

∆++

+

+∆++===

11

1ε ε ε



17





which is satisfied if (never true)

bull Hence the forward-in-time centred-in-space approximation to theone-dimensional advection equation is unconditionally unstable

33

( ) 1sin1 le∆∆

∆minus xk

x

t ci

m

x

uc

t

u

part

partminus=

part

part



Alternative advection algorithm

bull The forward-in-time upwind approximation to the advectionequation would be

bull This has amplification factor

bull We can show that for all k m if

This is usually called the Courant-Friedrichs-Lewy condition (CFL)and C is the Courant number

34

x

uuc

t

uu k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus

+

1

1

( ) xk x

t c

x

t cG m∆minus

minus

∆

∆

∆

∆+= cos1121

x

uc

t

u

part

partminus=

part

part



18



bull What is the physics of advection

Transport of a property u at velocity c

bull Consider the numerics of the upwind algorithm

The shaded triangle indicates thecomputational zone which influencesthe solution at point A

bull Determined by spatial resolution andby time step

Numerics vs Physicst

x

∆t

∆ x

A

35



bull Does the real physical advection process which influences point A

actually pass through the shaded triangle

bull The triangular zone captures the physics of advection at velocityc1 but not the physics of advection at velocity c2

For the c2 case then the mesh amp time step will cause instability

Numerics vs Physics

∆t

∆ x

A

36



19



Physical explanation of stability boundaries

bull We deduce that is the stability criterion for this algorithm

This is satisfied when the numerical domain of influence includes thephysical domain of influence

ie when the numerics captures the correct flow of information withinthe flowfield

bull Consider the following cases

Advection problem using centred-differences in space

bull Unstable Advection-diffusion problem using centred-differences in space

bull Stable (conditional - which way does information travel in diffusiveproblems)

1le∆

∆

x

t c

37



Linear Burgersrsquo Equation

bull We can now consider the stability limits of numerical solutions ofthe linear Burgers equation

This equation represents the time-variation of a flow property U

which is both convected by a uniform stream of velocity c and

diffused by viscosity micro

bull The equation is linear but displays some realistic physics

Unsteadiness

Convection (advection)

Diffusion

2

2

x

u

x

uc

t

u

part

part=

part

part+

part

part micro

2

21

x

u

dx

dp

x

uu

t

u

part

part+minus=

part

part+

part

part

ρ

micro

ρ

1D u-momentum equation



20



Von Neumann Analysis of Linear Burgersrsquo Eq

bull The forward-in-time centred-in-space (FTCS) finite-difference

scheme for the Burgers equation is

bull The amplification factor for this algorithm is

(useful exercise to prove)

where and

39

xk iC xk DGmm

∆minusminus∆+= sin)1(cos21

( )( )

( )k

j

k

j

k

j

k

j

k

j

k

j

k

juuu

xuu

x

c

t

uu11211

1

22

minus+minus+

+

+minus∆

+minus∆

minus=∆

minus micro




bull This equation describes an ellipse

centred at (1 - 2D) with

a semi-minor axis of C

a semi-major axis of 2 D

bull The condition for stability is met

if the ellipse stays within the unit circle

40

xk iC xk DGmm


Im

Re

unit circle

G

2 D

C



21



Stability plot 1 linear Burgers FTCS scheme

41

Im

Re

unit circle

G

2 D

C

C lt 1 2D lt 1 C 2 lt 2D C lt 1 2D lt 1 C 2 gt 2D

The ellipse is tangent to the unit circle at the right-handside where the local radius of curvature is C 2 2D




42

C lt 1 2D gt 1 C 2 lt 2D

Im

Re

unit circle

G

2 D

C

C lt 1 2D lt 1 C 2 lt 2D



22



Stability limits linear Burgers FTCS scheme

bull The stability limits for the FTCS approximation for the linearBurgers equation can be written

bull This can also be expressed as

where the cell Reynolds number is defined as

43

C RC x 22 lele ∆

122

ltlt DC




bull For the paint spill simulation the two limits C 2 gt 2D and 2D gt 1

affect opposite ends of the mean flow speed lsquocrsquo range

bull Replacing the centred difference of the advection term with an

upwind difference removes the stability limit on so that just one stability limit exists

However there is an impact upon accuracy and upon the second

stability limit

bull All this will be explored in the practical work

44

D

C

R x =∆





Numerical Stability Summary

bull Perturbations in a numerical solution introduced by round-off (orcoding) error may be amplified by the algorithm

Dynamic instability may be managed by reducing the time step

Static instability may be managed by changing the algorithm

bull Non-linear equations are difficult to analyse for stability but thevon Neumann technique yields quantitative stability boundariesfor linear algorithms

These may be applied with correction factors to similar schemes fornon-linear problems

bull Stability problems are related to a failure to capture theinformation flow in the physical domain

45



2




bull We use the example of a paint spill in the Thames

bull We are interested in the concentration of the paint spill as theriver carries it downstream

bull We want to look at the paint concentration profile u at different

points along the river rather than at different times

bull We want to see how the mean flow rate c of the Thames affects

the concentration profiles

For example in the different seasons of the year

bull We want to use the same amount of compute time in each case

The lower the mean flow rate c the larger the time step ∆t

3





4

983085983088983086983090983088

983088983086983088983088

983088983086983090983088

983088983086983092983088

983088983086983094983088

983088983086983096983088

983089983086983088983088

983089983086983090983088

983088983086983088983088 983088983086983093983088 983089983086983088983088 983089983086983093983088 983090983086983088983088

983125

983128

983116983145983150983141983137983154 983106983157983154983143983141983154983155 983109983153983157983137983156983145983151983150

983125 983137983156 983155983156983141983152 983088

983125 983137983156 983155983156983141983152 983093983088

983125 983137983156 983155983156983141983152 983089983088983088

983125 983137983156 983155983156983141983152 983089983093983088

983125 983137983156 983155983156983141983152 983090983088983088



3





5

983085983088983086983090983088

983088983086983088983088

983088983086983090983088

983088983086983092983088

983088983086983094983088

983088983086983096983088

983089983086983088983088

983089983086983090983088

983088983086983088983088 983088983086983093983088 983089983086983088983088 983089983086983093983088 983090983086983088983088

983125

983128

983116983145983150983141983137983154 983106983157983154983143983141983154983155 983109983153983157983137983156983145983151983150

983125 983137983156 983155983156983141983152 983088

983125 983137983156 983155983156983141983152 983093983088

983125 983137983156 983155983156983141983152 983089983088983088

983125 983137983156 983155983156983141983152 983089983093983088

983125 983137983156 983155983156983141983152 983090983088983088

983085983088983086983090983088

983088983086983088983088

983088983086983090983088

983088983086983092983088

983088983086983094983088

983088983086983096983088

983089983086983088983088

983089983086983090983088

983088983086983088983088 983088983086983093983088 983089983086983088983088 983089983086983093983088 983090983086983088983088

983125

983128

983116983145983150983141983137983154 983106983157983154983143983141983154983155 983109983153983157983137983156983145983151983150

983125 983137983156 983155983156983141983152 983088

983125 983137983156 983155983156983141983152 983093983088

983125 983137983156 983155983156983141983152 983089983088983088

983125 983137983156 983155983156983141983152 983089983093983088

983125 983137983156 983155983156983141983152 983090983088983088



983085983088983086983090983088

983088983086983088983088

983088983086983090983088

983088983086983092983088

983088983086983094983088

983088983086983096983088

983089983086983088983088

983089983086983090983088

983088983086983088983088 983088983086983093983088 983089983086983088983088 983089983086983093983088 983090983086983088983088

983125

983128

983116983145983150983141983137983154 983106983157983154983143983141983154983155 983109983153983157983137983156983145983151983150

983125 983137983156 983155983156983141983152 983088

983125 983137983156 983155983156983141983152 983093983088

983125 983137983156 983155983156983141983152 983089983088983088

983125 983137983156 983155983156983141983152 983089983093983088

983125 983137983156 983155983156983141983152 983090983088983088



6



4




983085983089983086983093983088

983085983089983086983088983088

983085983088983086983093983088

983088983086983088983088

983088983086983093983088

983089983086983088983088

983089983086983093983088

983090983086983088983088

983090983086983093983088

983088983086983088983088 983088983086983093983088 983089983086983088983088 983089983086983093983088 983090983086983088983088

983125

983128

983116983145983150983141983137983154 983106983157983154983143983141983154983155 983109983153983157983137983156983145983151983150

983125 983137983156 983155983156983141983152 983088

983125 983137983156 983155983156983141983152 983093983088

983125 983137983156 983155983156983141983152 983089983088983088

983125 983137983156 983155983156983141983152 983089983093983088

983125 983137983156 983155983156983141983152 983090983088983088


7




(again)


8

983085983088983086983090983088

983088983086983088983088

983088983086983090983088

983088983086983092983088

983088983086983094983088

983088983086983096983088

983089983086983088983088

983089983086983090983088

983088983086983088983088 983088983086983093983088 983089983086983088983088 983089983086983093983088 983090983086983088983088

983125

983128

983116983145983150983141983137983154 983106983157983154983143983141983154983155 983109983153983157983137983156983145983151983150

983125 983137983156 983155983156983141983152 983088

983125 983137983156 983155983156983141983152 983093983088

983125 983137983156 983155983156983141983152 983089983088983088

983125 983137983156 983155983156983141983152 983089983093983088

983125 983137983156 983155983156983141983152 983090983088983088



5




983085983088983086983094983088

983085983088983086983092983088

983085983088983086983090983088

983088983086983088983088

983088983086983090983088

983088983086983092983088

983088983086983094983088

983088983086983096983088

983089983086983088983088

983089983086983090983088

983089983086983092983088

983088983086983088983088 983088983086983093983088 983089983086983088983088 983089983086983093983088 983090983086983088983088

