Mixing
Approximating finite differences by higher order derivatives of continuous functions
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In a mathematical physics exercise, some discrete fields can conveniently be expressed using finite difference schemes as follows:
beginalign
A_6 &= phi_i+1+phi_i-1+4phi_i , , \
A_7 &= phi_i+2+phi_i-2-2phi_i , , \
A_8 &= phi_i+2-phi_i-2 , ,
endalign
wherein $phi_i equiv phi(x_i)$ such that $x_i = x_0 + ih$ are the node positions.
Moreover, $h$ represents a uniform grid spacing.
The goal is to approximate these groups by continuous functions following a standard discrete-to-continuum approach.
Here is what I have tried:
beginalign
A_6 &= phi_i+1+phi_i-1-2phi_i + 6phi_i approx h^2 phi''(x)+6phi(x) , , \
A_7 &= phi_i+2-4phi_i+1+6phi_i-4phi_i-1+phi_i+2
+4 left( phi_i+1+phi_i-1-2phi_i right)
approx h^2 left( h^2phi''''(x)+4 phi''(x) right) , , \
A_8 &= 2 left( fracphi_i+22-phi_i-1+phi_i+1- fracphi_i-22 +phi_i-1-phi_i+1 right)
approx 2 hleft( h^2phi'''(x) -2phi'(x)right) , .
endalign
I'd like to know whether these approximations are unique and/or whether a better accuracy can be reached using different approaches.
Any help or comment is very welcome.
Here is a link to the wiki article on finite difference coefficients
real-analysis differential-equations derivatives finite-differences finite-difference-methods
asked Jul 14 at 15:15
Mickhausen St Ilgen Sandhausen
92419
add a comment |Â
up vote
2
down vote
favorite
In a mathematical physics exercise, some discrete fields can conveniently be expressed using finite difference schemes as follows:
beginalign
A_6 &= phi_i+1+phi_i-1+4phi_i , , \
A_7 &= phi_i+2+phi_i-2-2phi_i , , \
A_8 &= phi_i+2-phi_i-2 , ,
endalign
wherein $phi_i equiv phi(x_i)$ such that $x_i = x_0 + ih$ are the node positions.
Moreover, $h$ represents a uniform grid spacing.
The goal is to approximate these groups by continuous functions following a standard discrete-to-continuum approach.
Here is what I have tried:
beginalign
A_6 &= phi_i+1+phi_i-1-2phi_i + 6phi_i approx h^2 phi''(x)+6phi(x) , , \
A_7 &= phi_i+2-4phi_i+1+6phi_i-4phi_i-1+phi_i+2
+4 left( phi_i+1+phi_i-1-2phi_i right)
approx h^2 left( h^2phi''''(x)+4 phi''(x) right) , , \
A_8 &= 2 left( fracphi_i+22-phi_i-1+phi_i+1- fracphi_i-22 +phi_i-1-phi_i+1 right)
approx 2 hleft( h^2phi'''(x) -2phi'(x)right) , .
endalign
I'd like to know whether these approximations are unique and/or whether a better accuracy can be reached using different approaches.
Any help or comment is very welcome.
Here is a link to the wiki article on finite difference coefficients
real-analysis differential-equations derivatives finite-differences finite-difference-methods
asked Jul 14 at 15:15
Mickhausen St Ilgen Sandhausen
92419
1
Just insert the Taylor expansion at $x_i$.
– LutzL
Jul 14 at 18:43
@LutzL Thanks for the comment. Can you be more precise please? Taylor expansion of what?
– Mickhausen St Ilgen Sandhausen
Jul 14 at 18:52
2
Write $phi_i+r = exp(r h D)phi(x)$ where $D$ is the differential operator and expand in powers of $h D$. E.g. $phi_i+1 + phi_i-1= 2 cosh(h D)phi(x) = 2 phi(x) + h^2 phi''(x)+cdots$.
– Count Iblis
Jul 14 at 19:19
@CountIblis Thanks iblis for your clarifications -- i got it :)
– Mickhausen St Ilgen Sandhausen
Jul 14 at 19:27
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
In a mathematical physics exercise, some discrete fields can conveniently be expressed using finite difference schemes as follows:
beginalign
A_6 &= phi_i+1+phi_i-1+4phi_i , , \
A_7 &= phi_i+2+phi_i-2-2phi_i , , \
A_8 &= phi_i+2-phi_i-2 , ,
endalign
wherein $phi_i equiv phi(x_i)$ such that $x_i = x_0 + ih$ are the node positions.
Moreover, $h$ represents a uniform grid spacing.
The goal is to approximate these groups by continuous functions following a standard discrete-to-continuum approach.
Here is what I have tried:
beginalign
A_6 &= phi_i+1+phi_i-1-2phi_i + 6phi_i approx h^2 phi''(x)+6phi(x) , , \
A_7 &= phi_i+2-4phi_i+1+6phi_i-4phi_i-1+phi_i+2
+4 left( phi_i+1+phi_i-1-2phi_i right)
approx h^2 left( h^2phi''''(x)+4 phi''(x) right) , , \
A_8 &= 2 left( fracphi_i+22-phi_i-1+phi_i+1- fracphi_i-22 +phi_i-1-phi_i+1 right)
approx 2 hleft( h^2phi'''(x) -2phi'(x)right) , .
endalign
I'd like to know whether these approximations are unique and/or whether a better accuracy can be reached using different approaches.
Any help or comment is very welcome.
