Fokker–Planck Mass Transport Model

Set up a Fokker–Planck equation:

∂c(t,x)

∂t

+

masstransportmodel

︷

∇·

0	0
0	-d

∇c(x)+

y

2ξωy+x

c(x)

=0

Set up the mass transport model variables

vars

:

In[1]:=

vars={c[t,x,y],t,{x,y}};

Set up a rectangular domain:

In[2]:=

Ω=Rectangle[{-10,-10},{10,10}];

Specify the model parameter's species diffusivity

d

, the convection velocity term and parameters:

In[3]:=

pars="ModelForm""Conservative","DiffusionCoefficient"{{0,0},{0,d}},"MassConvectionVelocity"{y,-2ξωy-x},d1/10,ω1,ξ1/20,μ{5,5},σ1/9*IdentityMatrix[2];

Set up the initial conditions:

In[4]:=

ics=c[0,x,y]PDF[MultinormalDistribution[μ,σ]/.pars,{x,y}];

Set up the equation:

In[5]:=

eqn={MassTransportPDEComponent[vars,pars]0,ics}

Out[5]=



∇

{x,y}

·{0,0},0,-

1

10

.

∇

{x,y}

c[t,x,y]+

∇

{x,y}

·yc[t,x,y],-x-

y

10

c[t,x,y]+

(1,0,0)

c

[t,x,y]0,c[0,x,y]

9

1

2

-9

2

(-5+x)

-9

2

(-5+y)





2π



In this case, a first-order mesh is sufficient to solve the Fokker–Planck equation. Using a higher mesh order will result in increased memory consumption.

While solving,

NDSolve

will warn about the convection-dominant nature of the PDE:

In[6]:=

cfun=Monitor[NDSolveValue[eqn,c,{t,0,100},{x,y}∈Ω,Method{"PDEDiscretization"{"MethodOfLines",{"SpatialDiscretization"{"FiniteElement","MeshOptions"{"MaxCellMeasure"0.05,"MeshOrder"1}}}}},EvaluationMonitor(monitor=Row[{"t = ",CForm[t]}])],monitor]

NDSolveValue

:The PDE is convection dominated and the result may not be stable. Adding artificial diffusion may help.

Out[6]=

InterpolatingFunction



Domain: {{0.,100.},{-10.,10.},{-10.,10.}}

Output: scalar

Data not saved. Save now



Visualize the expected solution at the point

{0,0}

:

In[7]:=

Plotcfun[t,0,0],{t,0,100},



Out[7]=

Wolfram Language Example Repository

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Fokker–Planck Mass Transport Model

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