983125

983128

983116983145983150983141983137983154 983106983157983154983143983141983154983155 983109983153983157983137983156983145983151983150

983125 983137983156 983155983156983141983152 983088

983125 983137983156 983155983156983141983152 983093983088

983125 983137983156 983155983156983141983152 983089983088983088

983125 983137983156 983155983156983141983152 983089983093983088

983125 983137983156 983155983156983141983152 983090983088983088


9





What is going on

10



6



Numerical Errors










12



7





13

-02

-01

0

01

02

0 02 04 06 08 1

x

U N






14


-02

-01

0

01

02

0 02 04 06 08 1

x

U D



8





round-off error

15


-0001

0

0001

0002

0 02 04 06 08 1

x

ε ε









16



9







17

2

2

x

u

t

u

part

part=

part

part micro

( ) ( )k

j

k

j

k

j

k

j

k

juuu

xt

uu112

1

2minus+

+

+minus∆

=∆

minus micro








ε is the error

to obtain

18

( ) ( )k

j

k

j

k

j

k

j

k

juuu

xt

uu112

1

2minus+

+

+minus∆

=∆

minus micro

k

j

k

j

k

j Du ε +=

( ) ( )

( ) ( )k

j

k

j

k

j

k

j

k

j

k

j

k

j

k

j

k

j

k

j

x D D D

xt t

D D112112

11

22minus+minus+

++

+minus∆

++minus∆

=∆

minus+

∆

minusε ε ε

micro micro ε ε



10









19

ε ε ε j

k

j

k

j

k += +

1∆





20

Error at time k

-0002

-0001

0

0001

0002

0 02 04 06 08 1

x

ε εε ε

j minus 1 + 1

( ) ( )k

j

k

j

k

j

k

j x

t 112

2minus+

+minus∆


ε



11





may be greater than


21

( )

micro ∆

∆

t

x2

k

jε 2








22

x

uc

t

u

part

partminus=

part

part

( )k

j

k

j

k

j

k

juu

x

c

t

uu11

1

2 minus+

+

minus∆

minus=∆

minus



12





23

Error at time k

-0002

-0001

0

0001

0002

0 02 04 06 08 1

x

ε εε ε

j minus 1 + 1

( )k

j

k

j

k

j x

t c11

2 minus+

minus∆







24

k

jε



13









25










26



14




27






where



28

( ) sum=

+=

2

1

N

m

xik at met xε

L



15










29

k

j

k

j ε ε le+1

11

le=+

k

j

k j

Gε

ε


jmk xik at k

j e +

=ε





remembering that



30

2

11

12

xt

k

j

k

j

k

j

k

j

k

j

∆

+minus=

∆

minusminus+

+ ε ε ε micro

ε ε jmk

xik at k

j e +

=ε

2

2

x

u

t

u

part

part=

part

part micro

( ) ( ) ( ) x xik at k

j

x xik at k

j

xik t t ak

j jm

k jm

k jm

k

eee ∆minus+

minus

∆++

+

+∆++=== 11

1ε ε ε



16









31

( )1

2sin41

2

2 le

∆

∆

∆minus

xk

x

t m micro

( )1

2sin41

2

2 le

∆

∆

∆minus

xk

x

t m micro

( ) 12sin41

2

2 minusge

∆

∆

∆minus

xk

x

t m micro

2

2

x

u

t

u

part

part=

part

part micro








32

jmk xik at k

j e +

=ε x

ct

k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus+

+

2

11

1 ε ε ε ε

x

uc

t

u

part

partminus=

part

part

( ) ( ) ( ) x xik at k

j

x xik at k

j

xik t t ak

j jm

k jm

k jm

k

eee ∆minus+

minus

∆++

+

+∆++===

11

1ε ε ε



17







33

( ) 1sin1 le∆∆

∆minus xk

x

t ci

m

x

uc

t

u

part

partminus=

part

part








34

x

uuc

t

uu k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus

+

1

1

( ) xk x

t c

x

t cG m∆minus

minus

∆

∆

∆

∆+= cos1121

x

uc

t

u

part

partminus=

part

part



18









x

∆t

∆ x

A

35







Numerics vs Physics

∆t

∆ x

A

36



19











1le∆

∆

x

t c

37









Unsteadiness


Diffusion

2

2

x

u

x

uc

t

u

part

part=

part

part+

part

part micro

2

21

x

u

dx

dp

x

uu

t

u

part

part+minus=

part

part+

part

part

ρ

micro

ρ




20








where and

39

xk iC xk DGmm


( )( )

( )k

j

k

j

k

j

k

j

k

j

k

j

k

juuu

xuu

x

c

t

uu11211

1

22

minus+minus+

+

+minus∆

+minus∆

minus=∆

minus micro










40

xk iC xk DGmm


Im

Re

unit circle

G

2 D

C



21




41

Im

Re

unit circle

G

2 D

C






42


Im

Re

unit circle

G

2 D

C




22







43

C RC x 22 lele ∆

122

ltlt DC









stability limit


44

D

C

R x =∆












45



3





5

983085983088983086983090983088

983088983086983088983088

983088983086983090983088

983088983086983092983088

983088983086983094983088

983088983086983096983088

983089983086983088983088

983089983086983090983088

983088983086983088983088 983088983086983093983088 983089983086983088983088 983089983086983093983088 983090983086983088983088

983125

983128

983116983145983150983141983137983154 983106983157983154983143983141983154983155 983109983153983157983137983156983145983151983150

983125 983137983156 983155983156983141983152 983088

983125 983137983156 983155983156983141983152 983093983088

983125 983137983156 983155983156983141983152 983089983088983088

983125 983137983156 983155983156983141983152 983089983093983088

983125 983137983156 983155983156983141983152 983090983088983088

983085983088983086983090983088

983088983086983088983088

983088983086983090983088

983088983086983092983088

983088983086983094983088

983088983086983096983088

983089983086983088983088

983089983086983090983088

983088983086983088983088 983088983086983093983088 983089983086983088983088 983089983086983093983088 983090983086983088983088

983125

983128

983116983145983150983141983137983154 983106983157983154983143983141983154983155 983109983153983157983137983156983145983151983150

983125 983137983156 983155983156983141983152 983088

983125 983137983156 983155983156983141983152 983093983088

983125 983137983156 983155983156983141983152 983089983088983088

983125 983137983156 983155983156983141983152 983089983093983088

983125 983137983156 983155983156983141983152 983090983088983088



983085983088983086983090983088

983088983086983088983088

983088983086983090983088

983088983086983092983088

983088983086983094983088

983088983086983096983088

983089983086983088983088

983089983086983090983088

983088983086983088983088 983088983086983093983088 983089983086983088983088 983089983086983093983088 983090983086983088983088

983125

983128

983116983145983150983141983137983154 983106983157983154983143983141983154983155 983109983153983157983137983156983145983151983150

983125 983137983156 983155983156983141983152 983088

983125 983137983156 983155983156983141983152 983093983088

983125 983137983156 983155983156983141983152 983089983088983088

983125 983137983156 983155983156983141983152 983089983093983088

983125 983137983156 983155983156983141983152 983090983088983088



6



4




983085983089983086983093983088

983085983089983086983088983088

983085983088983086983093983088

983088983086983088983088

983088983086983093983088

983089983086983088983088

983089983086983093983088

983090983086983088983088

983090983086983093983088

983088983086983088983088 983088983086983093983088 983089983086983088983088 983089983086983093983088 983090983086983088983088

983125

983128

983116983145983150983141983137983154 983106983157983154983143983141983154983155 983109983153983157983137983156983145983151983150

983125 983137983156 983155983156983141983152 983088

983125 983137983156 983155983156983141983152 983093983088

983125 983137983156 983155983156983141983152 983089983088983088

983125 983137983156 983155983156983141983152 983089983093983088

983125 983137983156 983155983156983141983152 983090983088983088


7




(again)


8

983085983088983086983090983088

983088983086983088983088

983088983086983090983088

983088983086983092983088

983088983086983094983088

983088983086983096983088

983089983086983088983088

983089983086983090983088

983088983086983088983088 983088983086983093983088 983089983086983088983088 983089983086983093983088 983090983086983088983088

983125

983128

983116983145983150983141983137983154 983106983157983154983143983141983154983155 983109983153983157983137983156983145983151983150

983125 983137983156 983155983156983141983152 983088

983125 983137983156 983155983156983141983152 983093983088

983125 983137983156 983155983156983141983152 983089983088983088

983125 983137983156 983155983156983141983152 983089983093983088

983125 983137983156 983155983156983141983152 983090983088983088



5




983085983088983086983094983088

983085983088983086983092983088

983085983088983086983090983088

983088983086983088983088

983088983086983090983088

983088983086983092983088

983088983086983094983088

983088983086983096983088

983089983086983088983088

983089983086983090983088

983089983086983092983088

983088983086983088983088 983088983086983093983088 983089983086983088983088 983089983086983093983088 983090983086983088983088

983125

983128

983116983145983150983141983137983154 983106983157983154983143983141983154983155 983109983153983157983137983156983145983151983150

