1. Simple FD
Gerard T. Schuster

2. Outline
Basics
Accuracy
Stability
Dispersion
Fourier Finite Difference

3. Basics of Finite Differences
The discretization of differential operators is not unique
1st-order approximations are one-sided approximations, full lines in figure:
2nd-order difference has higher accuracy but does not involve values at grid point xj dotted line in figure
Demands on numerical stability as well as numerical accuracy can not be satisfied in all cases

4. Explicit Methods
Advance numerical solution in time from the old time level t(n) to t(n+1) with spatial difference operator H
Explicit methods:
Spatial derivatives are calculated using the values at the old time level U(n)
Extrapolating the solution into the future
Time step t is restricted, e.g. CFL-condition
Easy to implement, parallelization possible
Most numerical methods, operator splitting

5. Outline
Basics
Accuracy
Stability
Dispersion
Fourier Finite Difference

6. Accuracy I
The accuracy of difference approximations can be tested by simple Taylor-series expansion
For the right-sided approximation we obtain
Adding the two expressions we see the more accurate central differences

7. central difference on a non-equidistant grid
Accuracy II
right-sided, 1st - order
left-sided, 1st - order
central difference, 2nd - order
central difference on a non-equidistant grid

8. 2nd Derivatives
A simple approximation of the 2nd-derivative on an equidistant grid can be obtained by summing up the left- an right-sided finite difference expressions, subtract 2fj
Higher order approximations involve more grid points, e.g. 4-order expression
Higher order: Be careful with the corresponding boundary conditions

9. Accuracy of 2nd Derivatives
Simplest approximation of the 2nd-derivative on an equidistant grid
Again, simple test function: f = C eikx and same procedure as before
See again 2x0, when x0
Approximation is 2nd-order, waves are dispersive with ~k2
Such difference scheme for diffusion problems, Laplace and Poisson equation

10. Accuracy of waves
Compare finite difference expression with differential operator, e.g. central difference and 1st-derivative
Simple test function: f = C eikx
Only small values of kx are well represented by sin(kx)
k has to be small or the wavelength 2/k has to large
The smaller k the less diffusion, transport of waves is dispersive
Errors are always seen on the smallest scales, zigzag-pattern

11. Outline
Basics
Accuracy
Stability
Dispersion
Fourier Finite Difference

12. Stability
The stability of any numerical method is measured by the amplification of numerical errors in time
n is the error at time step tn and g is the so-called amplification factor (or amplification matrix for systems of equations):
Numerical stability requires that the amplification of errors remains limited, i.e.
Only for simple, linear discretizations allow a rigorous analysis of the amplification matrix, non-linear systems yield to non-linear numerical schemes

13. Calculation of Stability
Taking a 1-dimensional interval X = [X1,X2], dividing it into (J-1)-subintervals where a function f(x) is defined on this interval X at distinct grid points xj with 1j J and f(xj)=fj
Fourier decomposition of f(x) on X, assume an equidistant grid with x
Fourier modes are orthogonal, therefore single modes can be analyzed to calculate stability of the numerical scheme

14. Example: Von Neumann Calculation of Instability
∂P/∂t=-c∂P/∂x (1)
∂^2P/∂t^2=-c∂(∂P/∂t)/∂x = c^2∂^2P/∂x^ (2)
∂P/∂t = (P P )/dt ;∂P/∂x=(P - P )/(2dx) (3)
j j j j-1
t t t t
Substitute (3) into (1) we get
P = P -cdt/(2dx)(P - P ) (4)
j j j j-1
t t t t
Substitute P = e e into (3) we get
ikjdx -iwt
e e = e cdt/(2dx)(e e )e (5)
ikjdx -iw(t+dt)
ikjdx -iwt
ik(j+1)dx ik(j-1)dx iwt
Divide by e we get
ikjdx – iwt
g= e = icdt/(dx)sin{kdx} ; |g|=sqrt{1+(cdt/dxsin{kdx})^2}
Stability demands |g|<1, which is impossible. Above is unstable because errors grow with time
– iwdt
Amplification
factor because
Pjn=gnPj0

