# explicit scheme stability restriction

Looking at the plain heat equation $u_t=u_{xx}$ the explicit scheme for it would look like the following iteration: $$u_{m,n+1}=\rho u_{m-1,n}+(1-2\rho)u_{m,n}+\rho u_{m+1,n}$$ I noticed this equation resembles some probability equation: basically If I look at it as: $$u_{m,n+1}=p_1 u_{m-1,n}+p_2u_{m,n}+p_3u_{m+1,n}$$ I notice $p_1+p_2+p_3=1$ and if I restrict those numbers to be between $[0,1]$ if as in case of probabilities it matches the stability criteria. Is there some sort of connection between probability and FDM here? It looks to me as $u_{m,n+1}$ is some sort of expectations but I can't make my argument complete so I would appreciate some help on that.

• Well, kind of, but not like you think. First, the explicit stability criterion restricts the value of $\rho$, which doesn't enter your probability interpretation at all. But diffusion (which is described by the heat equation) has a very close connection with random walks, and both can be thought of as limits of discrete equations describing "quantized" diffusion and random walks, respectively. Jul 1, 2016 at 17:49
• Your comments on connections to probability remind me of making estimates based on a Markov Process model. One might be able to view any deterministic dynamics as a Markov Process where you have perfect knowledge of the transition that will take place given some prior state. Jul 1, 2016 at 17:52