983125 983137983156 983155983156983141983152 983088

983125 983137983156 983155983156983141983152 983093983088

983125 983137983156 983155983156983141983152 983089983088983088

983125 983137983156 983155983156983141983152 983089983093983088

983125 983137983156 983155983156983141983152 983090983088983088


9





What is going on

10



6



Numerical Errors










12



7





13

-02

-01

0

01

02

0 02 04 06 08 1

x

U N






14


-02

-01

0

01

02

0 02 04 06 08 1

x

U D



8





round-off error

15


-0001

0

0001

0002

0 02 04 06 08 1

x

ε ε









16



9







17

2

2

x

u

t

u

part

part=

part

part micro

( ) ( )k

j

k

j

k

j

k

j

k

juuu

xt

uu112

1

2minus+

+

+minus∆

=∆

minus micro








ε is the error

to obtain

18

( ) ( )k

j

k

j

k

j

k

j

k

juuu

xt

uu112

1

2minus+

+

+minus∆

=∆

minus micro

k

j

k

j

k

j Du ε +=

( ) ( )

( ) ( )k

j

k

j

k

j

k

j

k

j

k

j

k

j

k

j

k

j

k

j

x D D D

xt t

D D112112

11

22minus+minus+

++

+minus∆

++minus∆

=∆

minus+

∆

minusε ε ε

micro micro ε ε



10









19

ε ε ε j

k

j

k

j

k += +

1∆





20

Error at time k

-0002

-0001

0

0001

0002

0 02 04 06 08 1

x

ε εε ε

j minus 1 + 1

( ) ( )k

j

k

j

k

j

k

j x

t 112

2minus+

+minus∆


ε



11





may be greater than


21

( )

micro ∆

∆

t

x2

k

jε 2








22

x

uc

t

u

part

partminus=

part

part

( )k

j

k

j

k

j

k

juu

x

c

t

uu11

1

2 minus+

+

minus∆

minus=∆

minus



12





23

Error at time k

-0002

-0001

0

0001

0002

0 02 04 06 08 1

x

ε εε ε

j minus 1 + 1

( )k

j

k

j

k

j x

t c11

2 minus+

minus∆







24

k

jε



13









25










26



14




27






where



28

( ) sum=

+=

2

1

N

m

xik at met xε

L



15










29

k

j

k

j ε ε le+1

11

le=+

k

j

k j

Gε

ε


jmk xik at k

j e +

=ε





remembering that



30

2

11

12

xt

k

j

k

j

k

j

k

j

k

j

∆

+minus=

∆

minusminus+

+ ε ε ε micro

ε ε jmk

xik at k

j e +

=ε

2

2

x

u

t

u

part

part=

part

part micro

( ) ( ) ( ) x xik at k

j

x xik at k

j

xik t t ak

j jm

k jm

k jm

k

eee ∆minus+

minus

∆++

+

+∆++=== 11

1ε ε ε



16









31

( )1

2sin41

2

2 le

∆

∆

∆minus

xk

x

t m micro

( )1

2sin41

2

2 le

∆

∆

∆minus

xk

x

t m micro

( ) 12sin41

2

2 minusge

∆

∆

∆minus

xk

x

t m micro

2

2

x

u

t

u

part

part=

part

part micro








32

jmk xik at k

j e +

=ε x

ct

k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus+

+

2

11

1 ε ε ε ε

x

uc

t

u

part

partminus=

part

part

( ) ( ) ( ) x xik at k

j

x xik at k

j

xik t t ak

j jm

k jm

k jm

k

eee ∆minus+

minus

∆++

+

+∆++===

11

1ε ε ε



17







33

( ) 1sin1 le∆∆

∆minus xk

x

t ci

m

x

uc

t

u

part

partminus=

part

part








34

x

uuc

t

uu k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus

+

1

1

( ) xk x

t c

x

t cG m∆minus

minus

∆

∆

∆

∆+= cos1121

x

uc

t

u

part

partminus=

part

part



18









x

∆t

∆ x

A

35







Numerics vs Physics

∆t

∆ x

A

36



19











1le∆

∆

x

t c

37









Unsteadiness


Diffusion

2

2

x

u

x

uc

t

u

part

part=

part

part+

part

part micro

2

21

x

u

dx

dp

x

uu

t

u

part

part+minus=

part

part+

part

part

ρ

micro

ρ




20








where and

39

xk iC xk DGmm


( )( )

( )k

j

k

j

k

j

k

j

k

j

k

j

k

juuu

xuu

x

c

t

uu11211

1

22

minus+minus+

+

+minus∆

+minus∆

minus=∆

minus micro










40

xk iC xk DGmm


Im

Re

unit circle

G

2 D

C



21




41

Im

Re

unit circle

G

2 D

C






42


Im

Re

unit circle

G

2 D

C




22







43

C RC x 22 lele ∆

122

ltlt DC









stability limit


44

D

C

R x =∆












45



4




983085983089983086983093983088

983085983089983086983088983088

983085983088983086983093983088

983088983086983088983088

983088983086983093983088

983089983086983088983088

983089983086983093983088

983090983086983088983088

983090983086983093983088

983088983086983088983088 983088983086983093983088 983089983086983088983088 983089983086983093983088 983090983086983088983088

983125

983128

983116983145983150983141983137983154 983106983157983154983143983141983154983155 983109983153983157983137983156983145983151983150

983125 983137983156 983155983156983141983152 983088

983125 983137983156 983155983156983141983152 983093983088

983125 983137983156 983155983156983141983152 983089983088983088

983125 983137983156 983155983156983141983152 983089983093983088

983125 983137983156 983155983156983141983152 983090983088983088


7




(again)


8

983085983088983086983090983088

983088983086983088983088

983088983086983090983088

983088983086983092983088

983088983086983094983088

983088983086983096983088

983089983086983088983088

983089983086983090983088

983088983086983088983088 983088983086983093983088 983089983086983088983088 983089983086983093983088 983090983086983088983088

983125

983128

983116983145983150983141983137983154 983106983157983154983143983141983154983155 983109983153983157983137983156983145983151983150

983125 983137983156 983155983156983141983152 983088

983125 983137983156 983155983156983141983152 983093983088

983125 983137983156 983155983156983141983152 983089983088983088

983125 983137983156 983155983156983141983152 983089983093983088

983125 983137983156 983155983156983141983152 983090983088983088



5




983085983088983086983094983088

983085983088983086983092983088

983085983088983086983090983088

983088983086983088983088

983088983086983090983088

983088983086983092983088

983088983086983094983088

983088983086983096983088

983089983086983088983088

983089983086983090983088

983089983086983092983088

983088983086983088983088 983088983086983093983088 983089983086983088983088 983089983086983093983088 983090983086983088983088

983125

983128

983116983145983150983141983137983154 983106983157983154983143983141983154983155 983109983153983157983137983156983145983151983150

983125 983137983156 983155983156983141983152 983088

983125 983137983156 983155983156983141983152 983093983088

983125 983137983156 983155983156983141983152 983089983088983088

983125 983137983156 983155983156983141983152 983089983093983088

983125 983137983156 983155983156983141983152 983090983088983088


9





What is going on

10



6



Numerical Errors










12



7





13

-02

-01

0

01

02

0 02 04 06 08 1

x

U N






14


-02

-01

0

01

02

0 02 04 06 08 1

x

U D



8





round-off error

15


-0001

0

0001

0002

0 02 04 06 08 1

x

ε ε









16



9







17

2

2

x

u

t

u

part

part=

part

part micro

( ) ( )k

j

k

j

k

j

k

j

k

juuu

xt

uu112

1

2minus+

+

+minus∆

=∆

minus micro








ε is the error

to obtain

18

( ) ( )k

j

k

j

k

j

k

j

k

juuu

xt

uu112

1

2minus+

+

+minus∆

=∆

minus micro

k

j

k

j

k

j Du ε +=

( ) ( )

( ) ( )k

j

k

j

k

j

k

j

k

j

k

j

k

j

k

j

k

j

k

j

x D D D

xt t

D D112112

11

22minus+minus+

++

+minus∆

++minus∆

=∆

minus+

∆

minusε ε ε

micro micro ε ε



10









19

ε ε ε j

k

j

k

j

k += +

1∆





20

Error at time k

-0002

-0001

0

0001

0002

0 02 04 06 08 1

x

ε εε ε

j minus 1 + 1

( ) ( )k

j

k

j

k

j

k

j x

t 112

2minus+

+minus∆


ε



11





may be greater than


21

( )

micro ∆

∆

t

x2

k

jε 2








22

x

uc

t

u

part

partminus=

part

part

( )k

j

k

j

k

j

k

juu

x

c

t

uu11

1

2 minus+

+

minus∆

minus=∆

minus



12





23

Error at time k

-0002

-0001

0

0001

0002

0 02 04 06 08 1

x

ε εε ε

j minus 1 + 1

( )k

j

k

j

k

j x

t c11

2 minus+

minus∆







24

k

jε



13









25










26



14




27






where



28

( ) sum=

+=

2

1

N

m

xik at met xε

L



15










29

k

j

k

j ε ε le+1

11

le=+

k

j

k j

Gε

ε


jmk xik at k

j e +

=ε





remembering that



30

2

11

12

xt

k

j

k

j

k

j

k

j

k

j

∆

+minus=

∆

minusminus+

+ ε ε ε micro

ε ε jmk

xik at k

j e +

=ε

2

2

x

u

t

u

part

part=

part

part micro

( ) ( ) ( ) x xik at k

j

x xik at k

j

xik t t ak

j jm

k jm

k jm

k

eee ∆minus+

minus

∆++

+

+∆++=== 11

1ε ε ε



16









31

( )1

2sin41

2

2 le

∆

∆

∆minus

xk

x

t m micro

( )1

2sin41

2

2 le

∆

∆

∆minus

xk

x

t m micro

( ) 12sin41

2

2 minusge

∆

∆

∆minus

xk

x

t m micro

2

2

x

u

t

u

part

part=

part

part micro








32

jmk xik at k

j e +

=ε x

ct

k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus+

+

2

11

1 ε ε ε ε

x

uc

t

u

part

partminus=

part

part

( ) ( ) ( ) x xik at k

j

x xik at k

j

xik t t ak

j jm

k jm

k jm

k

eee ∆minus+

minus

∆++

+

+∆++===

11

1ε ε ε



17







33

( ) 1sin1 le∆∆

∆minus xk

x

t ci

m

x

uc

t

u

part

partminus=

part

part








34

x

uuc

t

uu k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus

+

1

1

( ) xk x

t c

x

t cG m∆minus

minus

∆

∆

∆

∆+= cos1121

x

uc

t

u

part

partminus=

part

part



18









x

∆t

∆ x

A

35







Numerics vs Physics

∆t

∆ x

A

36



19











1le∆

∆

x

t c

37









Unsteadiness


Diffusion

2

2

x

u

x

uc

t

u

part

part=

part

part+

part

part micro

2

21

x

u

dx

dp

x

uu

t

u

part

part+minus=

part

part+

part

part

ρ

micro

ρ




20








where and

39

xk iC xk DGmm


( )( )