15. Example: Von Neumann Calculation of Stability
∂P/∂t=-c∂P/∂x (1)
∂^2P/∂t^2=-c∂(∂P/∂t)/∂x = c^2∂^2P/∂x^ (2)
∂P/∂t = (P P )/dt ;∂P/∂x=(P - P )/(2dx) (3)
j j j j-1
t t t t
Substitute (3) into (1) we get
t t t t
P = 0.5(P + P )
j j j-1
t t t
P = P -cdt/(2dx)(P - P ) ; (4)
j j j j-1
Substitute P = e e into (4) we get
ikjdx -iwt
eikjdx-iw(t+dt)= {cos(kdx) - icdt/(dx)sin[kdx] }eikjdx-iwt;
or dividing by eikjdx-iwt
g= e = cos(kdx) - icdt/(dx)sin[kdx] ;
Stability demands |g|<1, which it is if cdt/dx<1. Above is conditionally stable because errors damps with time if CFL holds (a necessary but not suficient condition for
stability). As along as accuracy of FD does not exceed highest order derivative, CFL for symmetric stencils is both sufficient and necessary.
– iwdt
Amplification
factor because
Pjn=gnPj0
|g|=sqrt{cos(kdx)^2 +(cdt/dxsin{kdx})^2}

16. Example: Von Neumann Calculation of Stability
t t t t
P = 0.5(P + P )
j j j-1
t t t
P = P -cdt/(2dx)(P - P ); (4)
j j j+1 j-1
j j j j j j j-1
P = P -cdt/(2dx)(P - P ) + 0.5dx^2(P P P )/dx^2
t t t t t t t
This is a Laplacian, and emulates a diffusion term that damps errors;
this is why Lax worked. Less accuracy but better stability

17. Outline
Basics
Accuracy
Stability
Dispersion
Fourier Finite Difference

18. Problems: Dispersion
Dispersion: The numerical approximation has artificial dispersion,
in other words, the wave speed becomes frequency dependent.
You have to find a frequency bandwidth where this effect is small.
The solution is to use a sufficient number of grid points per wavelength.
True velocity

19. Snapshot Example

20. Seismogram Dispersion

21. Outline Basics Accuracy Stability Dispersion
Pseudo-Spectral or Fourier Finite-Difference Modeling

22. What is a Pseudo-spectral Method? Numerical Methods in Geophysics
Spectral solutions to time-dependent PDEs are formulated in the frequency-wavenumber domain and solutions are obtained in terms of spectra (e.g. seismograms). This technique is particularly interesting for geometries where partial solutions in the -k domain can be obtained analytically (e.g. for layered models).
In the pseudo-spectral approach - in a finite-difference like manner - the PDEs are solved pointwise in physical space (x-t). However, the space derivatives are calculated using orthogonal functions (e.g. Fourier Integrals, Chebyshev polynomials). They are either evaluated using matrix-matrix multiplications or the fast Fourier transform (FFT).
Numerical Methods in Geophysics
The Fourier Method

23. MATLAB Code FD Approx. to 1-D Acoustic Wave Equation
a = (c*dt/dx)^2;
for t=1:nt
f = force(:,:,t);
dfuture = dpres - 2* dpast + a*del(dpres);
dfuture = dfuture + f;
dpast=dpresent;
dpresent=dfuture;
Dp2=fft(p);Dp2=k^2*Dp2;dp2=ifft(Dp2);
dp2
end

24. Fourier Derivatives
.. let us recall the definition of the derivative using Fourier integrals ...
... we could either ...
1) perform this calculation in the space domain by convolution
2) actually transform the function f(x) in the k-domain and back
Numerical Methods in Geophysics
The Fourier Method