Here is a link to the wiki article on finite difference coefficients
real-analysis differential-equations derivatives finite-differences finite-difference-methods
asked Jul 14 at 15:15
Mickhausen St Ilgen Sandhausen
92419
In a mathematical physics exercise, some discrete fields can conveniently be expressed using finite difference schemes as follows:
beginalign
A_6 &= phi_i+1+phi_i-1+4phi_i , , \
A_7 &= phi_i+2+phi_i-2-2phi_i , , \
A_8 &= phi_i+2-phi_i-2 , ,
endalign
wherein $phi_i equiv phi(x_i)$ such that $x_i = x_0 + ih$ are the node positions.
Moreover, $h$ represents a uniform grid spacing.
The goal is to approximate these groups by continuous functions following a standard discrete-to-continuum approach.
Here is what I have tried:
beginalign
A_6 &= phi_i+1+phi_i-1-2phi_i + 6phi_i approx h^2 phi''(x)+6phi(x) , , \
A_7 &= phi_i+2-4phi_i+1+6phi_i-4phi_i-1+phi_i+2
+4 left( phi_i+1+phi_i-1-2phi_i right)
approx h^2 left( h^2phi''''(x)+4 phi''(x) right) , , \
A_8 &= 2 left( fracphi_i+22-phi_i-1+phi_i+1- fracphi_i-22 +phi_i-1-phi_i+1 right)
approx 2 hleft( h^2phi'''(x) -2phi'(x)right) , .
endalign
I'd like to know whether these approximations are unique and/or whether a better accuracy can be reached using different approaches.
Any help or comment is very welcome.
Here is a link to the wiki article on finite difference coefficients
real-analysis differential-equations derivatives finite-differences finite-difference-methods
asked Jul 14 at 15:15
Mickhausen St Ilgen Sandhausen
92419
edited Jul 14 at 18:58
asked Jul 14 at 15:15
Mickhausen St Ilgen Sandhausen
92419
asked Jul 14 at 15:15
Mickhausen St Ilgen Sandhausen
92419
asked Jul 14 at 15:15
Mickhausen St Ilgen Sandhausen
92419
92419
1
Just insert the Taylor expansion at $x_i$.
– LutzL
Jul 14 at 18:43
@LutzL Thanks for the comment. Can you be more precise please? Taylor expansion of what?
– Mickhausen St Ilgen Sandhausen
Jul 14 at 18:52
2
Write $phi_i+r = exp(r h D)phi(x)$ where $D$ is the differential operator and expand in powers of $h D$. E.g. $phi_i+1 + phi_i-1= 2 cosh(h D)phi(x) = 2 phi(x) + h^2 phi''(x)+cdots$.
– Count Iblis
Jul 14 at 19:19
@CountIblis Thanks iblis for your clarifications -- i got it :)
– Mickhausen St Ilgen Sandhausen
Jul 14 at 19:27
add a comment |Â
1
Just insert the Taylor expansion at $x_i$.
– LutzL
Jul 14 at 18:43
@LutzL Thanks for the comment. Can you be more precise please? Taylor expansion of what?
– Mickhausen St Ilgen Sandhausen
Jul 14 at 18:52
2
Write $phi_i+r = exp(r h D)phi(x)$ where $D$ is the differential operator and expand in powers of $h D$. E.g. $phi_i+1 + phi_i-1= 2 cosh(h D)phi(x) = 2 phi(x) + h^2 phi''(x)+cdots$.
– Count Iblis
Jul 14 at 19:19
@CountIblis Thanks iblis for your clarifications -- i got it :)
– Mickhausen St Ilgen Sandhausen
Jul 14 at 19:27
1
1
Just insert the Taylor expansion at $x_i$.
– LutzL
Jul 14 at 18:43
Just insert the Taylor expansion at $x_i$.
– LutzL
Jul 14 at 18:43
@LutzL Thanks for the comment. Can you be more precise please? Taylor expansion of what?
– Mickhausen St Ilgen Sandhausen
Jul 14 at 18:52
@LutzL Thanks for the comment. Can you be more precise please? Taylor expansion of what?
– Mickhausen St Ilgen Sandhausen
Jul 14 at 18:52
2
2
Write $phi_i+r = exp(r h D)phi(x)$ where $D$ is the differential operator and expand in powers of $h D$. E.g. $phi_i+1 + phi_i-1= 2 cosh(h D)phi(x) = 2 phi(x) + h^2 phi''(x)+cdots$.
– Count Iblis
Jul 14 at 19:19
Write $phi_i+r = exp(r h D)phi(x)$ where $D$ is the differential operator and expand in powers of $h D$. E.g. $phi_i+1 + phi_i-1= 2 cosh(h D)phi(x) = 2 phi(x) + h^2 phi''(x)+cdots$.
– Count Iblis
Jul 14 at 19:19
@CountIblis Thanks iblis for your clarifications -- i got it :)
– Mickhausen St Ilgen Sandhausen
Jul 14 at 19:27
@CountIblis Thanks iblis for your clarifications -- i got it :)
– Mickhausen St Ilgen Sandhausen
Jul 14 at 19:27
add a comment |Â
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1
Just insert the Taylor expansion at $x_i$.
– LutzL
Jul 14 at 18:43
@LutzL Thanks for the comment. Can you be more precise please? Taylor expansion of what?
– Mickhausen St Ilgen Sandhausen
Jul 14 at 18:52
2
Write $phi_i+r = exp(r h D)phi(x)$ where $D$ is the differential operator and expand in powers of $h D$. E.g. $phi_i+1 + phi_i-1= 2 cosh(h D)phi(x) = 2 phi(x) + h^2 phi''(x)+cdots$.
– Count Iblis
Jul 14 at 19:19
@CountIblis Thanks iblis for your clarifications -- i got it :)
– Mickhausen St Ilgen Sandhausen
Jul 14 at 19:27