As is known in the study of the initial value problem for ODE, the initial value are given with some errors. The question is that there existe the numerical methods in using of which the errors obtained in the initial value do not increase.
If you're using an adaptive solver and the tolerance is low enough, then the numerical error will be less than the error in the initial condition which should suffice. This isn't always true since many adaptive solvers use local residuals, but adaptive methods on global residuals make this rigorous.
The smaller the truncation error of the numerical method which has a relatively good stability, the error in the initial condition do not increase
It depends on the nature of your ODE. Take something as simple as y'=y, y(0)=y_0 and compare the solution with perturbed IV: u(0) = y_0+e_0.
You will find that the error increases exponentially and so does the error of your numerical solution.
However, if the error of a perturbed system does not increase, then you could look for numerical solvers that mimick that behavior. Classical example is implicit Euler vs explicit Euler.
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