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The local truncation error (defined as the error made in one step) of the backward euler method is , using the big o notation The error at a specific time is. The local truncation error of the euler method is the error made in a single step It is the difference between the numerical solution after one step, , and the exact solution at time.
The error caused by choosing a finite number of rectangles as opposed to an infinite number of them is a truncation error in the mathematical process of integration. An expression of general interest is the local truncation error of a method That is, it is the quantity if refers to the exact value and to the numerical approximation. There are also accompanying requirements if one requires the method to have a certain order p, meaning that the local truncation error is o (hp+1)
Unlike truncation error which only indicates that error is due to taking small step sizes, local truncation error give us error in terms of complexity theory
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