Why do we need higher order operators?
Taylor Operators
One-sided operators
High-order Taylor Extrapolation
Runge-Kutta Method
Truncated Fourier Operators - Derivative and Interpolation
Accuracy of high-order schemes derivatives
Numerical Methods in Geophysics:High-Order Operators
Numerical Methods in Geophysics High-order operators
Why do we need higher-order operators?
For realistic problems the first and second order methodsare not accurate enough. So far we only used information from the nearest neighbouring grid points. Could we improve the accuracy of the derivative operators by using more information (on both sides)?
Length of operator Derivative
n g n g n g n g n g
Derivative
Numerical Methods in Geophysics High-order operators
Taylor Operators
... like so often we look at Taylor series ...Remember how we derived the second-order scheme
a f a f a f d x+ ≈ + '
b f b f b f d x− ≈ − '
⇒ + ≈ + + −+ −a f b f a b f a b f d x( ) ( ) '
the solution to this equation for a and b leads to a system of equation which can be cast in matrix form
Interpolation Derivative
01
=−=+
baba
dxbaba
/10
=−=+
Numerical Methods in Geophysics High-order operators
Taylor Operators
... in matrix form ...
Numerical Methods in Geophysics High-order operators
⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛− 0
111
11ba
Interpolation Derivative
... so that the solution for the weights is ...
⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛− dxb
a/10
1111
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−
=⎟⎟⎠
⎞⎜⎜⎝
⎛−
01
1111 1
ba
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−
=⎟⎟⎠
⎞⎜⎜⎝
⎛−
dxba
/10
1111 1
Taylor Operators
... and the result ...
Interpolation Derivative
Can we generalise this idea to longer operators?
⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛2/12/1
ba
⎟⎟⎠
⎞⎜⎜⎝
⎛−
=⎟⎟⎠
⎞⎜⎜⎝
⎛1
12
1dxb
a
Let us start by extending the Taylor expansion beyond f(x±dx):
Numerical Methods in Geophysics High-order operators
Taylor Operators
'''!3
)2(''!2)2(')2()2(
32
fdxfdxfdxfdxxf −+−≈−*a |
*b |
*c |
*d |
'''!3)(''
!2)(')()(
32
fdxfdxfdxfdxxf −+−≈−
'''!3)(''
!2)(')()(
32
fdxfdxfdxfdxxf +++≈+
'''!3
)2(''!2)2(')2()2(
32
fdxfdxfdxfdxxf +++≈+
... 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 ...
Numerical Methods in Geophysics High-order operators
Taylor Operators
≈+++ +++−−− dfcfbfaf
++++ )( dcbaf
+++−− )22(' dcbadxf
++++ )222
2(''2 dcbafdx
)68
61
61
68('''3 dcbafdx ++−−
... we can now ask for the coefficients a,b,c,d, so that theleft-hand-side yields either f,f’,f’’,f’’’ ...
Numerical Methods in Geophysics High-order operators
Taylor Operators
... if you want the interpolated value ...
1=+++ dcba
022 =++−− dcba
0222
2 =+++ dcba
068
61
61
68
=++−− dcba
... you need to solve the matrix system ...
Numerical Methods in Geophysics High-order operators
Taylor Operators
... Interpolation ...
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−−
−−
0001
6/86/16/16/822/12/1221121111
dcba
... with the result after inverting the matrix on the lhs ...
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−
−
=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
6/13/23/26/1
dcba
Numerical Methods in Geophysics High-order operators
Taylor Operators
... first derivative ...
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−−
−−
00/10
6/86/16/16/822/12/1221121111
dx
dcba
... with the result ...
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−
−=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
6/13/43/4
6/1
21dx
dcba
Numerical Methods in Geophysics High-order operators
Taylor Operators
... third derivative ...
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−−
−−
3/1000
6/86/16/16/822/12/1221121111
dxdcba
... with the result ...
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−
−
=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
2/11
12/1
13dx
dcba
... why did it not work for the 2nd derivative ?
