PDE with RecurrenceTable

• To: mathgroup at smc.vnet.net
• Subject: [mg124740] PDE with RecurrenceTable
• From: Alexei Boulbitch <Alexei.Boulbitch at iee.lu>
• Date: Fri, 3 Feb 2012 02:11:36 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com

```Dear Community,

I am trying to make a simple numeric FEM solution of parabolic PDE. It seems that RecurrenceTable function is designed exactly for such a job. Indeed, in the Help/RecurrenceTable/Scope/Partial Difference Equations one finds an example. My problem is that this example is only one, and I would say, it is not basic enough.

My question: do you know some other examples of the use of the RecurrenceTable for this sort of equations??

I would like to explain: I already went through the MathGroup archive and have seen numerous posts recommending various sophisticated FEM packages. My question is not about them. I want to learn to make simple programs of this sort myself to fast test an equation at hand.

For example, here is a classical equation of temperature conductivity:

(\[PartialD]u(x,t))/\[PartialD]t=a^2*(\[PartialD]^2u(x,t))/\[PartialD]x^2

with a=0.1, the boundary conditions: u[0,t]==1 and u[1,t]==0 and the initial condition u[x,0]== Cos[3 \[Pi]*x/2];

using the explicit method on the rectangular lattice, taken from a textbook:

u[j,k+1]=\[Sigma]*u[j+1,k]+(1-2*\[Sigma])*u[j,k]+\[Sigma]*u[j-1,k];

\[Sigma]=(a^2*\[Tau])/h^2;

Tau and h are temporal and spatial step sizes.

This is the code:

a = 0.1;

h = 0.1;

\[Tau] = 0.0001;

\[Sigma] = a^2*\[Tau]/h^2;

lst2 = RecurrenceTable[{u[j,

k + 1] == \[Sigma]*u[j + 1, k] + (1 - 2*\[Sigma])*

u[j, k] + \[Sigma]*u[j - 1, k], u[j, 0] == Cos[3 Pi*j/20.],

u[0, k] == 1, u[10, k] == 0}, u, {j, 0, 10}, {k, 0, 100}];

Show[{

ListPlot3D[lst2,

AxesLabel -> {Style["t", 16, Italic], Style["x", 16, Italic],

Style["u", 16, Italic]}, PlotStyle -> Blue],

Plot3D[0, {t, 0, 100}, {x, 0, 10},

PlotStyle -> {Yellow, Opacity[0.4]}]

}]

The solution obtained this way, however, does not show evolution. It is clear that with increasing t the temperature, u, should forget its initial form and approach to a straight line. What is wrong?

Thank you, Alexei

Alexei BOULBITCH, Dr., habil.

IEE S.A.

ZAE Weiergewan,

11, rue Edmond Reuter,

L-5326 Contern, LUXEMBOURG

Office phone :  +352-2454-2566

Office fax:       +352-2454-3566

mobile phone:  +49 151 52 40 66 44

e-mail: alexei.boulbitch at iee.lu<mailto:alexei.boulbitch at iee.lu>

```

• Prev by Date: Re: Derivative of experimental data
• Next by Date: Re: Derivative of experimental data
• Previous by thread: Re: Funny Behavior of Module
• Next by thread: Re: PDE with RecurrenceTable