( )k

j

k

j

k

j

k

j

k

j

k

j

k

juuu

xuu

x

c

t

uu11211

1

22

minus+minus+

+

+minus∆

+minus∆

minus=∆

minus micro










40

xk iC xk DGmm


Im

Re

unit circle

G

2 D

C



21




41

Im

Re

unit circle

G

2 D

C






42


Im

Re

unit circle

G

2 D

C




22







43

C RC x 22 lele ∆

122

ltlt DC









stability limit


44

D

C

R x =∆












45



5




983085983088983086983094983088

983085983088983086983092983088

983085983088983086983090983088

983088983086983088983088

983088983086983090983088

983088983086983092983088

983088983086983094983088

983088983086983096983088

983089983086983088983088

983089983086983090983088

983089983086983092983088

983088983086983088983088 983088983086983093983088 983089983086983088983088 983089983086983093983088 983090983086983088983088

983125

983128

983116983145983150983141983137983154 983106983157983154983143983141983154983155 983109983153983157983137983156983145983151983150

983125 983137983156 983155983156983141983152 983088

983125 983137983156 983155983156983141983152 983093983088

983125 983137983156 983155983156983141983152 983089983088983088

983125 983137983156 983155983156983141983152 983089983093983088

983125 983137983156 983155983156983141983152 983090983088983088


9





What is going on

10



6



Numerical Errors










12



7





13

-02

-01

0

01

02

0 02 04 06 08 1

x

U N






14


-02

-01

0

01

02

0 02 04 06 08 1

x

U D



8





round-off error

15


-0001

0

0001

0002

0 02 04 06 08 1

x

ε ε









16



9







17

2

2

x

u

t

u

part

part=

part

part micro

( ) ( )k

j

k

j

k

j

k

j

k

juuu

xt

uu112

1

2minus+

+

+minus∆

=∆

minus micro








ε is the error

to obtain

18

( ) ( )k

j

k

j

k

j

k

j

k

juuu

xt

uu112

1

2minus+

+

+minus∆

=∆

minus micro

k

j

k

j

k

j Du ε +=

( ) ( )

( ) ( )k

j

k

j

k

j

k

j

k

j

k

j

k

j

k

j

k

j

k

j

x D D D

xt t

D D112112

11

22minus+minus+

++

+minus∆

++minus∆

=∆

minus+

∆

minusε ε ε

micro micro ε ε



10









19

ε ε ε j

k

j

k

j

k += +

1∆





20

Error at time k

-0002

-0001

0

0001

0002

0 02 04 06 08 1

x

ε εε ε

j minus 1 + 1

( ) ( )k

j

k

j

k

j

k

j x

t 112

2minus+

+minus∆


ε



11





may be greater than


21

( )

micro ∆

∆

t

x2

k

jε 2








22

x

uc

t

u

part

partminus=

part

part

( )k

j

k

j

k

j

k

juu

x

c

t

uu11

1

2 minus+

+

minus∆

minus=∆

minus



12





23

Error at time k

-0002

-0001

0

0001

0002

0 02 04 06 08 1

x

ε εε ε

j minus 1 + 1

( )k

j

k

j

k

j x

t c11

2 minus+

minus∆







24

k

jε



13









25










26



14




27






where



28

( ) sum=

+=

2

1

N

m

xik at met xε

L



15










29

k

j

k

j ε ε le+1

11

le=+

k

j

k j

Gε

ε


jmk xik at k

j e +

=ε





remembering that



30

2

11

12

xt

k

j

k

j

k

j

k

j

k

j

∆

+minus=

∆

minusminus+

+ ε ε ε micro

ε ε jmk

xik at k

j e +

=ε

2

2

x

u

t

u

part

part=

part

part micro

( ) ( ) ( ) x xik at k

j

x xik at k

j

xik t t ak

j jm

k jm

k jm

k

eee ∆minus+

minus

∆++

+

+∆++=== 11

1ε ε ε



16









31

( )1

2sin41

2

2 le

∆

∆

∆minus

xk

x

t m micro

( )1

2sin41

2

2 le

∆

∆

∆minus

xk

x

t m micro

( ) 12sin41

2

2 minusge

∆

∆

∆minus

xk

x

t m micro

2

2

x

u

t

u

part

part=

part

part micro








32

jmk xik at k

j e +

=ε x

ct

k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus+

+

2

11

1 ε ε ε ε

x

uc

t

u

part

partminus=

part

part

( ) ( ) ( ) x xik at k

j

x xik at k

j

xik t t ak

j jm

k jm

k jm

k

eee ∆minus+

minus

∆++

+

+∆++===

11

1ε ε ε



17







33

( ) 1sin1 le∆∆

∆minus xk

x

t ci

m

x

uc

t

u

part

partminus=

part

part








34

x

uuc

t

uu k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus

+

1

1

( ) xk x

t c

x

t cG m∆minus

minus

∆

∆

∆

∆+= cos1121

x

uc

t

u

part

partminus=

part

part



18









x

∆t

∆ x

A

35







Numerics vs Physics

∆t

∆ x

A

36



19











1le∆

∆

x

t c

37









Unsteadiness


Diffusion

2

2

x

u

x

uc

t

u

part

part=

part

part+

part

part micro

2

21

x

u

dx

dp

x

uu

t

u

part

part+minus=

part

part+

part

part

ρ

micro

ρ




20








where and

39

xk iC xk DGmm


( )( )

( )k

j

k

j

k

j

k

j

k

j

k

j

k

juuu

xuu

x

c

t

uu11211

1

22

minus+minus+

+

+minus∆

+minus∆

minus=∆

minus micro










40

xk iC xk DGmm


Im

Re

unit circle

G

2 D

C



21




41

Im

Re

unit circle

G

2 D

C






42


Im

Re

unit circle

G

2 D

C




22







43

C RC x 22 lele ∆

122

ltlt DC









stability limit


44

D

C

R x =∆












45



6



Numerical Errors










12



7





13

-02

-01

0

01

02

0 02 04 06 08 1

x

U N






14


-02

-01

0

01

02

0 02 04 06 08 1

x

U D



8





round-off error

15


-0001

0

0001

0002

0 02 04 06 08 1

x

ε ε









16



9







17

2

2

x

u

t

u

part

part=

part

part micro

( ) ( )k

j

k

j

k

j

k

j

k

juuu

xt

uu112

1

2minus+

+

+minus∆

=∆

minus micro








ε is the error

to obtain

18

( ) ( )k

j

k

j

k

j

k

j

k

juuu

xt

uu112

1

2minus+

+

+minus∆

=∆

minus micro

k

j

k

j

k

j Du ε +=

( ) ( )

( ) ( )k

j

k

j

k

j

k

j

k

j

k

j

k

j

k

j

k

j

k

j

x D D D

xt t

D D112112

11

22minus+minus+

++

+minus∆

++minus∆

=∆

minus+

∆

minusε ε ε

micro micro ε ε



10









19

ε ε ε j

k

j

k

j

k += +

1∆





20

Error at time k

-0002

-0001

0

0001

0002

0 02 04 06 08 1

x

ε εε ε

j minus 1 + 1

( ) ( )k

j

k

j

k

j

k

j x

t 112

2minus+

+minus∆


ε



11





may be greater than


21

( )

micro ∆

∆

t

x2

k

jε 2








22

x

uc

t

u

part

partminus=

part

part

( )k

j

k

j

k

j

k

juu

x

c

t

uu11

1

2 minus+

+

minus∆

minus=∆

minus



12





23

Error at time k

-0002

-0001

0

0001

0002

0 02 04 06 08 1

x

ε εε ε

j minus 1 + 1

( )k

j

k

j

k

j x

t c11

2 minus+

minus∆







24

k

jε



13









25










26



14




27






where



28

( ) sum=

+=

2

1

N

m

xik at met xε

L



15










29

k

j

k

j ε ε le+1

11

le=+

k

j

k j

Gε

ε


jmk xik at k

j e +

=ε





remembering that



30

2

11

12

xt

k

j

k

j

k

j

k

j

k

j

∆

+minus=

∆

minusminus+

+ ε ε ε micro

ε ε jmk

xik at k

j e +

=ε

2

2

x

u

t

u

part

part=

part

part micro

( ) ( ) ( ) x xik at k

j

x xik at k

j

xik t t ak

j jm

k jm

k jm

k

eee ∆minus+

minus

∆++

+

+∆++=== 11

1ε ε ε



16









31

( )1

2sin41

2

2 le

∆

∆

∆minus

xk

x

t m micro

( )1

2sin41

2

2 le

∆

∆

∆minus

xk

x

t m micro

( ) 12sin41

2

2 minusge

∆

∆

∆minus

xk

x

t m micro

2

2

x

u

t

u

part

part=

part

part micro








32

jmk xik at k

j e +

=ε x

ct

k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus+

+

2

11

1 ε ε ε ε

x

uc

t

u

part

partminus=

part

part

( ) ( ) ( ) x xik at k

j

x xik at k

j

xik t t ak

j jm

k jm

k jm

k

eee ∆minus+

minus

∆++

+

+∆++===

11

1ε ε ε



17







33

( ) 1sin1 le∆∆

∆minus xk

x

t ci

m

x

uc

t

u

part

partminus=

part

part








34

x

uuc

t

uu k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus

+

1

1

( ) xk x

t c

x

t cG m∆minus

minus

∆

∆

∆

∆+= cos1121

x

uc

t

u

part

partminus=

part

part



18









x

∆t

∆ x

A

35







Numerics vs Physics

∆t

∆ x

A

36



19











1le∆

∆

x

t c

37









Unsteadiness


Diffusion

2

2

x

u

x

uc

t

u

part

part=

part

part+

part

part micro

2

21

x

u

dx

dp

x

uu

t

u

part

part+minus=

part

part+

part

part

ρ

micro

ρ




20








where and

39

xk iC xk DGmm


( )( )

