Re: Problems in Nintegrate
- To: mathgroup at smc.vnet.net
- Subject: [mg121376] Re: Problems in Nintegrate
- From: Andrew Moylan <amoylan at wolfram.com>
- Date: Tue, 13 Sep 2011 07:18:22 -0400 (EDT)
- Delivered-to: l-mathgroup@mail-archive0.wolfram.com
Hi,
These are difficult numerical integrations. You could spend some time tuning NIntegrate (and your set-up code including NDSolve) to get faster / more accurate results (starting with: manually splitting the integration range at trouble points and/or using EvaluationMonitor to see where NIntegrate is struggling).
But here is a quick and dirty solution:
* Increase the PrecisionGoal and AccuracyGoal for NDSolve since you will be reusing the result in another numerical function
* Method -> {"SymbolicPreprocessing",
"InterpolationPointsSubdivision" -> False} speeds things up a bit in this case
* Crank up the MaxRecursion and let NIntegrate grind out a satisfactory solution: MaxRecursion -> 20
Your code, modified:
Remove["Global`*"];
\[Rho] = 8910;(*mass density:kg/m^3*)v = 4.08;(*sound \
velocity:nm/ps*)\[Beta] = 1.3 10^-5;(*linear expansion: /K*)B =
1.8 10^11;(*bulk modulus:Pa*)c =
3 10^5;(*light speed:nm/ps*)\[Lambda] = 800; \[Omega] =
2 \[Pi] c/\[Lambda];(*light wavelength:nm*)Ce =
1.065 10^3;(*electron heat cap.at 300 K:3.19 10^5 J/m^3K*)Cl =
3.95 10^6;(*lattice heat cap. :J/m^3K=26.1 J/mol.K*)g =
4.4 10^5;(*coupling constant:4.4 10^17 W/m^3.K*)K =
91 10^6;(*thermal conductivity:91 W/m.K->91 10^18*)\[Xi]1 = \
13.5;(*pump absorption depth:nm*)\[Xi]2 = 14.5;(*probe absorption \
depth:nm*)Dl =
2.3 10^-5;(*diffusivity:m^2/s*)R = 0.4;(*reflection at interface*)I0 \
= 1.05 10^10;(*2.77 10^13 J/m^2.pulse (ps)->2.77 10^22*)PulseWidth = \
0.2;(*200 fs*)S[t_] := I0 Exp[-t^2/(2 PulseWidth)^2];
pow[z_, t_] :=
1/\[Xi]1 S[
t] Exp[-z/\[Xi]1];(*W/m^3*)L = 1000;(*sample \
thickness:nm*)solution =
NDSolve[{Ce Te[z, t] D[Te[z, t], t] ==
K D[Te[z, t], z, z] - g (Te[z, t] - Tl[z, t]) + pow[z, t],
Cl D[Tl[z, t], t] == g (Te[z, t] - Tl[z, t]),
Te[z, -100] == Tl[z, -100] == 300,
Te[L, t] == 300, (D[Te[z, t], z] /. z -> 0) == 0}, {Te, Tl}, {z,
0, L}, {t, -100, 100}, MaxSteps -> Infinity,
MaxStepSize -> {1, 0.1}, PrecisionGoal -> 10, AccuracyGoal -> 10][[
1]]
solz = {Te[z, t], Tl[z, t]} /. solution /. z -> 0;
Plot[solz /. t -> tau, {tau, -2, 100}, PlotRange -> All]
dTl = (D[Tl[z, t] /. solution, t] /. {z -> Abs[zz - v (t1 - t2)],
t -> t2});
\[Eta][z_,
t_] := -((3 B \[Beta])/(2 \[Rho] v^2)) NIntegrate[
Evaluate[
Sign[zz - v (t1 - t2)] dTl /. {zz -> z, t1 -> t}], {t2, -100,
100}, AccuracyGoal -> 4, Exclusions -> {t2 == 0},
PrecisionGoal -> 4, MaxRecursion -> 20,
Method -> {"SymbolicPreprocessing",
"InterpolationPointsSubdivision" -> False}];
vals = Table[{t, \[Eta][1, t]}, {t, 1, 20, 0.5}]
ListPlot[vals]
I hope this helps,
Cheers,
Andrew Moylan
Wolfram Research
----- Original Message -----
> From: "Jiwan Kim" <hwoarang.kim at gmail.com>
> To: mathgroup at smc.vnet.net
> Sent: Monday, September 12, 2011 1:21:38 AM
> Subject: Problems in Nintegrate
>
> Hello, Mathgroup.
