what's the difference in accuracy between solving a stiff problem by ode15s and ode45?

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Zixuan Li
Zixuan Li 2019-3-26
回答： Josh Meyer 2019-3-27
It seems ode45 can also solve a stiff problem, but takes more time ( there are more data on finer grid as well, but the curve is still differennt from that of ode15s). Does this mean ode45 can do better in terms of accuracy?

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John D'Errico
John D'Errico 2019-3-26
Actually, in many cases, sufficiently stiff problems see a solver like ODE45 grind to a complete halt, unable to pass a point where the step size needs to get so small that no effective progress is deemed possible. Yes, ODE45 can push through some problem regions. Not all stiff problems are equally nasty.
As for ODE45 producing a different result, when that happens, it usually means that it got the wrong result, not a more accurate one. So ODE45 may have gotten onto a completely different trajectory. It survived the stiff region, but to what effect, if the answer is wrong?
https://en.wikipedia.org/wiki/Stiff_equation

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Josh Meyer
Josh Meyer 2019-3-27
Stiffness is more of an issue of stability than accuracy. Nonstiff solvers can solve stiff problems, it just takes them a long time to do so.
I would also note that since ode45 is meant to be a general-purpose solver, it uses interpolation to output extra points per step, which leads to better plots. This is controlled by the 'Refine' option, which has a value of 4 for ode45 and 1 for all other solvers. So if you want ode15s to output some extra points per step as well, just use
opts = odeset('Refine',4);
[t,y] = ode15s(@odefcn,tspan,y0,opts);

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