( )k

j

k

j

k

j

k

j

k

j

k

j

k

juuu

xuu

x

c

t

uu11211

1

22

minus+minus+

+

+minus∆

+minus∆

minus=∆

minus micro










40

xk iC xk DGmm


Im

Re

unit circle

G

2 D

C



21




41

Im

Re

unit circle

G

2 D

C






42


Im

Re

unit circle

G

2 D

C




22







43

C RC x 22 lele ∆

122

ltlt DC









stability limit


44

D

C

R x =∆












45



7





13

-02

-01

0

01

02

0 02 04 06 08 1

x

U N






14


-02

-01

0

01

02

0 02 04 06 08 1

x

U D



8





round-off error

15


-0001

0

0001

0002

0 02 04 06 08 1

x

ε ε









16



9







17

2

2

x

u

t

u

part

part=

part

part micro

( ) ( )k

j

k

j

k

j

k

j

k

juuu

xt

uu112

1

2minus+

+

+minus∆

=∆

minus micro








ε is the error

to obtain

18

( ) ( )k

j

k

j

k

j

k

j

k

juuu

xt

uu112

1

2minus+

+

+minus∆

=∆

minus micro

k

j

k

j

k

j Du ε +=

( ) ( )

( ) ( )k

j

k

j

k

j

k

j

k

j

k

j

k

j

k

j

k

j

k

j

x D D D

xt t

D D112112

11

22minus+minus+

++

+minus∆

++minus∆

=∆

minus+

∆

minusε ε ε

micro micro ε ε



10









19

ε ε ε j

k

j

k

j

k += +

1∆





20

Error at time k

-0002

-0001

0

0001

0002

0 02 04 06 08 1

x

ε εε ε

j minus 1 + 1

( ) ( )k

j

k

j

k

j

k

j x

t 112

2minus+

+minus∆


ε



11





may be greater than


21

( )

micro ∆

∆

t

x2

k

jε 2








22

x

uc

t

u

part

partminus=

part

part

( )k

j

k

j

k

j

k

juu

x

c

t

uu11

1

2 minus+

+

minus∆

minus=∆

minus



12





23

Error at time k

-0002

-0001

0

0001

0002

0 02 04 06 08 1

x

ε εε ε

j minus 1 + 1

( )k

j

k

j

k

j x

t c11

2 minus+

minus∆







24

k

jε



13









25










26



14




27






where



28

( ) sum=

+=

2

1

N

m

xik at met xε

L



15










29

k

j

k

j ε ε le+1

11

le=+

k

j

k j

Gε

ε


jmk xik at k

j e +

=ε





remembering that



30

2

11

12

xt

k

j

k

j

k

j

k

j

k

j

∆

+minus=

∆

minusminus+

+ ε ε ε micro

ε ε jmk

xik at k

j e +

=ε

2

2

x

u

t

u

part

part=

part

part micro

( ) ( ) ( ) x xik at k

j

x xik at k

j

xik t t ak

j jm

k jm

k jm

k

eee ∆minus+

minus

∆++

+

+∆++=== 11

1ε ε ε



16









31

( )1

2sin41

2

2 le

∆

∆

∆minus

xk

x

t m micro

( )1

2sin41

2

2 le

∆

∆

∆minus

xk

x

t m micro

( ) 12sin41

2

2 minusge

∆

∆

∆minus

xk

x

t m micro

2

2

x

u

t

u

part

part=

part

part micro








32

jmk xik at k

j e +

=ε x

ct

k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus+

+

2

11

1 ε ε ε ε

x

uc

t

u

part

partminus=

part

part

( ) ( ) ( ) x xik at k

j

x xik at k

j

xik t t ak

j jm

k jm

k jm

k

eee ∆minus+

minus

∆++

+

+∆++===

11

1ε ε ε



17







33

( ) 1sin1 le∆∆

∆minus xk

x

t ci

m

x

uc

t

u

part

partminus=

part

part








34

x

uuc

t

uu k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus

+

1

1

( ) xk x

t c

x

t cG m∆minus

minus

∆

∆

∆

∆+= cos1121

x

uc

t

u

part

partminus=

part

part



18









x

∆t

∆ x

A

35







Numerics vs Physics

∆t

∆ x

A

36



19











1le∆

∆

x

t c

37









Unsteadiness


Diffusion

2

2

x

u

x

uc

t

u

part

part=

part

part+

part

part micro

2

21

x

u

dx

dp

x

uu

t

u

part

part+minus=

part

part+

part

part

ρ

micro

ρ




20








where and

39

xk iC xk DGmm


( )( )

( )k

j

k

j

k

j

k

j

k

j

k

j

k

juuu

xuu

x

c

t

uu11211

1

22

minus+minus+

+

+minus∆

+minus∆

minus=∆

minus micro










40

xk iC xk DGmm


Im

Re

unit circle

G

2 D

C



21




41

Im

Re

unit circle

G

2 D

C






42


Im

Re

unit circle

G

2 D

C




22







43

C RC x 22 lele ∆

122

ltlt DC









stability limit


44

D

C

R x =∆












45



8





round-off error

15


-0001

0

0001

0002

0 02 04 06 08 1

x

ε ε









16



9







17

2

2

x

u

t

u

part

part=

part

part micro

( ) ( )k

j

k

j

k

j

k

j

k

juuu

xt

uu112

1

2minus+

+

+minus∆

=∆

minus micro








ε is the error

to obtain

18

( ) ( )k

j

k

j

k

j

k

j

k

juuu

xt

uu112

1

2minus+

+

+minus∆

=∆

minus micro

k

j

k

j

k

j Du ε +=

( ) ( )

( ) ( )k

j

k

j

k

j

k

j

k

j

k

j

k

j

k

j

k

j

k

j

x D D D

xt t

D D112112

11

22minus+minus+

++

+minus∆

++minus∆

=∆

minus+

∆

minusε ε ε

micro micro ε ε



10









19

ε ε ε j

k

j

k

j

k += +

1∆





20

Error at time k

-0002

-0001

0

0001

0002

0 02 04 06 08 1

x

ε εε ε

j minus 1 + 1

( ) ( )k

j

k

j

k

j

k

j x

t 112

2minus+

+minus∆


ε



11





may be greater than


21

( )

micro ∆

∆

t

x2

k

jε 2








22

x

uc

t

u

part

partminus=

part

part

( )k

j

k

j

k

j

k

juu

x

c

t

uu11

1

2 minus+

+

minus∆

minus=∆

minus



12





23

Error at time k

-0002

-0001

0

0001

0002

0 02 04 06 08 1

x

ε εε ε

j minus 1 + 1

( )k

j

k

j

k

j x

t c11

2 minus+

minus∆







24

k

jε



13









25










26



14




27






where



28

( ) sum=

+=

2

1

N

m

xik at met xε

L



15










29

k

j

k

j ε ε le+1

11

le=+

k

j

k j

Gε

ε


jmk xik at k

j e +

=ε





remembering that



30

2

11

12

xt

k

j

k

j

k

j

k

j

k

j

∆

+minus=

∆

minusminus+

+ ε ε ε micro

ε ε jmk

xik at k

j e +

=ε

2

2

x

u

t

u

part

part=

part

part micro

( ) ( ) ( ) x xik at k

j

x xik at k

j

xik t t ak

j jm

k jm

k jm

k

eee ∆minus+

minus

∆++

+

+∆++=== 11

1ε ε ε



16









31

( )1

2sin41

2

2 le

∆

∆

∆minus

xk

x

t m micro

( )1

2sin41

2

2 le

∆

∆

∆minus

xk

x

t m micro

( ) 12sin41

2

2 minusge

∆

∆

∆minus

xk

x

t m micro

2

2

x

u

t

u

part

part=

part

part micro








32

jmk xik at k

j e +

=ε x

ct

k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus+

+

2

11

1 ε ε ε ε

x

uc

t

u

part

partminus=

part

part

( ) ( ) ( ) x xik at k

j

x xik at k

j

xik t t ak

j jm

k jm

k jm

k

eee ∆minus+

minus

∆++

+

+∆++===

11

1ε ε ε



17







33

( ) 1sin1 le∆∆

∆minus xk

x

t ci

m

x

uc

t

u

part

partminus=

part

part








34

x

uuc

t

uu k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus

+

1

1

( ) xk x

t c

x

t cG m∆minus

minus

∆

∆

∆

∆+= cos1121

x

uc

t

u

part

partminus=

part

part



18









x

∆t

∆ x

A

35







Numerics vs Physics

∆t

∆ x

A

36



19











1le∆

∆

x

t c

37









Unsteadiness


Diffusion

2

2

x

u

x

uc

t

u

part

part=

part

part+

part

part micro

2

21

x

u

dx

dp

x

uu

t

u

part

part+minus=

part

part+

part

part

ρ

micro

ρ




20








where and

39

xk iC xk DGmm


( )( )

