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Question: Coupled PDEs:Error, (in pdsolve/numeric/animate) unable to compute solution for t>HFloat(0.0): Newton iteration is not converging

Question:Coupled PDEs:Error, (in pdsolve/numeric/animate) unable to compute solution for t>HFloat(0.0): Newton iteration is not converging

Melvin Brown 124 Maple 2018
pde numeric pdsolve differential-equations
March 02 2020
0


 

Analysis of the semiclassical (SC) momentum rate equations

Plotting the ICs and BCs and examining sensitivity to the Re and Im forces

MRB: 24/2/2020, 27/2/2020, 2/3/2020.

We examine solution of the SC version of the momentum rate equations, in which O`ℏ`^2 terms for u(x, t) are removed. A high level of sensitivity to ICs and BCs makes solution finding difficult.

> 

restart;
> 

with(PDETools): with(CodeTools):with(plots):

We set up the initial conditions:

> 

ICu := {u(x, 0) = .1*sin(2*Pi*x)}; ICv := {v(x, 0) = .2*sin(Pi*x)};

{u(x, 0) = .1*sin(2*Pi*x)}

  

{v(x, 0) = .2*sin(Pi*x)}

 (1)
> 

plot([0.1*sin(2*Pi*x),0.2*sin(Pi*x)],x = 0..2, title="ICs:\n u(x,0) (red), v(x,0) (blue)",color=[red,blue],gridlines=true);  

 

The above initial conditions represent a positive velocity field v(x, 0) (blue) and a colliding momentum field u(x, t)(red).

 

Here are the BCs

> 

BCu := {u(0,t) = 0.5*(1-cos(2*Pi*t))};

{u(0, t) = .5-.5*cos(2*Pi*t)}

 (2)
> 

BCv := {v(0,t) = 0.5*sin(2*Pi*t),v(2,t)=-0.5*sin(2*Pi*t)};  

{v(0, t) = .5*sin(2*Pi*t), v(2, t) = -.5*sin(2*Pi*t)}

 (3)
> 

plot([0.5*(1-cos(2*Pi*t)),0.5*sin(2*Pi*t),-0.5*sin(2*Pi*t)],t=0..1,color=[red,blue,blue],linestyle=[dash,dash,dot],title="BCs:\n u(0,t) (red-dash),\n v(0,t) (blue-dash), v(1,t) (blue-dot)",gridlines=true);
> 

 

 

We can now set up the PDEs for the semiclassical case.

> 

hBar:= 1:m:= 1:Fu:= 0.2:Fv:= 0.1:#1.0,0.2
> 

pdeu := diff(u(x,t),t)+u(x,t)/m*(diff(u(x,t),x)) = Fu;

diff(u(x, t), t)+u(x, t)*(diff(u(x, t), x)) = .2

 (4)
> 

pdev := diff(v(x,t),t)+u(x,t)/m*(diff(v(x,t),x))-hBar*(diff(u(x,t),x$2))/(2*m)+v(x,t)*(diff(u(x,t),x))/m = Fv;  

diff(v(x, t), t)+u(x, t)*(diff(v(x, t), x))-(1/2)*(diff(diff(u(x, t), x), x))+v(x, t)*(diff(u(x, t), x)) = .1

 (5)
> 

ICu:={u(x,0) = 0.1*sin(2*Pi*x)};  

{u(x, 0) = .1*sin(2*Pi*x)}

 (6)
> 

ICv:={v(x,0) = 0.2*sin(Pi*x/2)};  

{v(x, 0) = .2*sin((1/2)*Pi*x)}

 (7)
> 

IC := ICu union ICv;  

{u(x, 0) = .1*sin(2*Pi*x), v(x, 0) = .2*sin((1/2)*Pi*x)}

 (8)
> 

BCu := {u(0,t) = 0.5*(1-cos(2*Pi*t)), D[1](u)(2,t) = 0.1*cos(2*Pi*t)};

{u(0, t) = .5-.5*cos(2*Pi*t), (D[1](u))(2, t) = .1*cos(2*Pi*t)}

 (9)
> 

BCv := {v(0,t) = 0.2*(1-cos(2*Pi*t))};  

{v(0, t) = .2-.2*cos(2*Pi*t)}

 (10)
> 

BC := BCu union BCv;  

{u(0, t) = .5-.5*cos(2*Pi*t), v(0, t) = .2-.2*cos(2*Pi*t), (D[1](u))(2, t) = .1*cos(2*Pi*t)}

 (11)

We now set up the PDE solver:

> 

pds := pdsolve({pdeu,pdev},{BC[],IC[]},time = t,range = 0..2,numeric);#'numeric' solution

_m2592591229440

 (12)
> 

Cp:=pds:-animate({[u, color = red, linestyle = dash],[v,color = blue,linestyle = dash]},t = 30,frames = 400,numpoints = 400,title="Semiclassical momentum equations solution for Re and Im momenta u(x,t) (red) and v(x,t) (blue) \n under respective constant positive forces [0.2, 0.1] \n with sinusoidal boundary conditions at x = 0, 1 and sinusoidal initial conditions: \n time = %f ", gridlines = true,linestyle=solid):Cp;

Error, (in pdsolve/numeric/animate) unable to compute solution for t>HFloat(0.0):
Newton iteration is not converging

  

Cp

 (13)

Observations on the quantum case:

The classical equation for u(x, t) is independent of the equation for v(x, t).  u(x, t) (red) is a solution of the classical Burgers equation subject to a force 0.2, but u(x, t) is NOT influenced by v(x, t).  On the otherhand, v(x, t) (blue) is a solution of the quantum dynamics equation subject to force 0.1 and is influenced by u(x, t).   This one way causality (u " implies v")  is a feature of the semiclassical case, and it emphasises the controlling influence of the classical u(x, t), which modulates the quantum solution for v(x, t).  Causally, we have u" implies v".

 

The initial conditions are of low momentum amplitude:"+/-"0.1 for the classical u(x, 0) (red) field and`&+-`(0).2 for v(x, 0) (blue)  but their influence is soon washed out by the boundary conditions "u(0,t) ~1, v(0,t)~0.5" and "v(1,t)~0.5" that drive the momentum dynamics.

 

The temporal frequency of the boundary condition on the v-field is twice that of the classical u-field. This is evident in the above blue transient plot. Moreover, the">=0" boundary condition on the classical u-momentum (red), drives that field in the positive direction, initially overtaking the quantum v(x, t) field, as consistent with the applied forces [0.2, 0.1]. NULLAlthough initially of greater amplitude than the classical u(x, t)field, the v(x, t) momentum field is asymptotically of the same amplitude as the u(x, t) field, but has greater spatial and temporal frequency, owing to the boundary conditions.

 

Referring to the semiclassical momentum rate equations, we note that the classical field u(x, t) (red) modulates the quantum momentum rate equation for v(x, t).

> 

``

 

 

 


 

Download SC-plots.mw

I am having difficulty getting solutions to a pair of PDEs.  Would anyone like to cast an eye over the attached file, incase I am missing something.

Thanks

Melvin
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