Numerical Methods in Geophysics High-order operators
Taylor Operators
''''!4)2('''
!3)2(''
!2)2(')2()2(
432
fdxfdxfdxfdxfdxxf +−+−≈−*a |
*b |
*c |
*d |
*e |
''''!4)('''
!3)(''
!2)(')()(
432
fdxfdxfdxfdxfdxxf +−+−≈−
fxf =)(
''''!4)('''
!3)(''
!2)(')()(
432
fdxfdxfdxfdxfdxxf ++++≈+
''''!4)2('''
!3)2(''
!2)2(')2()2(
432
fdxfdxfdxfdxfdxxf ++++≈+
... note that we had to add the 4th derivatives which will give us the required constraints on the coefficients a,b,c,d,e
Numerical Methods in Geophysics High-order operators
Taylor Operators
≈++++ +++−−− efdfcfbfaf
+++++ )( edcbaf
++++−− )202(' edbadxf
+++++ )22
02
2(''2 edbafdx
++++−− )68
610
61
68('''3 edbafdx
)32
2410
241
32(''''4 edbafdx ++++
... so finally we end up with the system ...
Numerical Methods in Geophysics High-order operators
Taylor Operators
... if you want the second derivative ...
0=++++ edcba
0202 =+++−− edba
122
02
2 =++++ edba
068
610
61
68
=+++−− edba
032
2410
241
32
=++++ edba
... you need to solve the matrix system ...
... could we find interpolation weights like this ?
Numerical Methods in Geophysics High-order operators
Taylor Operators
... second derivative ...
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
=
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
−−
−−
00
/100
3/224/1024/13/26/86/106/16/8
22/102/122101211111
2dx
edcba
... with the result ...
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
−
−
−
=
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
12/13/42/5
3/412/1
12dx
edcba
Numerical Methods in Geophysics High-order operators
Taylor Operators
... Fornberg1 (1996) gives a closed-form expression for the first derivative weights ...
⎪⎩
⎪⎨⎧
−+−
=
+
0)!2/()!2/(
)!2/()1( 21
1, jpjpj
pw
j
jpif j= 1, 2,..., p/2
if j= 0
... where p(even) is the order of accuracy, j is the x-position of the weight.
1Fornberg, B., A practical guide to pseudospectral methods, Cambridge University Press.
Numerical Methods in Geophysics High-order operators
One-sided operators
Before we look at how the operators look like as they grow longerwe investigate whether we can approximate a derivative near a physical boundary ....
derivative
domain boundary
Let us follow the same route as before and use Taylor series.Let’s start with a first order scheme.
Numerical Methods in Geophysics High-order operators
One-sided operators
... so we have to look for information on one side only
a f a f a f d x+ ≈ + '
)2(' dxbfbfbf +≈++
dxfbafbabfaf ')2()( +++≈+⇒ +++
the solution to this equation for a and b leads to a system of equations which can be cast in matrix form
Derivative
dxbaba
/120
=+=+
⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛10
2111
ba
... and the solution is ...
Numerical Methods in Geophysics High-order operators
One-sided operators
⎟⎟⎠
⎞⎜⎜⎝
⎛ −=⎟⎟
⎠
⎞⎜⎜⎝
⎛111
dxba
This is our well known definition of the centered derivative, butit will be defined not right at the boundary but dx/2 away from it!
Derivative
Let us extend this to the right and find higher-order operators
Numerical Methods in Geophysics High-order operators
One-sided operators
*a |
*b |
*c |
*d | '''!3)3(''
!2)3(')3(
'''!3)2(''
!2)2(')2(
'''!3)(''
!2)(')(
32
32
32
fdxfdxfdxff
fdxfdxfdxff
fdxfdxfdxff
ff
+++≈
+++≈
+++≈
=
+++
++
+
... again we multiply by our coefficients and add everything up ...
