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Numeric Solutions to The Heat Equation

I have been reading a lot on Lucia’s blog about two box models which are essentially an approximation of the heat equation with basis functions which are constant over a box.

The heat equation is given by:

\frac{\partial u}{\partial t} -k\left(\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}\right)=0

or equivalently:

\frac{\partial u}{\partial t} = k \nabla^2 u

The fundamental solutions or Greens functions (also see main body theory) are of the form:

\Phi(\mathbf{x},t) = \Phi(x_1,t)\Phi(x_2,t)\dots\Phi(x_n,t)=\frac{1}{(4\pi k t)^{n/2}}e^{-\mathbf{x}\cdot\mathbf{x}/4kt}.

This suggests my choice of a negative exponential basis in my post (Lagrangian Mechanics and the Heat Equation) was not two bad a choice although, Guasian functions will decay slightly faster then negative exponentials.

Not all solutions are based on on fundamental solutions for instance in the post (Approximations used in Crank-Nicolson method for solving PDEs numerically) I read that the Crank-Nicolson method was the standard method of soliving the Heat equation numericaly.

For instance in 1-D

\frac{\partial u}{\partial t}=k \frac{\partial^2u}{\partial x^2}

the Cranck Nicholson Method is given by:

\frac{u_{i}^{n + 1} - u_{i}^{n}}{\Delta t} = \frac{k}{2 (\Delta x)^2}\left((u_{i + 1}^{n + 1} - 2 u_{i}^{n + 1} + u_{i - 1}^{n + 1}) + (u_{i + 1}^{n} - 2 u_{i}^{n} + u_{i - 1}^{n})\right)

It should be noted that this method produces a difference equation. The values at the next time step can be solved for analytically, using cramer’s rule (see also Invertible matrix, analytic solutions). The frequency domain characteristics can be explored using the z transform. Where the frequency response is given by evaluating the z transform along the unit circle.

Also note that numeric error can be reduced when computing future time steps by either recursive squaring:
or by using matrix decomposition.

For other numeric method of solve partial differential equations see (Numerical partial differential equations), which I posted in the thread (Preparation for PDEs).

Further with regards to crank Nicholson, there is no time dependency on the right hand side of the equations so other methods, can be used to descretize the heat equations such as using Laplace transforms or the matrix exponential.

With regards to Lucia’s blog

My understanding as posted in (Arthur’s Case 2 (I think)) that the main focus of Lucia’s blog posts is to test the model chosen by Tamino:

lucia (Comment#19822) September 12th, 2009 at 9:23 pm

What I mean is– when testing the two box model, you don’t switch to the diffusive model even if it’s more inherently sensible and intelligent. That’s because to test “X” you must test “X”. You can’t test “Y” even if “Y” seems more likely to pass the test.

This is fine but I think that a wider discussion is warranted about how this model is just a simplified version of the heat equations and what principles of modeling and differential equations can be useful to obtain better solutions.

September 15, 2009 Posted by | Math | 3 Comments