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These can be derived from the definition of the truncation error itself.
The definition of the global truncation error is also unchanged.
An expression of general interest is the local truncation error of a method.
We therefore have a truncation error of 0.01.
See Truncation error (numerical integration) for more on this.
This means that, in this case, the local truncation error is proportional to the step size.
Additionally, truncation error can also become an issue.
This is true in general, also for other equations; see the section Global truncation error for more details.
The above definitions are particularly relevant in situations where truncation errors are not important.
The total truncation error, considering both sources, is upper bounded by:
The local truncation error of the Euler method is error made in a single step.
There will always be some form of corrupting noise, even if it is present as round-off or truncation error.
The truncation error can be twice the maximum error in rounding.
Thus, it is to be expected that the global truncation error will be proportional to .
The remainder term of a Taylor polynomial is convenient for analyzing the local truncation error.
Truncation errors in numerical integration are of two kinds:
More formally, the global truncation error, , at time is defined by:
This shows that for small , the local truncation error is approximately proportional to .
The relation between local and global truncation errors is slightly different than in the simpler setting of one-step methods.
There are also accompanying requirements if we require the method to have a certain order p, meaning that the local truncation error is O(h).
Robert, A., 1971: Truncation errors in a filtered barotropic model.
The main causes of error are round-off error and truncation error.
What does it mean when we say that the truncation error is created when we approximate a mathematical procedure?
This means that the local truncation error (the error made in one step) is , using the big O notation.
The global truncation error satisfies the recurrence relation:
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