25. The Fast Fourier Transform Numerical Methods in Geophysics
... the latter approach became interesting with the introduction of the
Fast Fourier Transform (FFT). What’s so fast about it ?
The FFT originates from a paper by Cooley and Tukey (1965, Math.
Comp. vol ) which revolutionised all fields where Fourier
transforms where essential to progress.
The discrete Fourier Transform can be written as
Numerical Methods in Geophysics
The Fourier Method

26. The Fast Fourier Transform
... this can be written as matrix-vector products ...
for example the inverse transform yields ...
.. where ...
Numerical Methods in Geophysics
The Fourier Method

27. The Fast Fourier Transform
... the FAST bit is recognising that the full matrix - vector multiplication
can be written as a few sparse matrix - vector multiplications
(for details see for example Bracewell, the Fourier Transform and its
applications, MacGraw-Hill) with the effect that:
Number of multiplications
full matrix FFT
N Nlog2N
this has enormous implications for large scale problems.
Note: the factorisation becomes particularly simple and effective
when N is a highly composite number (power of 2).
Numerical Methods in Geophysics
The Fourier Method

28. The Fast Fourier Transform
Number of multiplications
Problem full matrix FFT Ratio full/FFT
1D (nx=512) x x
1D (nx=2096)
1D (nx=8384)
.. the right column can be regarded as the speedup of an algorithm
when the FFT is used instead of the full system.
Numerical Methods in Geophysics
The Fourier Method

29. Acoustic Wave Equation - Fourier Method
let us take the acoustic wave equation with variable density
the left hand side will be expressed with our
standard centered finite-difference approach
... leading to the extrapolation scheme ...
Numerical Methods in Geophysics
The Fourier Method

30. Acoustic Wave Equation - Fourier Method
where the space derivatives will be calculated using the Fourier Method.
The highlighted term will be calculated as follows:
multiply by 1/
... then extrapolate ...
Numerical Methods in Geophysics
The Fourier Method

31. Acoustic Wave Equation - 3D
.. where the following algorithm applies to each space dimension ...
Numerical Methods in Geophysics
The Fourier Method

32. ... with the usual Ansatz we obtain ... leading to
Fourier Method - Dispersion and Stability
... with the usual Ansatz
we obtain
... leading to
Numerical Methods in Geophysics
The Fourier Method

33. What are the consequences?
Fourier Method - Dispersion and Stability
What are the consequences?
a) when dt << 1, sin-1 (kcdt/2) kcdt/2 and w/k=c
-> practically no dispersion
b) the argument of asin must be smaller than one.
Numerical Methods in Geophysics
The Fourier Method

34. Fourier Method - Comparison with FD - 10Hz
Example of acoustic 1D wave simulation.
FD 3 -point operator
red-analytic; blue-numerical; green-difference
Numerical Methods in Geophysics
The Fourier Method

35. Fourier Method - Comparison with FD - 10Hz
Example of acoustic 1D wave simulation.
FD 5 -point operator
red-analytic; blue-numerical; green-difference
Numerical Methods in Geophysics
The Fourier Method

36. Fourier Method - Comparison with FD - 10Hz
Example of acoustic 1D wave simulation.
Fourier operator
red-analytic; blue-numerical; green-difference
Numerical Methods in Geophysics
The Fourier Method

37. Fourier Method - Comparison with FD - 20Hz
Example of acoustic 1D wave simulation.
FD 3 -point operator
red-analytic; blue-numerical; green-difference
Numerical Methods in Geophysics
The Fourier Method

38. Fourier Method - Comparison with FD - 20Hz
Example of acoustic 1D wave simulation.
FD 5 -point operator
red-analytic; blue-numerical; green-difference
Numerical Methods in Geophysics
The Fourier Method

39. Fourier Method - Comparison with FD - 20Hz
Example of acoustic 1D wave simulation.
Fourier operator
red-analytic; blue-numerical; green-difference
Numerical Methods in Geophysics
The Fourier Method

40. Fourier Method - Comparison with FD - Table
Difference (%) between numerical and analytical solution
as a function of propagating frequency
Simulation time
5.4s
7.8s
33.0s
Numerical Methods in Geophysics
The Fourier Method