Re: Options for DSolve
- To: mathgroup at smc.vnet.net
- Subject: [mg123424] Re: Options for DSolve
- From: Oliver Ruebenkoenig <ruebenko at wolfram.com>
- Date: Wed, 7 Dec 2011 06:12:56 -0500 (EST)
- Delivered-to: l-mathgroup@mail-archive0.wolfram.com
- References: <201112060812.DAA18378@smc.vnet.net>
Hello Martin,
On Tue, 6 Dec 2011, Martin wrote:
> Hello,
>
> I have a question concerning the options for Mathematica's DSolve. I
> want to find the InterpolatingFunction for a function depending on two
> independent variables U[k,phi] in the range {k,1,500} and {phi,
> 1600,2500}. I try to solve a single differential equation for U[k,phi]
> involving the first derivative with respect to k, and the first and
> the second derivative with respect to phi.
> I use DSolve in the following form:
> NDSolve[{ differential equation , initial condition }, U , {k,
> 1,500} , {phi,1600,2500} , Options]
> My initial condition is U[500,phi]= 2.5 phi^2 .
>
> The method worked for simpler differential equations involving also
> two independent variables. The actual one however is a bit more
> complicated, involving for example Coth and Tanh. Although Mathematica
> finds a solution, it cannot be correct, since adding options like
> PrecisionGoal change the result dramatically. So I tried to improve
> the result using such options, but I don't know what is the right
> strategy. I considered
> PrecisionGoal ,
> AccuracyGoal ,
> MaxStepSize.
>
> Using PrecisionGoal and AccuracyGoal simultaneously, at PrecisionGoal-
>> 18, AccuracyGoal->10 I obtain the following warnings:
>
> NDSolve::mxsst: Using maximum number of grid points 10000 allowed by
> the MaxPoints or MinStepSize options for independent variable \
> [CurlyPhi].
> NDSolve::ndtol: Tolerances requested by the AccuracyGoal and
> PrecisionGoal options could not be achieved at k == 500.`.
>
> So I decided to decrease the StepSize (which means increasing the
> number of grid points, right?) by
> MaxStepSize->0.01 which results in the warnings:
>
> NDSolve::eerri: Warning: Estimated initial error on the specified
> spatial grid in the direction of independent variable \[CurlyPhi]
> exceeds prescribed error tolerance
> NDSolve::ndtol: Tolerances requested by the AccuracyGoal and
> PrecisionGoal options could not be achieved at k == 500.`
>
> Could someone please explain to me what is a meaningful strategy in
> such a case?
> A further question: does MaxStepSize change the number of grid points
> used in the computation? How can I tell Mathematica how much
> GridPoints I want to use?
>
> Here's the concrete example (copy- and paste-able into notebook):
>
>
> T = 45
> mu = 254
> g = 3.2
>
> sol = NDSolve[{k*D[U[k, \[CurlyPhi]], k] ==
> k^5/(12*\[Pi]^2)*(3/Sqrt[
> k^2 + 2*D[U[k, \[CurlyPhi]], \[CurlyPhi]]]*
> Coth[Sqrt[k^2 + 2*D[U[k, \[CurlyPhi]], \[CurlyPhi]]]/(2*T)] +
> 1/Sqrt[k^2 + 2*D[U[k, \[CurlyPhi]], \[CurlyPhi]] +
> 4*\[CurlyPhi]*D[U[k, \[CurlyPhi]], {\[CurlyPhi], 2}]]*
> Coth[Sqrt[
> k^2 + 2*D[U[k, \[CurlyPhi]], \[CurlyPhi]] +
> 4*\[CurlyPhi]*D[U[k, \[CurlyPhi]], {\[CurlyPhi], 2}]]/(
> 2*T)] - (2*3*2)/Sqrt[
> k^2 + g^2*\[CurlyPhi]]*(Tanh[(
> Sqrt[k^2 + g^2*\[CurlyPhi]] - mu)/(2*T)] +
> Tanh[(Sqrt[k^2 + g^2*\[CurlyPhi]] + mu)/(2*T)])),
> U[500, \[CurlyPhi]] == 10/4*\[CurlyPhi]^2},
> U, {k, 1, 500}, {\[CurlyPhi], 1600, 2500}, PrecisionGoal -> 18,
> AccuracyGoal -> 10]
>
>
> Thank you,
> Martin
>
>
Some general comments:
Have a look in the tutorial/NDSolvePDE that has a section on Controlling
the Spatial Grid Selection. That might be helpful. Also you could change
your g = 3.2 to g = 16/5; There is also a message about insufficient
boundary conditions. Adding some U[k,1600] and U[k,2500] would be good.
I hope this gets you started.
Oliver
- References:
- Options for DSolve
- From: Martin <mgrahl@gmx.net>
- Options for DSolve