# Stability of PDEs

I am currently trying to solve some PDEs with FiPy. At page 56, the manual mentions (https://www.ctcms.nist.gov/fipy/download/fipy-3.0.pdf).

The largest stable timestep that can be taken for this explicit 1D diffusion problem is $$∆t ≤ ∆x^ 2/(2D)$$.

A few questions that might appear basic but I'm having a really hard time getting my head around:

1. Why is that inequality? How does that scale for 2D and 3D problems?

2. Shouldn't units come into play here? For example, if I converted by time-step from hours to minutes, then clearly the magnitude of the value would change, even though the actual time represented would remain the same? Or does the fact I am also scaling the Diffusivity (which depends both on the units of space and time) ensures the inequality remains sound?

• Do you know what stability and convergence mean in the context of PDEs? Have you looked up von Neumann analysis? – Kyle Kanos Apr 2 '19 at 20:32
• Would Computational Science be a better home for this question? – Qmechanic Apr 2 '19 at 20:42
• @Qmechanic I don't think so. I am using a computer framework to (try to) solve this, but the questions are strictly related to be physics. If I had removed the preamble the question would still stand. – DarioP Apr 2 '19 at 20:47
• @DarioP There's actually nothing about physics in here -- it's purely computational science. That inequality, and the ratio etc., all arise from numerical discretization of the problem. It's not inherent to the PDE itself. Maybe question 2 would be on topic here, because it has to do with units of a physical property. – tpg2114 Apr 2 '19 at 21:46
• As @kyle-kanos partially pointed out this is a problem with numerical stability for which von Neumann analysis can lead to the above inequality. Any book on numerical PDEs would have a number of ways to perform this analysis. – Kyle Mandli Apr 3 '19 at 13:41