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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. $phi_i-1=phi_i+h_i-1fracdphidx|_i+frach_i-1^22!fracd^2phidx^2|_i+frach_i-1^33!fracd^3phidx^3|_i+...$


2. $phi_i+1=phi_i+h_ifracdphidx|_i+frach_i^22!fracd^2phidx^2|_i+frach_i^33!fracd^3phidx^3|_i+...$

Combining these in a linear combination, then solving for the first derivative
$$fracdphidx|_i=fracAphi_i-1+Bphi_i+1-(A+B)phi_iAh_i-1+Bh_i-frac12left(fracAh_i-1^2+Bh_i^2Ah_i-1+Bh_iright)fracd^2phidx^2|_i-frac16left(fracAh_i-1^3+Bh_i^3Ah_i-1+Bh_iright)fracd^3phidx^3|_i+/-...$$

According to my notes from class, I should be getting a final answer of
$$fracdphidx|_i=fracphi_i+1-phi_i-1x_i+1-x_i-1-fracleft(h_iright)^2-left(h_i-1right)^22left(x_i+1-x_i-1right)fracd^2phidx^2|_i-fracleft(h_iright)^3-left(h_i-1right)^36left(x_i+1-x_i-1right)fracd^3phidx^3|_i+O(h^3).$$

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

$endgroup$

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. $phi_i-1=phi_i+h_i-1fracdphidx|_i+frach_i-1^22!fracd^2phidx^2|_i+frach_i-1^33!fracd^3phidx^3|_i+...$


2. $phi_i+1=phi_i+h_ifracdphidx|_i+frach_i^22!fracd^2phidx^2|_i+frach_i^33!fracd^3phidx^3|_i+...$

Combining these in a linear combination, then solving for the first derivative
$$fracdphidx|_i=fracAphi_i-1+Bphi_i+1-(A+B)phi_iAh_i-1+Bh_i-frac12left(fracAh_i-1^2+Bh_i^2Ah_i-1+Bh_iright)fracd^2phidx^2|_i-frac16left(fracAh_i-1^3+Bh_i^3Ah_i-1+Bh_iright)fracd^3phidx^3|_i+/-...$$

According to my notes from class, I should be getting a final answer of
$$fracdphidx|_i=fracphi_i+1-phi_i-1x_i+1-x_i-1-fracleft(h_iright)^2-left(h_i-1right)^22left(x_i+1-x_i-1right)fracd^2phidx^2|_i-fracleft(h_iright)^3-left(h_i-1right)^36left(x_i+1-x_i-1right)fracd^3phidx^3|_i+O(h^3).$$

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

$endgroup$

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. $phi_i-1=phi_i+h_i-1fracdphidx|_i+frach_i-1^22!fracd^2phidx^2|_i+frach_i-1^33!fracd^3phidx^3|_i+...$


2. $phi_i+1=phi_i+h_ifracdphidx|_i+frach_i^22!fracd^2phidx^2|_i+frach_i^33!fracd^3phidx^3|_i+...$

Combining these in a linear combination, then solving for the first derivative
$$fracdphidx|_i=fracAphi_i-1+Bphi_i+1-(A+B)phi_iAh_i-1+Bh_i-frac12left(fracAh_i-1^2+Bh_i^2Ah_i-1+Bh_iright)fracd^2phidx^2|_i-frac16left(fracAh_i-1^3+Bh_i^3Ah_i-1+Bh_iright)fracd^3phidx^3|_i+/-...$$

According to my notes from class, I should be getting a final answer of
$$fracdphidx|_i=fracphi_i+1-phi_i-1x_i+1-x_i-1-fracleft(h_iright)^2-left(h_i-1right)^22left(x_i+1-x_i-1right)fracd^2phidx^2|_i-fracleft(h_iright)^3-left(h_i-1right)^36left(x_i+1-x_i-1right)fracd^3phidx^3|_i+O(h^3).$$

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