Numerical Methods in Geophysics High-order operators
One-sided operators
≈+++ ++++++ dfcfbfaf
++++ )( dcbaf
++++ )320(' dcbdxf
++++ )292
210(''2 dcbfdx
)6
2734
610('''3 dcbfdx +++
... to obtain the derivatives we have to solve the system ...
Numerical Methods in Geophysics High-order operators
One-sided operators
... if you want the first derivative ...
0=+++ dcba
1320 =+++ dcb
0292
210 =+++ dcb
06
2734
610 =+++ dcb
... you need to solve the matrix system ...
Numerical Methods in Geophysics High-order operators
One-sided operators
... Interpolation ...
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
00/10
6/273/46/102/922/10
32101111
dx
dcba
... with the result after inverting the matrix on the lhs ...
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−
−
=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
3/12/3
36/11
1dx
dcba
Numerical Methods in Geophysics High-order operators
Taylor Operators
-5 0 5-1.5
-1
-0.5
0
0.5
1
1.5Order of accuracy: 2
-5 0 5-1.5
-1
-0.5
0
0.5
1
1.5Order of accuracy: 4
-5 0 5-1.5
-1
-0.5
0
0.5
1
1.5Order of accuracy: 8
-5 0 5-1.5
-1
-0.5
0
0.5
1
1.5Order of accuracy: 16
2-point 4-point
8-point 16-point
Numerical Methods in Geophysics High-order operators
0 2 4 6
-1
0
1
2
0 2 4
-2
0
2
4
0 2 4 6
-5
0
5
0 2 4
-10
0
10
Taylor Operators - one sided
3-point5-point
7-point 8-point
Note the exploding coefficients with increasing operator length
Numerical Methods in Geophysics High-order operators
Taylor Operators - Summary
Finite-difference operators with high-order accuracy can be derived using Taylor series. For two-sided operators the coefficients rapidly decrease. For one-sided operators the coefficients get larger with increasing operator length.
Now that we improved the accuracy of the space derivatives, how can we improve the accuracy of the time extrapolation?
Let’s look at the Taylor scheme ...
Numerical Methods in Geophysics High-order operators
High-order Taylor extrapolation
Let us look at the acoustic wave equation
Scpcp zx2222 )( +∂+∂=
.. we now know how to accurately calculate the r.h.s. of this equation...our standard FD scheme for the time extrapolation yields
pdtdttptpdttp 2)()(2)( +−−=+
... extending this to higher orders leads to the scheme ...
nN
n
n
pn
dtdttptpdttp 2
1
2
)!2(2)()(2)( ∑
=
+−−=+
... this has interesting consequences as we only need the even orders of the time derivative, which we can easily calculate ...
Numerical Methods in Geophysics High-order operators
High-order Taylor extrapolation
... since ...
Scpcp zxt22222 )( +∂+∂=∂
... we have also ...
Scpcp ttzxt2222224 )( ∂+∂∂+∂=∂
... or ...
Scpcp ttzxt4242226 )( ∂+∂∂+∂=∂
... so we can loop through our algorithm as long as we want (N times) to achieve higher-order accuracy in the time-extrapolation scheme ...
nN
n
n
pn
dtdttptpdttp 2
1
2
)!2(2)()(2)( ∑
=
+−−=+
.. however we have to be careful how the spatial and temporal operators behave and whether the accuracy of the solution to the pde actually improves!
Numerical Methods in Geophysics High-order operators
High-order extrapolation
... often we have to extrapolate a first order system ...
),( tTfTt =∂
... or ...
τρ xt x
u ∂=∂)(
1
... and we initially used a simple scheme like ...
),(dt1 jjjj tTfTT +≈+
... this scheme is also known as the Euler scheme and is of little practical use ...
Numerical Methods in Geophysics High-order operators
High-order extrapolation
... how about predicting a value and then averaging ....
),(dt1*
jjjj tTfTT +≈+
this is our first guess (equivalent to the Euler scheme) and now we use this value to improve our solution ...
[ ]),(),(21
1*
11 +++ ++= jjjjjj tTftTfdtTT
Numerical Methods in Geophysics High-order operators
High-order extrapolation
... leading to a general algorithm like ...