( )k

j

k

j

k

j

k

j

k

j

k

j

k

juuu

xuu

x

c

t

uu11211

1

22

minus+minus+

+

+minus∆

+minus∆

minus=∆

minus micro










40

xk iC xk DGmm


Im

Re

unit circle

G

2 D

C



21




41

Im

Re

unit circle

G

2 D

C






42


Im

Re

unit circle

G

2 D

C




22







43

C RC x 22 lele ∆

122

ltlt DC









stability limit


44

D

C

R x =∆












45



9







17

2

2

x

u

t

u

part

part=

part

part micro

( ) ( )k

j

k

j

k

j

k

j

k

juuu

xt

uu112

1

2minus+

+

+minus∆

=∆

minus micro








ε is the error

to obtain

18

( ) ( )k

j

k

j

k

j

k

j

k

juuu

xt

uu112

1

2minus+

+

+minus∆

=∆

minus micro

k

j

k

j

k

j Du ε +=

( ) ( )

( ) ( )k

j

k

j

k

j

k

j

k

j

k

j

k

j

k

j

k

j

k

j

x D D D

xt t

D D112112

11

22minus+minus+

++

+minus∆

++minus∆

=∆

minus+

∆

minusε ε ε

micro micro ε ε



10









19

ε ε ε j

k

j

k

j

k += +

1∆





20

Error at time k

-0002

-0001

0

0001

0002

0 02 04 06 08 1

x

ε εε ε

j minus 1 + 1

( ) ( )k

j

k

j

k

j

k

j x

t 112

2minus+

+minus∆


ε



11





may be greater than


21

( )

micro ∆

∆

t

x2

k

jε 2








22

x

uc

t

u

part

partminus=

part

part

( )k

j

k

j

k

j

k

juu

x

c

t

uu11

1

2 minus+

+

minus∆

minus=∆

minus



12





23

Error at time k

-0002

-0001

0

0001

0002

0 02 04 06 08 1

x

ε εε ε

j minus 1 + 1

( )k

j

k

j

k

j x

t c11

2 minus+

minus∆







24

k

jε



13









25










26



14




27






where



28

( ) sum=

+=

2

1

N

m

xik at met xε

L



15










29

k

j

k

j ε ε le+1

11

le=+

k

j

k j

Gε

ε


jmk xik at k

j e +

=ε





remembering that



30

2

11

12

xt

k

j

k

j

k

j

k

j

k

j

∆

+minus=

∆

minusminus+

+ ε ε ε micro

ε ε jmk

xik at k

j e +

=ε

2

2

x

u

t

u

part

part=

part

part micro

( ) ( ) ( ) x xik at k

j

x xik at k

j

xik t t ak

j jm

k jm

k jm

k

eee ∆minus+

minus

∆++

+

+∆++=== 11

1ε ε ε



16









31

( )1

2sin41

2

2 le

∆

∆

∆minus

xk

x

t m micro

( )1

2sin41

2

2 le

∆

∆

∆minus

xk

x

t m micro

( ) 12sin41

2

2 minusge

∆

∆

∆minus

xk

x

t m micro

2

2

x

u

t

u

part

part=

part

part micro








32

jmk xik at k

j e +

=ε x

ct

k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus+

+

2

11

1 ε ε ε ε

x

uc

t

u

part

partminus=

part

part

( ) ( ) ( ) x xik at k

j

x xik at k

j

xik t t ak

j jm

k jm

k jm

k

eee ∆minus+

minus

∆++

+

+∆++===

11

1ε ε ε



17







33

( ) 1sin1 le∆∆

∆minus xk

x

t ci

m

x

uc

t

u

part

partminus=

part

part








34

x

uuc

t

uu k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus

+

1

1

( ) xk x

t c

x

t cG m∆minus

minus

∆

∆

∆

∆+= cos1121

x

uc

t

u

part

partminus=

part

part



18









x

∆t

∆ x

A

35







Numerics vs Physics

∆t

∆ x

A

36



19











1le∆

∆

x

t c

37









Unsteadiness


Diffusion

2

2

x

u

x

uc

t

u

part

part=

part

part+

part

part micro

2

21

x

u

dx

dp

x

uu

t

u

part

part+minus=

part

part+

part

part

ρ

micro

ρ




20








where and

39

xk iC xk DGmm


( )( )

( )k

j

k

j

k

j

k

j

k

j

k

j

k

juuu

xuu

x

c

t

uu11211

1

22

minus+minus+

+

+minus∆

+minus∆

minus=∆

minus micro










40

xk iC xk DGmm


Im

Re

unit circle

G

2 D

C



21




41

Im

Re

unit circle

G

2 D

C






42


Im

Re

unit circle

G

2 D

C




22







43

C RC x 22 lele ∆

122

ltlt DC









stability limit


44

D

C

R x =∆












45



10









19

ε ε ε j

k

j

k

j

k += +

1∆





20

Error at time k

-0002

-0001

0

0001

0002

0 02 04 06 08 1

x

ε εε ε

j minus 1 + 1

( ) ( )k

j

k

j

k

j

k

j x

t 112

2minus+

+minus∆


ε



11





may be greater than


21

( )

micro ∆

∆

t

x2

k

jε 2








22

x

uc

t

u

part

partminus=

part

part

( )k

j

k

j

k

j

k

juu

x

c

t

uu11

1

2 minus+

+

minus∆

minus=∆

minus



12





23

Error at time k

-0002

-0001

0

0001

0002

0 02 04 06 08 1

x

ε εε ε

j minus 1 + 1

( )k

j

k

j

k

j x

t c11

2 minus+

minus∆







24

k

jε



13









25










26



14




27






where



28

( ) sum=

+=

2

1

N

m

xik at met xε

L



15










29

k

j

k

j ε ε le+1

11

le=+

k

j

k j

Gε

ε


jmk xik at k

j e +

=ε





remembering that



30

2

11

12

xt

k

j

k

j

k

j

k

j

k

j

∆

+minus=

∆

minusminus+

+ ε ε ε micro

ε ε jmk

xik at k

j e +

=ε

2

2

x

u

t

u

part

part=

part

part micro

( ) ( ) ( ) x xik at k

j

x xik at k

j

xik t t ak

j jm

k jm

k jm

k

eee ∆minus+

minus

∆++

+

+∆++=== 11

1ε ε ε



16









31

( )1

2sin41

2

2 le

∆

∆

∆minus

xk

x

t m micro

( )1

2sin41

2

2 le

∆

∆

∆minus

xk

x

t m micro

( ) 12sin41

2

2 minusge

∆

∆

∆minus

xk

x

t m micro

2

2

x

u

t

u

part

part=

part

part micro








32

jmk xik at k

j e +

=ε x

ct

k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus+

+

2

11

1 ε ε ε ε

x

uc

t

u

part

partminus=

part

part

( ) ( ) ( ) x xik at k

j

x xik at k

j

xik t t ak

j jm

k jm

k jm

k

eee ∆minus+

minus

∆++

+

+∆++===

11

1ε ε ε



17







33

( ) 1sin1 le∆∆

∆minus xk

x

t ci

m

x

uc

t

u

part

partminus=

part

part








34

x

uuc

t

uu k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus

+

1

1

( ) xk x

t c

x

t cG m∆minus

minus

∆

∆

∆

∆+= cos1121

x

uc

t

u

part

partminus=

part

part



18









x

∆t

∆ x

A

35







Numerics vs Physics

∆t

∆ x

A

36



19











1le∆

∆

x

t c

37









Unsteadiness


Diffusion

2

2

x

u

x

uc

t

u

part

part=

part

part+

part

part micro

2

21

x

u

dx

dp

x

uu

t

u

part

part+minus=

part

part+

part

part

ρ

micro

ρ




20








where and

39

xk iC xk DGmm


( )( )

