Truncation Error and Second Order Accuracy.

In summary, by multiplying the first equation by -4, we can eliminate u'' and simplify the equations to solve for u'. The final equation is u' = ((3u_i-4u_(i-1)+u_(i-2))/2h)+O(h^4).
  • #1
mm391
66
0
By combining the two equations i should be able to solve for u' and get rid of u'':

u_(i-1) = u_i + (-h)*u' + 1/2 * (-h)^2 * u'' + O(h^4)

u_(i-2) = u_i + (-2h)*u' + 1/2 * (-2h)^2 * u'' + O(h^4)

But i keep getting stuck and can't come up with the answer below.

Can anyone help me please.

u'=((3u_i-4u_(i-1)+u_(i-2))/2h)+O(h^4)
 
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  • #2
multiply the first equation by -4

-4u_(i-1) = -4u_i + (4h)*u' - 1/2 * (-2h)^2 * u'' + O(h^4)

u_(i-2) = u_i + (-2h)*u' + 1/2 * (-2h)^2 * u'' + O(h^4)

now solve for u'
 

Related to Truncation Error and Second Order Accuracy.

1. What is truncation error?

Truncation error is the error that occurs when a mathematical method or algorithm is used to approximate a solution to a problem. It is the difference between the actual value and the value obtained through the approximation.

2. How is truncation error related to accuracy?

Truncation error affects the accuracy of a numerical method or algorithm. A smaller truncation error indicates a more accurate approximation, while a larger truncation error indicates a less accurate approximation.

3. What is second order accuracy?

Second order accuracy refers to the ability of a numerical method or algorithm to produce a solution that is accurate up to the second order of the error term. This means that for every decrease in the step size, the error decreases by a factor of the square of the step size.

4. How is second order accuracy different from first order accuracy?

First order accuracy refers to the ability of a numerical method or algorithm to produce a solution that is accurate up to the first order of the error term. This means that for every decrease in the step size, the error decreases by a factor of the step size. Second order accuracy is a higher level of accuracy, as it decreases the error by a factor of the square of the step size.

5. Can a method or algorithm have higher than second order accuracy?

Yes, a method or algorithm can have higher than second order accuracy. Higher order accuracy means that the error decreases by a factor of the nth power of the step size, where n is the order of accuracy. However, higher order accuracy often comes with increased computational complexity and may not be necessary for all applications.

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