# 1.4 Convergence

## 1.4.1 Types of Errors

Measurable Outcome 1.5, Measurable Outcome 1.6

When we approximate the solution of ODEs numerically, there are two primary sources of error: rounding (or floating point) errors and truncation errors. Rounding errors are associated to the floating-point arithmetic that our computers use to perform calculations. Truncation errors, on the other hand, are errors we incur based on the numerical method; these errors would exist even in the absence of rounding errors. To some extent, we cannot control rounding errors (they are determined by the precision of the machine), but we can control truncation errors.

Exercise How are truncation errors introduced in the Forward Euler method of Section 1.2.4.

Exercise 1
Taylor series approximation of \(u^ n\)
Taylor series approximation of \(u^{n+1}\)
Taylor series approximation of \(u_ t^{n}\)
None of the above

Answer: The derivative \(u^{n}_ t\) is evaluated directly, and \(u^ n\) is given. The truncation error arises due to truncating the Taylor series of \(u^{n+1}\).