( )k

j

k

j

k

j

k

j

k

j

k

j

k

juuu

xuu

x

c

t

uu11211

1

22

minus+minus+

+

+minus∆

+minus∆

minus=∆

minus micro










40

xk iC xk DGmm


Im

Re

unit circle

G

2 D

C



21




41

Im

Re

unit circle

G

2 D

C






42


Im

Re

unit circle

G

2 D

C




22







43

C RC x 22 lele ∆

122

ltlt DC









stability limit


44

D

C

R x =∆












45



11





may be greater than


21

( )

micro ∆

∆

t

x2

k

jε 2








22

x

uc

t

u

part

partminus=

part

part

( )k

j

k

j

k

j

k

juu

x

c

t

uu11

1

2 minus+

+

minus∆

minus=∆

minus



12





23

Error at time k

-0002

-0001

0

0001

0002

0 02 04 06 08 1

x

ε εε ε

j minus 1 + 1

( )k

j

k

j

k

j x

t c11

2 minus+

minus∆







24

k

jε



13









25










26



14




27






where



28

( ) sum=

+=

2

1

N

m

xik at met xε

L



15










29

k

j

k

j ε ε le+1

11

le=+

k

j

k j

Gε

ε


jmk xik at k

j e +

=ε





remembering that



30

2

11

12

xt

k

j

k

j

k

j

k

j

k

j

∆

+minus=

∆

minusminus+

+ ε ε ε micro

ε ε jmk

xik at k

j e +

=ε

2

2

x

u

t

u

part

part=

part

part micro

( ) ( ) ( ) x xik at k

j

x xik at k

j

xik t t ak

j jm

k jm

k jm

k

eee ∆minus+

minus

∆++

+

+∆++=== 11

1ε ε ε



16









31

( )1

2sin41

2

2 le

∆

∆

∆minus

xk

x

t m micro

( )1

2sin41

2

2 le

∆

∆

∆minus

xk

x

t m micro

( ) 12sin41

2

2 minusge

∆

∆

∆minus

xk

x

t m micro

2

2

x

u

t

u

part

part=

part

part micro








32

jmk xik at k

j e +

=ε x

ct

k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus+

+

2

11

1 ε ε ε ε

x

uc

t

u

part

partminus=

part

part

( ) ( ) ( ) x xik at k

j

x xik at k

j

xik t t ak

j jm

k jm

k jm

k

eee ∆minus+

minus

∆++

+

+∆++===

11

1ε ε ε



17







33

( ) 1sin1 le∆∆

∆minus xk

x

t ci

m

x

uc

t

u

part

partminus=

part

part








34

x

uuc

t

uu k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus

+

1

1

( ) xk x

t c

x

t cG m∆minus

minus

∆

∆

∆

∆+= cos1121

x

uc

t

u

part

partminus=

part

part



18









x

∆t

∆ x

A

35







Numerics vs Physics

∆t

∆ x

A

36



19











1le∆

∆

x

t c

37









Unsteadiness


Diffusion

2

2

x

u

x

uc

t

u

part

part=

part

part+

part

part micro

2

21

x

u

dx

dp

x

uu

t

u

part

part+minus=

part

part+

part

part

ρ

micro

ρ




20








where and

39

xk iC xk DGmm


( )( )

( )k

j

k

j

k

j

k

j

k

j

k

j

k

juuu

xuu

x

c

t

uu11211

1

22

minus+minus+

+

+minus∆

+minus∆

minus=∆

minus micro










40

xk iC xk DGmm


Im

Re

unit circle

G

2 D

C



21




41

Im

Re

unit circle

G

2 D

C






42


Im

Re

unit circle

G

2 D

C




22







43

C RC x 22 lele ∆

122

ltlt DC









stability limit


44

D

C

R x =∆












45



12





23

Error at time k

-0002

-0001

0

0001

0002

0 02 04 06 08 1

x

ε εε ε

j minus 1 + 1

( )k

j

k

j

k

j x

t c11

2 minus+

minus∆







24

k

jε



13









25










26



14




27






where



28

( ) sum=

+=

2

1

N

m

xik at met xε

L



15










29

k

j

k

j ε ε le+1

11

le=+

k

j

k j

Gε

ε


jmk xik at k

j e +

=ε





remembering that



30

2

11

12

xt

k

j

k

j

k

j

k

j

k

j

∆

+minus=

∆

minusminus+

+ ε ε ε micro

ε ε jmk

xik at k

j e +

=ε

2

2

x

u

t

u

part

part=

part

part micro

( ) ( ) ( ) x xik at k

j

x xik at k

j

xik t t ak

j jm

k jm

k jm

k

eee ∆minus+

minus

∆++

+

+∆++=== 11

1ε ε ε



16









31

( )1

2sin41

2

2 le

∆

∆

∆minus

xk

x

t m micro

( )1

2sin41

2

2 le

∆

∆

∆minus

xk

x

t m micro

( ) 12sin41

2

2 minusge

∆

∆

∆minus

xk

x

t m micro

2

2

x

u

t

u

part

part=

part

part micro








32

jmk xik at k

j e +

=ε x

ct

k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus+

+

2

11

1 ε ε ε ε

x

uc

t

u

part

partminus=

part

part

( ) ( ) ( ) x xik at k

j

x xik at k

j

xik t t ak

j jm

k jm

k jm

k

eee ∆minus+

minus

∆++

+

+∆++===

11

1ε ε ε



17







33

( ) 1sin1 le∆∆

∆minus xk

x

t ci

m

x

uc

t

u

part

partminus=

part

part








34

x

uuc

t

uu k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus

+

1

1

( ) xk x

t c

x

t cG m∆minus

minus

∆

∆

∆

∆+= cos1121

x

uc

t

u

part

partminus=

part

part



18









x

∆t

∆ x

A

35







Numerics vs Physics

∆t

∆ x

A

36



19











1le∆

∆

x

t c

37









Unsteadiness


Diffusion

2

2

x

u

x

uc

t

u

part

part=

part

part+

part

part micro

2

21

x

u

dx

dp

x

uu

t

u

part

part+minus=

part

part+

part

part

ρ

micro

ρ




20








where and

39

xk iC xk DGmm


( )( )

( )k

j

k

j

k

j

k

j

k

j

k

j

k

juuu

xuu

x

c

t

uu11211

1

22

minus+minus+

+

+minus∆

+minus∆

minus=∆

minus micro










40

xk iC xk DGmm


Im

Re

unit circle

G

2 D

C



21




41

Im

Re

unit circle

G

2 D

C






42


Im

Re

unit circle

G

2 D

C




22







43

C RC x 22 lele ∆

122

ltlt DC









stability limit


44

D

C

R x =∆












45



13









25










26



14




27






where



28

( ) sum=

+=

2

1

N

m

xik at met xε

L



15










29

k

j

k

j ε ε le+1

11

le=+

k

j

k j

Gε

ε


jmk xik at k

j e +

=ε





remembering that



30

2

11

12

xt

k

j

k

j

k

j

k

j

k

j

∆

+minus=

∆

minusminus+

+ ε ε ε micro

ε ε jmk

xik at k

j e +

=ε

2

2

x

u

t

u

part

part=

part

part micro

( ) ( ) ( ) x xik at k

j

x xik at k

j

xik t t ak

j jm

k jm

k jm

k

eee ∆minus+

minus

∆++

+

+∆++=== 11

1ε ε ε



16









31

( )1

2sin41

2

2 le

∆

∆

∆minus

xk

x

t m micro

( )1

2sin41

2

2 le

∆

∆

∆minus

xk

x

t m micro

( ) 12sin41

2

2 minusge

∆

∆

∆minus

xk

x

t m micro

2

2

x

u

t

u

part

part=

part

part micro








32

jmk xik at k

j e +

=ε x

ct

k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus+

+

2

11

1 ε ε ε ε

x

uc

t

u

part

partminus=

part

part

( ) ( ) ( ) x xik at k

j

x xik at k

j

xik t t ak

j jm

k jm

k jm

k

eee ∆minus+

minus

∆++

+

+∆++===

11

1ε ε ε



17







33

( ) 1sin1 le∆∆

∆minus xk

x

t ci

m

x

uc

t

u

part

partminus=

part

part








34

x

uuc

t

uu k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus

+

1

1

( ) xk x

t c

x

t cG m∆minus

minus

∆

∆

∆

∆+= cos1121

x

uc

t

u

part

partminus=

part

part



18









x

∆t

∆ x

A

35







Numerics vs Physics

∆t

∆ x

A

36



19











1le∆

∆

x

t c

37









Unsteadiness


Diffusion

2

2

x

u

x

uc

t

u

part

part=

part

part+

part

part micro

2

21

x

u

dx

dp

x

uu

t

u

part

part+minus=

part

part+

part

part

ρ

micro

ρ




20








where and

39

xk iC xk DGmm


( )( )

( )k

j

k

j

k

j

k

j

k

j

k

j

k

juuu

xuu

x

c

t

uu11211

1

22

minus+minus+

+

+minus∆

+minus∆

minus=∆

minus micro










40

xk iC xk DGmm


Im

Re

unit circle

G

2 D

C



21




41

Im

Re

unit circle

G

2 D

C






42


Im

Re

unit circle

G

2 D

C




22







43

C RC x 22 lele ∆

122

ltlt DC









stability limit


44

D

C

R x =∆












45



14




27






where



28

( ) sum=

+=

2

1

N

m

xik at met xε

L



15










29

k

j

k

j ε ε le+1

11

le=+

k

j

k j

Gε

ε


jmk xik at k

j e +

=ε





remembering that



30

2

11

12

xt

k

j

k

j

k

j

k

j

k

j

∆

+minus=

∆

minusminus+

+ ε ε ε micro

ε ε jmk

xik at k

j e +

=ε

2

2

x

u

t

u

part

part=

part

part micro

( ) ( ) ( ) x xik at k

j

x xik at k

j

xik t t ak

j jm

k jm

k jm

k

eee ∆minus+

minus

∆++

+

+∆++=== 11

1ε ε ε



16









31

( )1

2sin41

2

2 le

∆

∆

∆minus

xk

x

t m micro

( )1

2sin41

2

2 le

∆

∆

∆minus

xk

x

t m micro

( ) 12sin41

2

2 minusge

∆

∆

∆minus

xk

x

t m micro

2

2

x

u

t

u

part

part=

part

part micro








32

jmk xik at k

j e +

=ε x

ct

k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus+

+

2

11

1 ε ε ε ε

x

uc

t

u

part

partminus=

part

part

( ) ( ) ( ) x xik at k

j

x xik at k

j

xik t t ak

j jm

k jm

k jm

k

eee ∆minus+

minus

∆++

+

+∆++===

11

1ε ε ε



17







33

( ) 1sin1 le∆∆

∆minus xk

x

t ci

m

x

uc

t

u

part

partminus=

part

part








34

x

uuc

t

uu k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus

+

1

1

( ) xk x

t c

x

t cG m∆minus

minus

∆

∆

∆

∆+= cos1121

x

uc

t

u

part

partminus=

part

part



18









x

∆t

∆ x

A

35







Numerics vs Physics

∆t

∆ x

A

36



19











1le∆

∆

x

t c

37









Unsteadiness


Diffusion

2

2

x

u

x

uc

t

u

part

part=

part

part+

part

part micro

2

21

x

u

dx

dp

x

uu

t

u

part

part+minus=

part

part+

part

part

ρ

micro

ρ




20








where and

39

xk iC xk DGmm


( )( )

