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Finite difference for non-uniform unstructured mesh/stencil

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$begingroup$

Below I have shown my non-uniform unstructured mesh (as in there is no pattern between the relative size of $h_i$ and $h_{i+1}$ etc.

I've made the following equations using nodes $i-1$, $i$, and $i+1$.

1. $phi_{i-1}=phi_i+h_{i-1}frac{dphi}{dx}|_i+frac{h_{i-1}^2}{2!}frac{d^2phi}{dx^2}|_i+frac{h_{i-1}^3}{3!}frac{d^3phi}{dx^3}|_i+...$


2. $phi_{i+1}=phi_i+h_{i}frac{dphi}{dx}|_i+frac{h_{i}^2}{2!}frac{d^2phi}{dx^2}|_i+frac{h_{i}^3}{3!}frac{d^3phi}{dx^3}|_i+...$

Combining these in a linear combination, then solving for the first derivative
$$frac{dphi}{dx}|_i=frac{Aphi_{i-1}+Bphi_{i+1}-(A+B)phi_i}{Ah_{i-1}+Bh_i}-frac{1}{2}left(frac{Ah_{i-1}^2+Bh_i^2}{Ah_{i-1}+Bh_i}right)frac{d^2phi}{dx^2}|_i-frac{1}{6}left(frac{Ah_{i-1}^3+Bh_i^3}{Ah_{i-1}+Bh_i}right)frac{d^3phi}{dx^3}|_i+/-...$$

According to my notes from class, I should be getting a final answer of
$$frac{dphi}{dx}|_i=frac{phi_{i+1}-phi_{i-1}}{x_{i+1}-x_{i-1}}-frac{left(h_iright)^2-left(h_{i-1}right)^2}{2left(x_{i+1}-x_{i-1}right)}frac{d^2phi}{dx^2}|_i-frac{left(h_iright)^3-left(h_{i-1}right)^3}{6left(x_{i+1}-x_{i-1}right)}frac{d^3phi}{dx^3}|_i+O(h^3).$$

Based on what I am expecting to get, and the fact that I want a second order accurate final equation I can say the following.

1. $Ah_{i-1}^2+Bh_i^2=0$


2. $A+B=0$

However, I can't get the expected answer. Can I get some help with the error I'm making?

numerical-methods finite-differences finite-difference-methods

share|cite|improve this question

asked Mar 20 at 18:37

WnGatRC456WnGatRC456

10811

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The Taylor expansion for $phi_{i-1}$ is incorrect. The distance between between the points $x_{i-1}$ and $x_i$ is $x_{i-1} - x_i = -h_{i-1}$, so you need to replace $h_{i-1}$ by $-h_{i-1}$ everywhere in this expansion. Does this get you to the final answer?
  $endgroup$
  – ekkilop
  Apr 3 at 13:52
  

  
  
  
  

add a comment | 

0

$begingroup$

Below I have shown my non-uniform unstructured mesh (as in there is no pattern between the relative size of $h_i$ and $h_{i+1}$ etc.

I've made the following equations using nodes $i-1$, $i$, and $i+1$.

1. $phi_{i-1}=phi_i+h_{i-1}frac{dphi}{dx}|_i+frac{h_{i-1}^2}{2!}frac{d^2phi}{dx^2}|_i+frac{h_{i-1}^3}{3!}frac{d^3phi}{dx^3}|_i+...$


2. $phi_{i+1}=phi_i+h_{i}frac{dphi}{dx}|_i+frac{h_{i}^2}{2!}frac{d^2phi}{dx^2}|_i+frac{h_{i}^3}{3!}frac{d^3phi}{dx^3}|_i+...$

Combining these in a linear combination, then solving for the first derivative
$$frac{dphi}{dx}|_i=frac{Aphi_{i-1}+Bphi_{i+1}-(A+B)phi_i}{Ah_{i-1}+Bh_i}-frac{1}{2}left(frac{Ah_{i-1}^2+Bh_i^2}{Ah_{i-1}+Bh_i}right)frac{d^2phi}{dx^2}|_i-frac{1}{6}left(frac{Ah_{i-1}^3+Bh_i^3}{Ah_{i-1}+Bh_i}right)frac{d^3phi}{dx^3}|_i+/-...$$

