CFD - Finite Difference schemes (Intro)

First Order Derivatives

Forward Difference

$\frac{\delta u}{\delta x} =  \frac{u_{i+1} - u_{i}}{\Delta x} + O(\Delta x)$

Derivation :
  $u_{i+1}=u_{i} + \left .\frac{\Delta x}{1 !} \frac{\delta u}{\delta x} \right \vert _{i} + \left .\frac{(\Delta x)^2}{2 !} \frac{\delta ^2u}{\delta x ^2} \right \vert _{i} + \left .\frac{(\Delta x)^3}{3 !} \frac{\delta ^3u}{\delta x ^3} \right \vert _{i} +O((\Delta x)^4) $
$\implies \frac{u_{i+1}-u_{i}}{\Delta x} = \left .\frac{1}{1 !} \frac{\delta u}{\delta x} \right \vert _{i} + \left .\frac{\Delta x}{2 !} \frac{\delta ^2u}{\delta x ^2} \right \vert _{i} + \left .\frac{(\Delta x)^2}{3 !} \frac{\delta ^3u}{\delta x ^3} \right \vert _{i} +O((\Delta x)^3) $
$\implies \frac{u_{i+1}-u_{i}}{\Delta x} = \left .\frac{1}{1 !} \frac{\delta u}{\delta x} \right \vert _{i} + O((\Delta x)) $

Backward Difference

$\frac{\delta u}{\delta x} =  \frac{u_{i} - u_{i-1}}{\Delta x} + O(\Delta x)$

Derivation :

  $u_{i-1}=u_{i} - \left .\frac{\Delta x}{1 !} \frac{\delta u}{\delta x} \right \vert _{i} + \left .\frac{(\Delta x)^2}{2 !} \frac{\delta ^2u}{\delta x ^2} \right \vert _{i} - \left .\frac{(\Delta x)^3}{3 !} \frac{\delta ^3u}{\delta x ^3} \right \vert _{i} +O((\Delta x)^4) $
$\implies \frac{u_{i}-u_{i-1}}{\Delta x} = \left .\frac{1}{1 !} \frac{\delta u}{\delta x} \right \vert _{i} + O((\Delta x)) $

Central Difference

$\frac{\delta u}{\delta x} =  \frac{u_{i+1} - u_{i-1}}{ 2\Delta x } + O((\Delta x)^2)$

Derivation :

$-u_{i+1}+u_{i} + \left .\frac{\Delta x}{1 !} \frac{\delta u}{\delta x} \right \vert _{i} + \left .\frac{(\Delta x)^2}{2 !} \frac{\delta ^2u}{\delta x ^2} \right \vert _{i} + \left .\frac{(\Delta x)^3}{3 !} \frac{\delta ^3u}{\delta x ^3} \right \vert _{i} +O((\Delta x)^4)  =0      -(1) $ 

  $-u_{i-1}+u_{i} - \left .\frac{\Delta x}{1 !} \frac{\delta u}{\delta x} \right \vert _{i} + \left .\frac{(\Delta x)^2}{2 !} \frac{\delta ^2u}{\delta x ^2} \right \vert _{i} - \left .\frac{(\Delta x)^3}{3 !} \frac{\delta ^3u}{\delta x ^3} \right \vert _{i} +O((\Delta x)^4) =0-(2)$ 

$(\alpha eqn_1 +\beta eqn_2) = ..... + 0*\left .\frac{(\Delta x)^2}{2 !} \frac{\delta ^2u}{\delta x ^2} \right \vert _{i}  +O((\Delta x)^3)  =0 -(3)$

Solving for $\alpha,\beta $ substituting in equation 3 , rearranging we get the above result. 

Similarly using the above method, we can obtain higher order accuracies for  forward, backward difference schemes as well.

Second Order Derivatives 

Forward Difference

$\frac{\delta^2 u}{\delta x^2} =  \frac{u_{i+2}-2u_{i+1} + u_{i}}{ \Delta x ^2} + O((\Delta x))$

Central Difference

$\frac{\delta^2 u}{\delta x^2} =  \frac{u_{i}-2u_{i-1} + u_{i-2}}{ \Delta x ^2} + O((\Delta x))$

Central Difference

$\frac{\delta^2 u}{\delta x^2} =  \frac{u_{i+1}-2u_{i} + u_{i-1}}{ \Delta x ^2} + O((\Delta x)^2)$

Note : Derive the above equations for your own practice.

Other methods of derivation

There are many methods using which the above schemes can be derived . One is that of the polynomial method ,  the other one is discussed for a simple case below :

 $$\frac{\delta ^n f}{\delta x ^n} = \frac{\delta ^{n-1} }{\delta x ^{n-1}} \left (\frac{\delta f}{\delta x }\right )    $$

Hence for a second order derivative (using forward difference) we get

$$\frac{\delta ^2 f}{\delta x ^2} = \frac{\delta  }{\delta x  } \left (\frac{f_{i+1} - f_{i}}{\Delta x} \right )   $$

$$\frac{\delta ^2 f}{\delta x ^2} = \left (\frac{\delta f_{i+1}  }{\delta x}-\frac{\delta f_{i}  }{\delta x}\right )\frac{1}{\Delta x}   $$

$$\frac{\delta ^2 f}{\delta x ^2} = \left (\frac{ f_{i+1}  -f_{i+1}  }{\Delta x}-\frac{ f_{i+1} - f_{i}  }{\Delta x}\right )\frac{1}{\Delta x}   $$

$$\frac{\delta ^2 f}{\delta x ^2} = \left (\frac{ f_{i+1}  - 2f_{i+1} +f_{i} }{\Delta x ^2} \right )$$

Polynomial method

Let there be a polynomial $f(x)   = ax^2+bx+c$ .

$\frac{\delta f}{\delta x} = 2ax+b    -(1)$

$\frac{\delta ^2 f}{\delta x ^2} = 2a  -(2)$

Let at $x =0 ,f(x) = f_{i},x = \Delta x ,f(x) = f_{i+1} ,x =2 \Delta x ,f(x) = f_{i+2} $.

Now solve for $a , b,c$ in terms of $f_{i},f_{i+1},f_{i+2},\Delta x$ 

Substituting $a , b$ back in equation 1, 2 will give you the appropriate derivatives.

Try the above calculations for practice.