# A lazy numerical solution to the diffusion equation

• MATLAB
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## Main Question or Discussion Point

So I want to write a short code to solve the diffusion equation and I want to be lazy and use the gradient function for the spacial differences, so for the second order derivative:
$$\frac{f(i+1)-2*f(i)+f(i-1)}{h^{2}}=\textrm{gradient}(\textrm{gradient}(f,h),h)$$

So the code I wold use is:

$$T=T+dt*D*\textrm{gradient}(\textrm{gradient}(f,h),h)$$

When I do this, I get erroneous results from my initial condition of a Gaussian for example. values start becoming negative which wrong. I don't know where I am going wrong.

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Orodruin
Staff Emeritus
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I think you will need to be more specific about what you are doing and what result you are trying to obtain relative to what you get. Also, posting your code would be helpful.

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My goal is solving the diffusion equation with arbitrary initial conditions and arbitrary boundary conditions.

One problem I noted was I wasn't using the CFL condition on time. Once I did that it worked fine. I have yet to check it works with the formulation above yet.

Orodruin
Staff Emeritus
Not satisfying the CFL condition of course would make your solution unstable. I once taught this to students and made a point out of not telling them about it as I was telling them how to discretise the heat equation. Then I purposely set $\Delta t$ just above the limit for stability to have the resulting animation start out as expected but break down after about ten seconds. That brought down some laughs as I said "... and we see that the solution behaves perfectly as expected ..." at the same time as it started to diverge and a cautionary tale to make sure students would take care to check their stability conditions and not blindly apply numerical recipes.