According to my notes from class, I should be getting a final answer of
$$frac{dphi}{dx}|_i=frac{phi_{i+1}-phi_{i-1}}{x_{i+1}-x_{i-1}}-frac{left(h_iright)^2-left(h_{i-1}right)^2}{2left(x_{i+1}-x_{i-1}right)}frac{d^2phi}{dx^2}|_i-frac{left(h_iright)^3-left(h_{i-1}right)^3}{6left(x_{i+1}-x_{i-1}right)}frac{d^3phi}{dx^3}|_i+O(h^3).$$

Based on what I am expecting to get, and the fact that I want a second order accurate final equation I can say the following.

1. $Ah_{i-1}^2+Bh_i^2=0$


2. $A+B=0$

However, I can't get the expected answer. Can I get some help with the error I'm making?

numerical-methods finite-differences finite-difference-methods

share|cite|improve this question

asked Mar 20 at 18:37

WnGatRC456WnGatRC456

10811

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The Taylor expansion for $phi_{i-1}$ is incorrect. The distance between between the points $x_{i-1}$ and $x_i$ is $x_{i-1} - x_i = -h_{i-1}$, so you need to replace $h_{i-1}$ by $-h_{i-1}$ everywhere in this expansion. Does this get you to the final answer?
  $endgroup$
  – ekkilop
  Apr 3 at 13:52
  

  
  
  
  

add a comment | 

$begingroup$

Below I have shown my non-uniform unstructured mesh (as in there is no pattern between the relative size of $h_i$ and $h_{i+1}$ etc.

I've made the following equations using nodes $i-1$, $i$, and $i+1$.

1. $phi_{i-1}=phi_i+h_{i-1}frac{dphi}{dx}|_i+frac{h_{i-1}^2}{2!}frac{d^2phi}{dx^2}|_i+frac{h_{i-1}^3}{3!}frac{d^3phi}{dx^3}|_i+...$


2. $phi_{i+1}=phi_i+h_{i}frac{dphi}{dx}|_i+frac{h_{i}^2}{2!}frac{d^2phi}{dx^2}|_i+frac{h_{i}^3}{3!}frac{d^3phi}{dx^3}|_i+...$

Combining these in a linear combination, then solving for the first derivative
$$frac{dphi}{dx}|_i=frac{Aphi_{i-1}+Bphi_{i+1}-(A+B)phi_i}{Ah_{i-1}+Bh_i}-frac{1}{2}left(frac{Ah_{i-1}^2+Bh_i^2}{Ah_{i-1}+Bh_i}right)frac{d^2phi}{dx^2}|_i-frac{1}{6}left(frac{Ah_{i-1}^3+Bh_i^3}{Ah_{i-1}+Bh_i}right)frac{d^3phi}{dx^3}|_i+/-...$$

According to my notes from class, I should be getting a final answer of
$$frac{dphi}{dx}|_i=frac{phi_{i+1}-phi_{i-1}}{x_{i+1}-x_{i-1}}-frac{left(h_iright)^2-left(h_{i-1}right)^2}{2left(x_{i+1}-x_{i-1}right)}frac{d^2phi}{dx^2}|_i-frac{left(h_iright)^3-left(h_{i-1}right)^3}{6left(x_{i+1}-x_{i-1}right)}frac{d^3phi}{dx^3}|_i+O(h^3).$$

Based on what I am expecting to get, and the fact that I want a second order accurate final equation I can say the following.

1. $Ah_{i-1}^2+Bh_i^2=0$


2. $A+B=0$

However, I can't get the expected answer. Can I get some help with the error I'm making?

numerical-methods finite-differences finite-difference-methods

share|cite|improve this question

asked Mar 20 at 18:37

WnGatRC456WnGatRC456

10811

$endgroup$

share|cite|improve this question

asked Mar 20 at 18:37

WnGatRC456WnGatRC456

10811

share|cite|improve this question

asked Mar 20 at 18:37

WnGatRC456WnGatRC456

10811

asked Mar 20 at 18:37

WnGatRC456WnGatRC456

10811

asked Mar 20 at 18:37

WnGatRC456WnGatRC456

10811