  In practical computation with Runge--Kutta methods, the stage equations are
not satisfied exactly, due to roundoff errors, algebraic solver errors, and so
forth. We show by example that propagation of such errors within a single step
can have catastrophic effects for otherwise practical and well-known methods.
We perform a general analysis of internal error propagation, emphasizing that
it depends significantly on how the method is implemented. We show that for a
fixed method, essentially any set of internal stability polynomials can be
obtained by modifying the implementation details. We provide bounds on the
internal error amplification constants for some classes of methods with many
stages, including strong stability preserving methods and extrapolation
methods. These results are used to prove error bounds in the presence of
roundoff or other internal errors.
