Transient Landau–Ginzburg Model

Solve the Landau–Ginzburg equations, used in the theory of superconductivity, split into a real

u

and imaginary part

v

:

∂u ∂t +∇(-∇u)+∇(-∇v)	=	u-(u-2v)( 2 u + 2 v )
∂v ∂t +∇(∇u)+∇(-∇v)	=	v-(u+2v)( 2 u + 2 v )

Set up the equations:

In[1]:=

eqn=

∇

{x}

·

-

∇

{x}

u[t,x]+

∇

{x}

·

-

∇

{x}

v[t,x]+(-u[t,x]+(u[t,x]-2v[t,x])(

2

u[t,x]

+

2

v[t,x]

))+

(1,0)

u

[t,x]0,

∇

{x}

·

∇

{x}

u[t,x]+

∇

{x}

·

-

∇

{x}

v[t,x]+(-v[t,x]+(2u[t,x]+v[t,x])(

2

u[t,x]

+

2

v[t,x]

))+

(1,0)

v

[t,x]0;

Set up the initial conditions:

In[2]:=

ics={u[0,x]1-Exp[-x^2],v[0,x]Exp[-(10-x)^2]};

Specify the boundary conditions:

In[3]:=

bcs={u[t,-50]1,u[t,50]1,v[t,-50]0,v[t,50]0}

Out[3]=

{u[t,-50]1,u[t,50]1,v[t,-50]0,v[t,50]0}

Solve the equations on a mesh with specified spacing:

In[4]:=

{ufun,vfun}=NDSolveValue[{eqn,ics,bcs},{u,v},{t,0,40},{x,-50,50},Method{"MethodOfLines","SpatialDiscretization"{"FiniteElement","MeshOptions"{"MaxCellMeasure"0.4}}}]

Out[4]=



InterpolatingFunction



Domain: {{0.,40.},{-50.,50.}}

Output: scalar

Data not saved. Save now



,

InterpolatingFunction



Domain: {{0.,40.},{-50.,50.}}

Output: scalar

Data not saved. Save now





Visualize the result:

In[5]:=

DensityPlotEvaluate[Sqrt[ufun[t,x]^2+vfun[t,x]^2]],{x,-50,50},{t,0,40},



Out[5]=

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