For example, if you needed to rep exp(x) by a series, you could not use an infinite number of terms, but would have to stop ("truncate") the series after a specific number of terms. It is common practice to indicate the approx accuracy of a finite difference scheme by noting the leading term NEGLECTED in such a Taylor series. More will be said about this later.
exp(x) = 1 + x/1! +x^{2}/2! +x^{3}/3! + ... + _{n=4} ^{inf.} x^{n}/n! + ...
The English language can sometimes be ambiguous. Sometimes the error indicated here is due to the "truncation" of the series of functions used in a spectral method. This does not make it "truncation error" as defined above it is still a matter of limited resolution. Put another way, a given grid interval, or smallest Galerkin expansion function used, determines the smallest variation you can resolve.
This is an error type that often occurs due to a misuse of a numeric scheme. For example, the two numerical methods: the simplest centered in time and the simplest centered in space finite differences can work in the wave equation, but will not work in the heat equation. The result always blows up. Not because the parts of the scheme are bad, but one might view it as the errors in each of the two parts interact in a pathological way.
The result of this example is typical of propagated errors in that unrealistically large, numerically infinite values are reached. The solution is useless. We refer to such a scheme as being UNSTABLE. Stable schemes either cause the errors to die out as the method continues or keeps the error within a certain bounds.
While the examples are of time dependent problems, a type of "propagated" error can also occur in diagnostic calculations when they involve either some kind of iteration or some other recursive operation (e.g. some matrix solution algorithms)
Also, you may be trying to represent a physical process using a mathematical formula that may not be quite right. A simple example would be using geostrophic velocities in a model instead of true velocities. I have combined these two different types of error because they are not a property of your computer or of your numerical sehcme. Nonetheless, they may be crucial to the usefulness of your answer.
With all these sources of error, you might wonder how anything can be calculated with confidence! The answer is that while numerical analysis computing opens up many problems for solution, getting that solution takes some work. Resolving and avoiding these problems is the main reason why numerical analysis is a large and highly technical field of endeavor.
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