'InitialStep' and 'MaxStep' in ode45

[John B. Thoo]

On Jul 26, 2009, at 1:36 AM, Thomas Treichl wrote:

> John B. Thoo schrieb:
>> Hi, everyone.  I have a question about 'InitialStep' and 'MaxStep'  
>> in  ode45.
>> I tried to solve an equation using
>>      [T, u] = ode45 (@F, [t0 tf], [u0, u0_t]);
>> and this is what I got:
>> octave-3.2.0:4> run_twoway
>> warning: Option "RelTol" not set, new value 0.000001 is used
>> warning: Option "AbsTol" not set, new value 0.000001 is used
>> warning: Option "InitialStep" not set, new value 1.200000 is used
>> warning: Option "MaxStep" not set, new value 1.200000 is used
>> error: Solving has not been successful. The iterative integration   
>> loop exited at time t = 5.818179 before endpoint at tend =  
>> 12.000000  was reached. This may happen if the stepsize grows  
>> smaller than  defined in vminstepsize. Try to reduce the value of  
>> "InitialStep" and/ or "MaxStep" with the command "odeset".
>> octave-3.2.0:4>
>> So I halved tend and set 'InitialStep' and 'MaxStep' as follows:
>>      options = odeset ('InitialStep', 1.0e-3, 'MaxStep', 1.0e-3, ...
>>                        'RelTol', 1.0e-6, 'AbsTol', 1.0e-6);
>>      [T, u] = ode45 (@F, [t0 tf], [u0, u0_t], options);
>> This time I got:
>> octave-3.2.0:4> run_twoway
>> error: Solving has not been successful. The iterative integration   
>> loop exited at time t = 5.553544 before endpoint at tend =  
>> 6.000000  was reached. This may happen if the stepsize grows  
>> smaller than  defined in vminstepsize. Try to reduce the value of  
>> "InitialStep" and/ or "MaxStep" with the command "odeset".
>> octave-3.2.0:4>
>> My question is this:
>>    If the first time 'InitialStep' and 'MaxStep' were set at 1.2   
>> resulted in ode45 exiting prematurely at time t = 5.818179, then  
>> why  does decreasing 'InitialStep' and 'MaxStep' result in the  
>> solver  exiting prematurely at a *shorter* time t = 5.553544?
>
> Hi John,
>
> this strongly depends on your ODE system. I think you have a  
> singularity somewhere at t=5.5..6.0, ie. that this is the third  
> possibility for your problem that ode23..ode78 can not solve for  
> you however you choose InitialStep and/or MaxStep. If your problem  
> is that kind of stiff then you maybe should use a solver that can  
> solve stiff problems, cf. description about the different solvers  
> in odepkg.pdf and the description about all the different options  
> that can be set.
>
> InitialStep and MaxStep in ode23..ode78 are not computed (as like  
> the solvers in Matlab do) but must always be given by the user. If  
> no value is given (eg. your first trial) then simply (tf-t0)/10 is  
> taken - this might be completely wrong and depends on which ODE  
> system should be solved.
>
> Now, if you choose another InitalStep then solving can completely  
> different (in your first trial we have t1_1=(tf-t0)/10, in your  
> second trial t1_2=1e-3, you see that t1_1 and t1_2 are already  
> different in your first InititialStep, it would be pure luck if you  
> would hit tx_1 = 5.818179 and tx_2 = 5.553544).
>
> Just for info, you normally should hit tx_1=tx_2 exactly if you  
> take fixed step sizes, eg.
>
>   [T, u] = ode45 (@F, [t0:tf], [u0, u0_t]);
>                    ## --> <-- pass a vector with time values
>
>> Another question:
>>    How should I choose to set 'InitialStep' and 'MaxStep'?  And   
>> should they always be set to the same value?
>
> Ok, I hope I could explain InitialStep, MaxStep is different: There  
> are errors computed every time you solve one delta time (AbsTol,  
> RelTol). The solver tries to increase the dt automatically every  
> time one time step was solved successfully (ie. computed error <  
> AbsTol and/or RelTol). Let's say we had dt_old = 1 and the solver  
> thinks dt_new = 2. If you set MaxStep = 1.5 then the user decreases  
> dt to 1.5 and takes control over the solver's estimation.
>
> *If* ode45 is the right choice for your problem (because of  
> stiffness) then not necessarily InitialStep and MaxStep have the  
> same values. It depends once again what kind of system you have -  
> might there be stability problems in your ODE system right at the  
> beginning of solving (make InitialStep smaller) or somewhere  in  
> the middle or the end (make MaxStep smaller).
>
> Best regards
>
>   Thomas

Dear Thomas,

Thanks for your reply.  I have read it, but I need to read it over to  
try to digest all that you've written.

As a point of interest, before I had received your reply, I reduced  
'InitialStep' and 'MaxStep' further to 1e-6.  Octave tried for 48  
hours to solve the system until I reluctantly stopped it (CTRL-C).   
During that time I couldn't do anything else on my computer:  
selecting another window or typing became very, very, very slow as to  
be practically impossible.  Actually, I wanted to suspend the  
computation temporarily, but found that CTRL-Z did not seem to work,  
so I resorted to CTRL-C.  Now I can again do other things (like read  
my email).

Anyhow, enough of my woeful tale.  Thanks again for your help.

---John.