)(21
),(),(
211
112
1
1
kkyy
kyxdxfkyxdxfk
dxxx
nn
nn
nn
nn
++=
+==
+=
+
+
+
For n=0,1,2,3,...N-1
End
predictor
corrector
called predictor-corrector or modified Euler or .... scheme ...... how does this apply to our cooling problem ?
Numerical Methods in Geophysics High-order operators
High-order extrapolation
)(21
),(),(
211
112
1
1
kkyy
kyxdxfkyxdxfk
dxxx
nn
nn
nn
nn
++=
+==
+=
+
+
+
For n=0,1,2,3,...N-1
End
τ/TdtdT
−=nnn TdtTT
τ−=+1
... this was the simplest scheme ...with the modified Euler scheme we get
)(21
)(
211
12
1
1
kkTT
kTdtk
Tdtk
dttt
nn
n
n
nn
++=
+−=
−=
+=
+
+
τ
τ
For n=0,1,2,3,...N-1
End
Numerical Methods in Geophysics High-order operators
High-order extrapolation
... the next more accurate scheme is the fourth order Runge-Kutta method, an extension of the predictor-corrector scheme ...
)22(61
),()2/,()2/,(
),(
43211
314
22/13
12/12
1
1
kkkkyy
kyxdxfkkyxdxfkkyxdxfk
yxdxfkdxxx
nn
nn
nn
nn
nn
nn
++++=
+=+=+=
=+=
+
+
+
+
+
For n=0,1,2,3,...N-1
End
Numerical Methods in Geophysics High-order operators
High-order extrapolation
... Matlab sample code ...
for i=1:nt,
t(i)=i*dt;T(i+1)=T(i)-dt/tau*T(i); % EulerTa(i+1)=exp(-dt*i/tau); % Analytical solutionTi(i+1)=T(i)*(1+dt/tau)^(-1); % implicitTm(i+1)=(1-dt/(2*tau))/(1+dt/(2*tau))*Tm(i); % mixed implicit-explicit
k1=-dt/tau*Te(i);k2=-dt/tau*(Te(i)+k1);Te(i+1)=Te(i)+1/2*(k1+k2); % predictor-corrector
k1=-dt/tau*Tr(i);k2=-dt/tau*(Tr(i)+k1/2);k3=-dt/tau*(Tr(i)+k2/2); k4=-dt/tau*(Tr(i)+k3);Tr(i+1)=Tr(i)+1/6*(k1+2*k2+2*k3+k4); % Runge-Kutta
end
... with the results ...
Numerical Methods in Geophysics High-order operators
High-order extrapolation
1 2 3 4 5 6 7 8 9-0.5
0
0.5
1
Time(s )
Tem
pera
ture
dt=1; tau=0.7
Comparison of low order implicit, mixed implicit-explicit (Crank-Nicholson), modified Euler (predictor-corrector), Runge-Kutta (fourth order)
for Newtonian Cooling
blue - Euler
red - Crank-Nicholson
black - analytic solution
magenta - Runge-Kutta
green - implicit
Runge-Kutta is the winner!
Numerical Methods in Geophysics High-order operators
High-order extrapolation
1 1.5 2 2.50.1
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.2
Time(s )
Tem
pera
ture
dt=0.5; tau=0.7
Numerical Methods in Geophysics High-order operators
Comparison of low order implicit, mixed implicit-explicit (Crank-Nicholson), modified Euler (predictor-corrector), Runge-Kutta (fourth order)
for Newtonian Cooling
blue - Euler
red - Crank-Nicholson
black - analytic solution
magenta - Runge-Kutta
green - implicit
Fourier Coefficients
We will now approach the problem of finding high-order space operators from a completely different viewpoint: Fourier Integrals.
Let us recall
∫
∫∞
∞−
∞
∞−
−
=
=
dxexfkF
dkekFxf
ikx
ikx
)()(
)(21)(π
... where f(x) is an arbitrary function and F(k) is its Fourier spectrum. Note that there are several different definitions, which distinguish themselves through normalisation constants and the sign convention in the exponent.