( )k

j

k

j

k

j

k

j

k

j

k

j

k

juuu

xuu

x

c

t

uu11211

1

22

minus+minus+

+

+minus∆

+minus∆

minus=∆

minus micro










40

xk iC xk DGmm


Im

Re

unit circle

G

2 D

C



21




41

Im

Re

unit circle

G

2 D

C






42


Im

Re

unit circle

G

2 D

C




22







43

C RC x 22 lele ∆

122

ltlt DC









stability limit


44

D

C

R x =∆












45



15










29

k

j

k

j ε ε le+1

11

le=+

k

j

k j

Gε

ε


jmk xik at k

j e +

=ε





remembering that



30

2

11

12

xt

k

j

k

j

k

j

k

j

k

j

∆

+minus=

∆

minusminus+

+ ε ε ε micro

ε ε jmk

xik at k

j e +

=ε

2

2

x

u

t

u

part

part=

part

part micro

( ) ( ) ( ) x xik at k

j

x xik at k

j

xik t t ak

j jm

k jm

k jm

k

eee ∆minus+

minus

∆++

+

+∆++=== 11

1ε ε ε



16









31

( )1

2sin41

2

2 le

∆

∆

∆minus

xk

x

t m micro

( )1

2sin41

2

2 le

∆

∆

∆minus

xk

x

t m micro

( ) 12sin41

2

2 minusge

∆

∆

∆minus

xk

x

t m micro

2

2

x

u

t

u

part

part=

part

part micro








32

jmk xik at k

j e +

=ε x

ct

k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus+

+

2

11

1 ε ε ε ε

x

uc

t

u

part

partminus=

part

part

( ) ( ) ( ) x xik at k

j

x xik at k

j

xik t t ak

j jm

k jm

k jm

k

eee ∆minus+

minus

∆++

+

+∆++===

11

1ε ε ε



17







33

( ) 1sin1 le∆∆

∆minus xk

x

t ci

m

x

uc

t

u

part

partminus=

part

part








34

x

uuc

t

uu k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus

+

1

1

( ) xk x

t c

x

t cG m∆minus

minus

∆

∆

∆

∆+= cos1121

x

uc

t

u

part

partminus=

part

part



18









x

∆t

∆ x

A

35







Numerics vs Physics

∆t

∆ x

A

36



19











1le∆

∆

x

t c

37









Unsteadiness


Diffusion

2

2

x

u

x

uc

t

u

part

part=

part

part+

part

part micro

2

21

x

u

dx

dp

x

uu

t

u

part

part+minus=

part

part+

part

part

ρ

micro

ρ




20








where and

39

xk iC xk DGmm


( )( )

( )k

j

k

j

k

j

k

j

k

j

k

j

k

juuu

xuu

x

c

t

uu11211

1

22

minus+minus+

+

+minus∆

+minus∆

minus=∆

minus micro










40

xk iC xk DGmm


Im

Re

unit circle

G

2 D

C



21




41

Im

Re

unit circle

G

2 D

C






42


Im

Re

unit circle

G

2 D

C




22







43

C RC x 22 lele ∆

122

ltlt DC









stability limit


44

D

C

R x =∆












45



16









31

( )1

2sin41

2

2 le

∆

∆

∆minus

xk

x

t m micro

( )1

2sin41

2

2 le

∆

∆

∆minus

xk

x

t m micro

( ) 12sin41

2

2 minusge

∆

∆

∆minus

xk

x

t m micro

2

2

x

u

t

u

part

part=

part

part micro








32

jmk xik at k

j e +

=ε x

ct

k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus+

+

2

11

1 ε ε ε ε

x

uc

t

u

part

partminus=

part

part

( ) ( ) ( ) x xik at k

j

x xik at k

j

xik t t ak

j jm

k jm

k jm

k

eee ∆minus+

minus

∆++

+

+∆++===

11

1ε ε ε



17







33

( ) 1sin1 le∆∆

∆minus xk

x

t ci

m

x

uc

t

u

part

partminus=

part

part








34

x

uuc

t

uu k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus

+

1

1

( ) xk x

t c

x

t cG m∆minus

minus

∆

∆

∆

∆+= cos1121

x

uc

t

u

part

partminus=

part

part



18









x

∆t

∆ x

A

35







Numerics vs Physics

∆t

∆ x

A

36



19











1le∆

∆

x

t c

37









Unsteadiness


Diffusion

2

2

x

u

x

uc

t

u

part

part=

part

part+

part

part micro

2

21

x

u

dx

dp

x

uu

t

u

part

part+minus=

part

part+

part

part

ρ

micro

ρ




20








where and

39

xk iC xk DGmm


( )( )

( )k

j

k

j

k

j

k

j

k

j

k

j

k

juuu

xuu

x

c

t

uu11211

1

22

minus+minus+

+

+minus∆

+minus∆

minus=∆

minus micro










40

xk iC xk DGmm


Im

Re

unit circle

G

2 D

C



21




41

Im

Re

unit circle

G

2 D

C






42


Im

Re

unit circle

G

2 D

C




22







43

C RC x 22 lele ∆

122

ltlt DC









stability limit


44

D

C

R x =∆












45



17







33

( ) 1sin1 le∆∆

∆minus xk

x

t ci

m

x

uc

t

u

part

partminus=

part

part








34

x

uuc

t

uu k

j

k

j

k

j

k

j

∆

minusminus=

∆

minusminus

+

1

1

( ) xk x

t c

x

t cG m∆minus

minus

∆

∆

∆

∆+= cos1121

x

uc

t

u

part

partminus=

part

part



18









x

∆t

∆ x

A

35







Numerics vs Physics

∆t

∆ x

A

36



19











1le∆

∆

x

t c

37









Unsteadiness


Diffusion

2

2

x

u

x

uc

t

u

part

part=

part

part+

part

part micro

2

21

x

u

dx

dp

x

uu

t

u

part

part+minus=

part

part+

part

part

ρ

micro

ρ




20








where and

39

xk iC xk DGmm


( )( )

( )k

j

k

j

k

j

k

j

k

j

k

j

k

juuu

xuu

x

c

t

uu11211

1

22

minus+minus+

+

+minus∆

+minus∆

minus=∆

minus micro










40

xk iC xk DGmm


Im

Re

unit circle

G

2 D

C



21




41

Im

Re

unit circle

G

2 D

C






42


Im

Re

unit circle

G

2 D

C




22







43

C RC x 22 lele ∆

122

ltlt DC









stability limit


44

D

C

R x =∆












45



18









x

∆t

∆ x

A

35







Numerics vs Physics

∆t

∆ x

A

36



19











1le∆

∆

x

t c

37









Unsteadiness


Diffusion

2

2

x

u

x

uc

t

u

part

part=

part

part+

part

part micro

2

21

x

u

dx

dp

x

uu

t

u

part

part+minus=

part

part+

part

part

ρ

micro

ρ




20








where and

39

xk iC xk DGmm


( )( )

( )k

j

k

j

k

j

k

j

k

j

k

j

k

juuu

xuu

x

c

t

uu11211

1

22

minus+minus+

+

+minus∆

+minus∆

minus=∆

minus micro










40

xk iC xk DGmm


Im

Re

unit circle

G

2 D

C



21




41

Im

Re

unit circle

G

2 D

C






42


Im

Re

unit circle

G

2 D

C




22







43

C RC x 22 lele ∆

122

ltlt DC









stability limit


44

D

C

R x =∆












45



19











1le∆

∆

x

t c

37









Unsteadiness


Diffusion

2

2

x

u

x

uc

t

u

part

part=

part

part+

part

part micro

2

21

x

u

dx

dp

x

uu

t

u

part

part+minus=

part

part+

part

part

ρ

micro

ρ




20








where and

39

xk iC xk DGmm


( )( )

( )k

j

k

j

k

j

k

j

k

j

k

j

k

juuu

xuu

x

c

t

uu11211

1

22

minus+minus+

+

+minus∆

+minus∆

minus=∆

minus micro










40

xk iC xk DGmm


Im

Re

unit circle

G

2 D

C



21




41

Im

Re

unit circle

G

2 D

C






42


Im

Re

unit circle

G

2 D

C




22







43

C RC x 22 lele ∆

122

ltlt DC









stability limit


44

D

C

R x =∆












45



20








where and

39

xk iC xk DGmm


( )( )

( )k

j

k

j

k

j

k

j

k

j

k

j

k

juuu

xuu

x

c

t

uu11211

1

22

minus+minus+

+

+minus∆

+minus∆

minus=∆

minus micro










40

xk iC xk DGmm


Im

Re

unit circle

G

2 D

C



21




41

Im

Re

unit circle

G

2 D

C






42


Im

Re

unit circle

G

2 D

C




22







43

C RC x 22 lele ∆

122

ltlt DC









stability limit


44

D

C

R x =∆












45



21




41

Im

Re

unit circle

G

2 D

C






42


Im

Re

unit circle

G

2 D

C




22







43

C RC x 22 lele ∆

122

ltlt DC









stability limit


44

D

C

R x =∆












45



22







43

C RC x 22 lele ∆

122

ltlt DC









stability limit


44

D

C

R x =∆












45












45

cfd numerical error

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