02 finite differences

7/30/2019 02 Finite Differences

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Finite Differences

Finite Difference Approximations

Simple geophysical partial differential equations

Finite differences - definitions

Finite-difference approximations to pdes Exercises

Acoustic wave equation in 2D

Seismometer equations

Diffusion-reaction equation

Finite differences and Taylor Expansion

Stability -> The Courant Criterion

Numerical dispersion

1Computational Seismology

dx

xfdxxffx

)()(


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Finite Differences

)(222

22

zyx

t

spcp

The acoustic

wave equation

- seismology- acoustics

- oceanography

- meteorology

Diffusion, advection,

Reaction

- geodynamics- oceanography

- meteorology

- geochemistry

- sedimentology

- geophysical fluid dynamics

P pressurec acoustic wave speed

s sources

pRCCCkCt v

C tracer concentrationk diffusivityv flow velocityR reactivityp sources

Partial Differential Equations in Geophysics


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Finite Differences

Finite differences

Finite volumes

- time-dependent PDEs

- seismic wave propagation- geophysical fluid dynamics

- Maxwells equations

- Ground penetrating radar

-> robust, simple concept, easy to

parallelize, regular grids, explicit method

Finite elements- static andtime-dependent PDEs

- seismic wave propagation- geophysical fluid dynamics

- all problems

-> implicit approach, matrix inversion, well founded,

irregular grids, more complex algorithms,

engineering problems

- time-dependent PDEs

- seismic wave propagation- mainly fluid dynamics

-> robust, simple concept, irregular grids, explicit

method

Numerical methods: properties


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Finite Differences

Particle-based

methods

Pseudospectral

methods

- lattice gas methods

- molecular dynamics- granular problems

- fluid flow

- earthquake simulations

-> very heterogeneous problems, nonlinear problems

Boundary element

methods

- problems with boundaries (rupture)

- based on analytical solutions- only discretization of planes

-> good for problems with special boundary conditions

(rupture, cracks, etc)

- orthogonal basis functions, special case of FD

- spectral accuracy of space derivatives- wave propagation, ground penetrating radar

->regular grids, explicit method, problems with

strongly heterogeneous media

Other numerical methods


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Finite Differences

What is a finite difference?

Common definitions of the derivative of f(x):

dx

xfdxxff

dxx

)()(lim

0

dxdxxfxff

dxx )()(lim

0

dx

dxxfdxxff

dxx

2

)()(lim

0

These are all correct definitions in the limit dx->0.

But we want dx to remain FINITE


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Finite Differences

What is a finite difference?

The equivalent approximat ions of the derivatives are:

dx

xfdxxffx

)()(

dx

dxxfxffx

)()(

dx

dxxfdxxffx

2

)()(

forward difference

backward difference

centered difference


7/27Finite Differences

Thebigquestion:

How good are the FD approximations?

This leads us to Taylor series....



Taylor Series

Taylor series are expansions of a function f(x) for somefinite distance dx to f(x+dx)

What happens, if we use this expression for

dx

xfdxxffx

)()(

?

...)(!4

)(!3

)(!2

)(dx)()( ''''4

'''3

''2

' xfdx

xfdx

xfdx

xfxfdxxf



Taylor Series

... that leads to :

The error of the first derivative using the forward

formulation is of order dx.

Is this the case for other formulations of the derivative?

Lets check!

)()(

...)(!3

)(!2

)(dx1)()(

'

'''3

''2

'

dxOxf

xfdx

xfdx

xfdxdx

xfdxxf



... with the centeredformulation we get:

The error of the first derivative using the centered

approximation is of order dx2

.

This is an important results: it DOES matter which formulation

we use. The centered scheme is more accurate!

Taylor Series

)()(

...)(!3

)(dx1)2/()2/(

2'

'''3

'

dxOxf

xfdx

xfdxdx

dxxfdxxf



xj 1 xj xj 1 xj 2 xj 3

f xj( )

dx h

desired x location

What is the (approximate) value of the function or its (first,

second ..) derivative at the desired location ?

How can we calculate the weights for the neighboring points?

x

f(x)

Alternative Derivation



Lets try Taylors Expansion

f x( )

dx

x

f(x)

dxxfxfdxxf )(')()( (1)

(2)

we are looking for something like

f x w f xi ji

index jj L

( ) ( )

( ),

( ) ( ) 1

Alternative Derivation

dxxfxfdxxf )(')()(



deriving the second-order scheme

af af af dx '

bf bf bf dx '

af bf a b f a b f dx( ) ( ) 'the solution to this equation for a and b leads to

a system of equations which can be cast in matrix form

dxba

ba

/1

0

0

1

ba

baInterpolation Derivative

2nd order weights



Taylor Operators

... in matrix form ...

