Choosing more stringent values for RelTol and AbsTol in ode45, for smooth functions such as sin(t) or cos(t), does not change the numerical solutions,
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I'm playing with the RelTol and AbsTol options for the ode45 solver, choosing more stringent values such as 1e-10 for both. However, for smooth solutions such as y(t) = sin(t), more stringent RelTol and AbsTol values do not bring back different solutions. Why is this? Is it because for smooth functions, the time-stepping errors are already so small that adjusting RelTol and AbsTol to be more stringent will not have an effect on the accuracy of solutions?
The ode45 solver uses approximately double the amount of time-steps (e.g. 500 vs. 1000 steps) with 1e-10 error tolerances set, with no different solutions for smooth functions.