Taylor series for getting different formulas

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SUMMARY

The discussion centers on the application of Taylor series in calculating the partial derivative \(\frac{\partial u}{\partial t}(t+k/2, x)\) and the challenges faced in achieving the correct order of the error term \(O(k^2)\). User ericm1234 suggests that instead of using Taylor series, one should apply the elementary definition of a derivative as a limit. Another participant proposes a reformulation of the expression to clarify the relationship between the terms involved.

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ericm1234
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I am trying to establish why, I'm assuming one uses taylor series,
\frac{\partial u}{\partial t}(t+k/2, x)= (u(t+k,x)-u(t,x))/k + O(k^2)

I have tried every possible combination of adding/subtracting taylor series, but either I can not get it exactly or my O(k^2) term doesn't work out (it's O(k^1) or O(k^3) )
 
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hi ericm1234! :smile:

no you don't need taylor, just use the elementary definition of derivative (as a limit) …

perhaps it's more obvious if you write (u(t+k,x)-u(t,x)) as (u(t+k,x)-u(t+k/2,x)) + (u(t+k/2,x)-u(t,x)) ? :wink:​
 

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