0

1

11

11

b

a

Interpolation Derivative

... so that the solution for the weights is ...

dxb

a

/1

0

11

11

0

1

11

11 1

b

a

dxb

a

/1

0

11

11 1



... and the result ...

Interpolation Derivative

Can we generalise this idea to longer operators?

2/1

2/1

b

a

1

1

2

1

dxb

a

Let us start by extending the Taylor expansion beyond f(xdx):

Interpolation and difference weights



'''!3

)2(''!2

)2(')2()2(32

fdxfdxfdxfdxxf *a |

*b |

*c |

*d |

... again we are looking for the coefficients a,b,c,d with whichthe function values at x(2)dx have to be multiplied in order

to obtain the interpolated value or the first (or second) derivative!

... Let us add up all these equations like in the previous case ...

'''!3

)(''

!2

)(')()(

32

fdx

fdx

fdxfdxxf

'''!3)(''

!2)(')()(

32

fdxfdxfdxfdxxf

'''!3

)2(''

!2

)2(')2()2(

32

fdx

fdx

fdxfdxxf

Higher order operators



dfcfbfaf

)( dcbaf

)22(' dcbadxf

)222

2(''2 d

cbafdx

)6

8

6

1

6

1

6

8

('''3

dcbafdx

... we can now ask for the coefficients a,b,c,d, so that the

left-hand-side yields either f,f,f,f ...

Higher order operators



1 dcba

022 dcba

0222

2 dcb

a

06

8

6

1

6

1

6

8 dcba

... if you want the interpolated value ...

... you need to solve the matrix system ...

Linear system



High-order interpolation

0

0

0

1

6/86/16/16/8

22/12/12

2112

1111

d

c

b

a

... with the result after inverting the matrix on the lhs ...

6/1

3/2

3/2

6/1

d

c

b

a

... Interpolation ...



0

0

/1

0

6/86/16/16/8

22/12/12

2112

1111

dx

d

c

b

a

... with the result ...

6/1

3/4

3/4

6/1

2

1

dx

d

c

b

a

... first derivative ...

First derivative



Our first FD algorithm (ac1d.m) !

)( 222

22

zyx

t spcp

P pressurec acoustic wave speeds sources

Problem: Solve the 1D acoustic wave equation using the finiteDifference method.

Solution:

2

2

22

)()(2

)()(2)()(

sdtdttptp

dxxpxpdxxpdx

dtcdttp



Problems: Stability

2

2

22

)()(2

)()(2)()(

sdtdttptp

dxxpxpdxxpdx

dtcdttp

1 dx

dtc

Stability: Careful analysis using harmonic functions shows thata stable numerical calculation is subject to special conditions(conditional stability). This holds for many numerical problems.(Derivation on the board).



Problems: Dispersion

2

2

22

)()(2

)()(2)()(

sdtdttptp

dxxpxpdxxpdx

dtcdttp

Dispersion: The numerical

approximation hasartificial dispersion,in other words, the wavespeed becomes frequencydependent (Derivation inthe board).

You have to find afrequency bandwidthwhere this effect is small.The solution is to use asufficient number of gridpoints per wavelength.

True velocity



Our first FD code!

2

2

22

)()(2

)()(2)()(

sdtdttptp

dxxpxpdxxpdx

dtcdttp

% Time stepping

for i=1:nt,

% FD

disp(sprintf(' Time step : %i',i));

for j=2:nx-1

d2p(j)=(p(j+1)-2*p(j)+p(j-1))/dx^2; % space derivative

end

pnew=2*p-pold+d2p*dt^2; % time extrapolation

pnew(nx/2)=pnew(nx/2)+src(i)*dt^2; % add source term

pold=p; % time levels

p=pnew;p(1)=0; % set boundaries pressure free

p(nx)=0;

% Display

plot(x,p,'b-')

title(' FD ')

drawnow

end



Snapshot Example

0 1000 2000 3000 4000 5000 6000

0

500

1000

1500

2000

2500

3000

Distance (km)

Time(s)

Velocity 5 km/s


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Finite Differences

Seismogram Dispersion


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Finite Differences - Summary

Conceptually the most simple of the numerical methods and

can be learned quite quickly

Depending on the physical problem FD methods are

conditionally stable (relation between time and space

increment) FD methods have difficulties concerning the accurate

implementation ofboundary conditions (e.g. free surfaces,

absorbing boundaries)

FD methods are usually explicit and therefore very easy to

implement and efficient on parallel computers FD methods work best on regular, rectangular grids

02 finite differences

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