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PDE solution using NDSolve not smooth

srinivas.gk

Posted 3 years ago

I am trying to solve 2 simultaneous PDE using NDSolve and the solution is not smooth. I think this has to do with setting of the computational grid, but I am not sure how to change it. PDEin:=-2.4669065517374813Derivative[0, 1][Tfin][\[Tau], z] + Derivative[1, 0][Tfin][\[Tau], z] == 0.0001770831817216026(-Tfin[\[Tau], z] + Tfo[\[Tau], z]) PDEo:=0.9327310058932085Derivative[0, 1][Tfo][\[Tau], z] + Derivative[1, 0][Tfo][\[Tau], z] == 0.07577538961255041(10 + 0.026z - Tfo[\[Tau], z]) + 0.00006695469437122687(Tfin[\[Tau], z] - Tfo[\[Tau], z]) IC:={Tfin[0, z] == 10 + 0.026z, Tfo[0, z] == 10 + 0.026z} tA=NDSolve[{PDEin,PDEo,IC,Tfo[\[Tau],0]==10,Tfin[\[Tau],L]==Tfo[\[Tau],L]},{Tfin,Tfo},{\[Tau],0,3600},{z,0,L}] t:=3000 Plot[Evaluate[{Tfin[t,z],Tfo[t,z]}/.tA],{z,0,L}]

I am trying to solve 2 simultaneous PDE using NDSolve and the solution is not smooth. I think this has to do with setting of the computational grid, but I am not sure how to change it.

PDEin:=-2.4669065517374813*Derivative[0, 1][Tfin][\[Tau], z] + Derivative[1, 0][Tfin][\[Tau], z] == 0.0001770831817216026*(-Tfin[\[Tau], z] + Tfo[\[Tau], z])
PDEo:=0.9327310058932085*Derivative[0, 1][Tfo][\[Tau], z] + Derivative[1, 0][Tfo][\[Tau], z] == 0.07577538961255041*(10 + 0.026*z - Tfo[\[Tau], z]) + 0.00006695469437122687*(Tfin[\[Tau], z] - Tfo[\[Tau], z])
IC:={Tfin[0, z] == 10 + 0.026*z, Tfo[0, z] == 10 + 0.026*z}
tA=NDSolve[{PDEin,PDEo,IC,Tfo[\[Tau],0]==10,Tfin[\[Tau],L]==Tfo[\[Tau],L]},{Tfin,Tfo},{\[Tau],0,3600},{z,0,L}]
t:=3000
Plot[Evaluate[{Tfin[t,z],Tfo[t,z]}/.tA],{z,0,L}]

POSTED BY: srinivas.gk

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Updating Name

Posted 3 years ago

Using StiffnessSwitching seems to have smoothened out the solution. Thank you for the suggestions. tA = NDSolve[{Subscript[PDE, in], Subscript[PDE, out], IC, Subscript[T, f, o][\[Tau], 0] == 10, Subscript[T, f, in][\[Tau], L] == Subscript[T, f, o][\[Tau], L]}, {Subscript[T, f, in], Subscript[T, f, o]}, {\[Tau], 0, 3600}, {z, 0, L}, Method -> "StiffnessSwitching"]

Using StiffnessSwitching seems to have smoothened out the solution. Thank you for the suggestions.

tA = NDSolve[{Subscript[PDE, in], Subscript[PDE, out], IC, 
   Subscript[T, f, o][\[Tau], 0] == 10, 
   Subscript[T, f, in][\[Tau], L] == 
    Subscript[T, f, o][\[Tau], L]}, {Subscript[T, f, in], Subscript[T,
    f, o]}, {\[Tau], 0, 3600}, {z, 0, L}, 
  Method -> "StiffnessSwitching"]

POSTED BY: Updating Name

srinivas.gk

Posted 3 years ago

Thanks for the suggestion Neil. I will take a look at the suggested link. The differential equations are for a heat balance between coaxial tubes with heat loss/gain from surroundings. This is why the wavy solution seems strange.

POSTED BY: srinivas.gk

Neil Singer

Neil Singer, AC Kinetics, Inc.

Posted 3 years ago

You have a precision issue because your equations have some rapid transients. L = 1000; PDEin := -2.4669065517374813* Derivative[0, 1][Tfin][\[Tau], z] + Derivative[1, 0][Tfin][\[Tau], z] == 0.0001770831817216026(-Tfin[\[Tau], z] + Tfo[\[Tau], z]) PDEo := 0.9327310058932085Derivative[0, 1][Tfo][\[Tau], z] + Derivative[1, 0][Tfo][\[Tau], z] == 0.07577538961255041(10 + 0.026z - Tfo[\[Tau], z]) + 0.00006695469437122687(Tfin[\[Tau], z] - Tfo[\[Tau], z]) IC := {Tfin[0, z] == 10 + 0.026z, Tfo[0, z] == 10 + 0.026*z} tA = NDSolve[ SetPrecision[{PDEin, PDEo, IC, Tfo[\[Tau], 0] == 10, Tfin[\[Tau], L] == Tfo[\[Tau], L]}, 20], {Tfin, Tfo}, {\[Tau], 0, 3600}, {z, 0, L}, WorkingPrecision -> 20]; gives Plot3D[Evaluate[{Tfin[t, z]} /. tA], {t, 0, 3000}, {z, 0, L}, PlotRange -> All] To get but you still get some "waviness" in the solution. I suggest you look at diving into the various settings for PDE solving using Method. Look at the settings in the documentation in Advanced Numerical Differential Equation Solving in the Wolfram Language if you need a better solution. You can also set the grid spacing, etc. there. Regards, Neil

You have a precision issue because your equations have some rapid transients.

L = 1000; PDEin := -2.4669065517374813*
    Derivative[0, 1][Tfin][\[Tau], z] + 
   Derivative[1, 0][Tfin][\[Tau], z] == 
  0.0001770831817216026*(-Tfin[\[Tau], z] + Tfo[\[Tau], z])
PDEo := 0.9327310058932085*Derivative[0, 1][Tfo][\[Tau], z] + 
   Derivative[1, 0][Tfo][\[Tau], z] == 
  0.07577538961255041*(10 + 0.026*z - Tfo[\[Tau], z]) + 
   0.00006695469437122687*(Tfin[\[Tau], z] - Tfo[\[Tau], z])
IC := {Tfin[0, z] == 10 + 0.026*z, Tfo[0, z] == 10 + 0.026*z}
tA = NDSolve[
   SetPrecision[{PDEin, PDEo, IC, Tfo[\[Tau], 0] == 10, 
     Tfin[\[Tau], L] == Tfo[\[Tau], L]}, 20], {Tfin, Tfo}, {\[Tau], 0,
     3600}, {z, 0, L}, WorkingPrecision -> 20];

gives

Plot3D[Evaluate[{Tfin[t, z]} /. tA], {t, 0, 3000}, {z, 0, L}, 
 PlotRange -> All]

To get

enter image description here

but you still get some "waviness" in the solution. I suggest you look at diving into the various settings for PDE solving using Method. Look at the settings in the documentation in Advanced Numerical Differential Equation Solving in the Wolfram Language if you need a better solution. You can also set the grid spacing, etc. there.

Regards,

Neil

POSTED BY: Neil Singer

Gianluca Gorni

Gianluca Gorni, University of Udine

Posted 3 years ago

You are not giving a value for `L`.

POSTED BY: Gianluca Gorni

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