Numerical Methods in Geophysics High-order operators
Fourier Coefficients
... how can we express the derivative of a function using these expressions?
∫
∫∞
∞−
−
∞
∞−
−
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂=∂
dkekikF
dkekFxf
ikx
ikxxx
)(
)()(
... because F(k) clearly does not depend on x. Let us define ...
ikkP −=)(
... note that we use capital letters to denote fields in the wavenumber domain ...
Numerical Methods in Geophysics High-order operators
Fourier Coefficients
... so that ...
∫∞
∞−
−=∂ dkekFkPxf ikxx )()()(
... now a bell rings ... and we remember the Convolution Theorem which says “a multiplication in the wavenumber domain is a convolution in the space domain” which can be expressed as
∫∞
∞−
−=∂ dxxfxxpxfx )'()'()(
... note the small letters as we are now in the space domain!
In the discrete and band-limited world this integral turns into a convolution sum.This is the most general way of describing a differential operator. It comprises all
the cases from two-point, local operators up to the exact spectral operator.
Numerical Methods in Geophysics High-order operators
Fourier Coefficients
.. we now want to express the operator p(x) in the space domain ...we first have to get rid of the infinities as we are in a discrete domain where
we have a maximum frqeuency (wavenumber), the Nyquist frequency.This band-limitation can be expressed using Heaviside functions.
⎩⎨⎧
=01
)(xHif x>0
if x<0
in our example the limitation in k can be expressed as
( ))()()( NyNy kkHkkHikkP −++=
we now have to transform this back into the space domain
∫∞
∞−
−= dkekPxp ikx)()(
Numerical Methods in Geophysics High-order operators
Fourier Coefficients
... to obtain ...
[ ])cos()sin(1)( 2 xkxkxkx
xp NyNyNy −=π
... in a staggered scheme we need to discretise space like ...
⎩⎨⎧
=+=+
dxkdxnx
Ny
n
/)2/1(2/1
π
... leading to ...
2))2/1(()1()(
dxnxp
n
n +−
=π
Numerical Methods in Geophysics High-order operators
Fourier Coefficients
2))2/1(()1()(
dxnxp
n
n +−
=π
... these are our differential weights ...
-20 -15 -10 -5 0 5 10 15 20-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
Numerical Methods in Geophysics High-order operators
to shorten the operator we taper with a Gaussian function
-20 -15 -10 -5 0 5 10 15 20-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
-20 -15 -10 -5 0 5 10 15 20
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
Numerical Methods in Geophysics High-order operators
Fourier Coefficients - Interpolation
... the same approach can be applied to the problem of interpolation ...
∫
∫
∫
∞
∞−
−
∞
∞−
−−
∞
∞−
+−
=
=
=+
dkekIkF
dkeekF
dkekFdxxf
ikx
ikxikdx
dxxik
)()(
)(
)()2/(
2/
)2/(
2/)( ikdxekI −=
i.e. I(k) is now our interpolation operator expressed in the wavenumber domain. Again we are looking for the equivalent representation
in the space domain, which we get by inverse Fourier Transform
Numerical Methods in Geophysics High-order operators
Fourier Coefficients - Interpolation
... in the band-limited world our operator is ...
( ))()()( 2/NyNy
ikdx kkHkkHekI −++= −
... which in the space domain yields ...