>
> By solving the coupled differential equation, I got Te[z,t] and
> Tl[z,t]
> solution in the following code.
> Then, I wanted to get the Eta[z,t] using NIntegrate function.
> I am using the ver. 7.0 and AMD dual core, 2GB memory.
> I confirmed that the my code below is working for higher version or a
> better
> computer.
> But it is not working with my computer and program ver.
> I have no idea for these errors...
> The representative error messages are like followings:
>
> * Solve::ifun: Inverse functions are being used by Solve, so some
> solutions
> may not be found; use Reduce for complete solution information. >>
> * NIntegrate::ncvb: NIntegrate failed to converge to prescribed
> accuracy
> after 9 recursive bisections in t2 near {t2} = {-0.69605}. NIntegrate
> obtained -8.17582 and 0.0012150431662771995` for the integral and
> error
> estimates. >>
>
> Is there anything wrong? could you help me ?
> Thank you in advance.
>
> Jiwan.
> _________________________________________________________
> Remove["Global`*"];
> \[Rho] = 8910;(* mass density : kg/m^3 *)
> v = 4.08;(* sound velocity : nm/ps *)
> \[Beta] = 1.3 10^-5;(* linear expansion : /K *)
> B = 1.8 10^11; (* bulk modulus : Pa *)
> c = 3 10^5; (* light speed : nm/ps *)
> \[Lambda] = 800; \[Omega] =
> 2 \[Pi] c/\[Lambda]; (* light wavelength : nm *)
> Ce = 1.065 10^3; (* electron heat cap. at 300 K : 3.19 10^5 J/m^3K *)
> Cl = 3.95 10^6; (* lattice heat cap. : J/m^3K = 26.1 J/mol.K *)
> g = 4.4 10^5; (* coupling constant : 4.4 10^17 W/m^3.K *)
> K = 91 10^6; (* thermal conductivity : 91 W/m.K -> 91 10^18 *)
> \[Xi]1 = 13.5; (* pump absorption depth: nm *)
> \[Xi]2 = 14.5; (* probe absorption depth: nm *)
> Dl = 2.3 10^-5; (* diffusivity : m^2/s *)
> R = 0.4; (* reflection at interface *)
> I0 = 1.05 10^10; (* 2.77 10^13 J/m^2.pulse(ps) -> 2.77 10^22 *)
> PulseWidth = 0.2 ; (* 200 fs *)
> S[t_] := I0 Exp[-t^2/(2 PulseWidth)^2];
> pow[z_, t_] := 1/\[Xi]1 S[t] Exp[-z/\[Xi]1]; (* W/m^3 *)
> L = 1000; (* sample thickness : nm *)
> solution =
> NDSolve[{Ce Te[z, t] D[Te[z, t], t] ==
> K D[Te[z, t], z, z] - g (Te[z, t] - Tl[z, t]) + pow[z, t],
> Cl D[Tl[z, t], t] == g (Te[z, t] - Tl[z, t]),
> Te[z, -100] == Tl[z, -100] == 300,
> Te[L, t] == 300, (D[Te[z, t], z] /. z -> 0) == 0}, {Te, Tl}, {z,
> 0, L}, {t, -100, 100}, MaxSteps -> Infinity,
> MaxStepSize -> {1, 0.1}][[1]]
> solz = {Te[z, t], Tl[z, t]} /. solution /. z -> 0;
> Plot[solz /. t -> tau, {tau, -2, 100}, PlotRange -> All]
> dTl = (D[Tl[z, t] /. solution, t] /. {z -> Abs[zz - v (t1 - t2)],
> t -> t2});
> \[Eta][z_,
> t_] := -((3 B \[Beta])/(2 \[Rho] v^2)) NIntegrate[
> Evaluate[
> Sign[zz - v (t1 - t2)] dTl /. {zz -> z, t1 -> t}], {t2, -100,
> 100}, AccuracyGoal -> 4, Exclusions -> {t2 == 0}];
> vals = Table[{t, \[Eta][1, t]}, {t, 1, 20, 0.5}]
> ListPlot[vals]
> ------------------------------------------------------------------------------
>
>
> -----------------------------------------------------------------------------
> Institute of Physics and Chemistry of Materials Strasbourg (IPCMS)
> Department of Ultrafast Optics and Nanophotonics (DON)
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>