Corrected some mistakes and made the text more comprehensible
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Eliad
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This is the procedure to derive formulations for Compact finite difference schemes.

First, assume the stencil and the general structure of the scheme. For the classical Padé scheme you have:

$$ \alpha f'_{n+1} + f'_{n} + \alpha f'_{n-1} = a \dfrac{f_{n+1} - f_{n-1}}{2 \Delta x} \qquad(1) $$

You write the Taylor series for the derivatives and the functions, e.g.,

$$ f'_{n+1}=f'_{n} + \Delta x f''_n + \dfrac{(\Delta x)^2}{2!} f^{(3)}_n + \cdots $$

You then replace the derivatives and the functions in Eq. (1) with their Taylor series. The system of equations to solve consists of the following equations:

  • Set $f'_{n}$'s and $f_{n}$'s coefficient equal to 1.

    Set $f'_{n}$'s coefficients on both sides to be equal. Here we have, $$ 2\alpha + 1 = a $$

  • Start with coefficients of the lowest powers of $\Delta x$ and set them to be equal to zero, until you get enough equations.

    $f_n$ coefficient could only appear on the right hand side and its coefficient must be set equal to zero. In this case, this does not give us any equations.

  • Start with coefficients of the lowest powers of $\Delta x$ and set them to be equal, until you get enough equations, e.g., $$ a = 2 \dfrac{3!}{2!} \alpha $$

This is the procedure to derive formulations for Compact finite difference schemes.

First, assume the stencil and the general structure of the scheme. For the classical Padé scheme you have:

$$ \alpha f'_{n+1} + f'_{n} + \alpha f'_{n-1} = a \dfrac{f_{n+1} - f_{n-1}}{2 \Delta x} \qquad(1) $$

You write the Taylor series for the derivatives and the functions, e.g.,

$$ f'_{n+1}=f'_{n} + \Delta x f''_n + \dfrac{(\Delta x)^2}{2!} f^{(3)}_n + \cdots $$

You then replace the derivatives and the functions in Eq. (1) with their Taylor series. The system of equations to solve consists of the following equations:

  • Set $f'_{n}$'s and $f_{n}$'s coefficient equal to 1.
  • Start with coefficients of the lowest powers of $\Delta x$ and set them to be equal to zero, until you get enough equations.

This is the procedure to derive formulations for Compact finite difference schemes.

First, assume the stencil and the general structure of the scheme. For the classical Padé scheme you have:

$$ \alpha f'_{n+1} + f'_{n} + \alpha f'_{n-1} = a \dfrac{f_{n+1} - f_{n-1}}{2 \Delta x} \qquad(1) $$

You write the Taylor series for the derivatives and the functions, e.g.,

$$ f'_{n+1}=f'_{n} + \Delta x f''_n + \dfrac{(\Delta x)^2}{2!} f^{(3)}_n + \cdots $$

You then replace the derivatives and the functions in Eq. (1) with their Taylor series. The system of equations to solve consists of the following equations:

  • Set $f'_{n}$'s coefficients on both sides to be equal. Here we have, $$ 2\alpha + 1 = a $$

  • $f_n$ coefficient could only appear on the right hand side and its coefficient must be set equal to zero. In this case, this does not give us any equations.

  • Start with coefficients of the lowest powers of $\Delta x$ and set them to be equal, until you get enough equations, e.g., $$ a = 2 \dfrac{3!}{2!} \alpha $$

added 32 characters in body
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Eliad
  • 239
  • 1
  • 8

This is the procedure to derive formulations for Compact finite difference schemes.

First, assume the stencil and the general structure of the scheme. For the classical Padé scheme you have:

$$ \alpha f'_{n+1} + f'_{n} + \alpha f'_{n-1} = a \dfrac{f_{n+1} - f_{n-1}}{2 \Delta x} \qquad(1) $$

You write the Taylor series for the derivatives and the functions, e.g.,

$$ f'_{n+1}=f'_{n} + \Delta x f''_n + \dfrac{(\Delta x)^2}{2!} f^{(3)}_n + \cdots $$

You then replace the derivatives and the functions in Eq. (1) with their Taylor series. The system of equations to solve consists of the following equations:

  • Set $f'_{n}$'s and $f_{n}$'s coefficient equal to 1.
  • Start with coefficients of the lowest powers of $\Delta x$ and set them to be equal to zero, until you get enough equations.

This is the procedure to derive formulations for Compact finite difference schemes.

First, assume the stencil and the general structure of the scheme. For the classical Padé scheme you have:

$$ \alpha f'_{n+1} + f'_{n} + \alpha f'_{n-1} = a \dfrac{f_{n+1} - f_{n-1}}{2 \Delta x} \qquad(1) $$

You write the Taylor series for the derivatives and the functions, e.g.,

$$ f'_{n+1}=f'_{n} + \Delta x f''_n + \dfrac{(\Delta x)^2}{2!} f^{(3)}_n + \cdots $$

You then replace the derivatives and the functions in Eq. (1) with their Taylor series. The system of equations to solve consists of the following equations:

  • Set $f'_{n}$'s and $f_{n}$'s coefficient equal to 1.
  • Start with coefficients of the lowest powers of $\Delta x$ and set them to be equal to zero.

This is the procedure to derive formulations for Compact finite difference schemes.

First, assume the stencil and the general structure of the scheme. For the classical Padé scheme you have:

$$ \alpha f'_{n+1} + f'_{n} + \alpha f'_{n-1} = a \dfrac{f_{n+1} - f_{n-1}}{2 \Delta x} \qquad(1) $$

You write the Taylor series for the derivatives and the functions, e.g.,

$$ f'_{n+1}=f'_{n} + \Delta x f''_n + \dfrac{(\Delta x)^2}{2!} f^{(3)}_n + \cdots $$

You then replace the derivatives and the functions in Eq. (1) with their Taylor series. The system of equations to solve consists of the following equations:

  • Set $f'_{n}$'s and $f_{n}$'s coefficient equal to 1.
  • Start with coefficients of the lowest powers of $\Delta x$ and set them to be equal to zero, until you get enough equations.
Source Link
Eliad
  • 239
  • 1
  • 8

This is the procedure to derive formulations for Compact finite difference schemes.

First, assume the stencil and the general structure of the scheme. For the classical Padé scheme you have:

$$ \alpha f'_{n+1} + f'_{n} + \alpha f'_{n-1} = a \dfrac{f_{n+1} - f_{n-1}}{2 \Delta x} \qquad(1) $$

You write the Taylor series for the derivatives and the functions, e.g.,

$$ f'_{n+1}=f'_{n} + \Delta x f''_n + \dfrac{(\Delta x)^2}{2!} f^{(3)}_n + \cdots $$

You then replace the derivatives and the functions in Eq. (1) with their Taylor series. The system of equations to solve consists of the following equations:

  • Set $f'_{n}$'s and $f_{n}$'s coefficient equal to 1.
  • Start with coefficients of the lowest powers of $\Delta x$ and set them to be equal to zero.