⎩⎨⎧
=+=+
dxkdxnx
Ny
n
/)2/1(2/1
π)2/())2/(sin(
)(dxx
dxxkxi Ny
+
+=
πdiscretising with
we obtain
)2/1()1()(+
−=
ndxxi
n
n π
Numerical Methods in Geophysics High-order operators
Fourier Coefficients - Interpolation
- 4 0 - 3 0 - 2 0 - 1 0 0 1 0 2 0- 0 . 8
- 0 . 6
- 0 . 4
- 0 . 2
0
0 . 2
0 . 4
0 . 6
0 . 8
1
1 . 2
Numerical Methods in Geophysics High-order operators
Fourier Coefficients - Interpolation
-30 -20 -10 0 10 20 30-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
the final 8-point operator
Numerical Methods in Geophysics High-order operators
High-order operators - Accuracy
... as mentioned earlier the derivative operator in the wavenumber domain is
∫
∫∞
∞−
−
∞
∞−
−
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂=∂
dkekikF
dkekFxf
ikx
ikxxx
)(
)()(
ikkP −=)(
-20 -15 -10 -5 0 5 10 15 20-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
... which in the space domain led to the convolutional operator p(x) looking likeexact shortened
-20 -15 -10 -5 0 5 10 15 20
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
)(xp )(~ xp
Numerical Methods in Geophysics High-order operators
High-order operators - Accuracy
... this suggests that we can now Fourier transform our shortened operator and compare it with the exact one in the k-domain ...
Numerical Methods in Geophysics High-order operators
kikpFFT ~))(~( −=
-20 -15 -10 -5 0 5 10 15 20
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
... which also means that we now have a numerical wavenumber as a function of the exact wavenumber we propagate in our system.
0 20 40 60 80 1000
0.5
1
1.5
2
2.5
3
3.5
% Nyquis t
Im (
P(k)
)
Exact derivative operator
High-order operators - Accuracy
... we can now compare some of our high-order Taylor operators with the exact operator in the wavenumber domain ...
1 1.5 2-1
-0.5
0
0.5
12 points
0 50 1000
1
2
3
4 k-domain
0 50 100-1.5
-1
-0.5
0 Max re l.e rr < 4/5 Nyquis t : 0.243039
% Nyquis t 0 50 100
-0.4
-0.3
-0.2
-0.1
0Cum e rror < 4/5 Nyquis t : 9.0835
% Nyquis t
2 Points
Numerical Methods in Geophysics High-order operators
High-order operators - Accuracy
... we can now compare some of our high-order Taylor operators with the exact operator in the wavenumber domain ...
1 2 3 4-2
-1
0
1
22 points
0 50 1000
1
2
3
4 k-domain
0 50 100-1
-0.5
0 Max re l.e rr < 4/5 Nyquis t : 0.13023
% Nyquis t 0 50 100
-0.3
-0.2
-0.1
0Cum e rror < 4/5 Nyquis t : 3.60571
% Nyquis t
4 Points
Numerical Methods in Geophysics High-order operators
High-order operators - Accuracy
... we can now compare some of our high-order Taylor operators with the exact operator in the wavenumber domain ...
0 5 10-2
-1
0
1
22 points
0 50 1000
1
2
3
4 k-domain
0 50 100-0.6
-0.4
-0.2
0 Max re l.e rr < 4/5 Nyquis t : 0.0606789
% Nyquis t 0 50 100
-0.2
-0.15
-0.1
-0.05
0Cum e rror < 4/5 Nyquis t : 1.5652
% Nyquis t
8 Points
Numerical Methods in Geophysics High-order operators
High-order operators - Accuracy
... we can now compare some of our high-order Taylor operators with the exact operator in the wavenumber domain ...
0 10 20-2
-1
0
1
22 points
0 50 1000
1
2
3
4 k-domain
0 50 100-0.6
-0.4
-0.2
0 Max re l.e rr < 4/5 Nyquis t : 0.0247515
% Nyquis t 0 50 100
-0.15
-0.1
-0.05
0Cum e rror < 4/5 Nyquis t : 0.95043
% Nyquis t
16 Points
Numerical Methods in Geophysics High-order operators
Fourier Coefficients
Summary
Finite-difference operators can be regarded as (truncated) spectral (global) operators in a general way. In pseudo-spectral methods, the space derivativesare calculated in the wavenumber domain.
The length of the operator determines its accuracy.
increasing length of operator2 nx
FD Spectralimproving accuracy
Numerical Methods in Geophysics High-order operators