# Stability condition for explicit/implicit via non negative coefficients

To make stability proofs simpler, I can consider an explicit scheme written as $$V(n+1,i)=aV(n,i-1)+bV(n,i)+cV(n,i+1)$$ and one can show that if $a,b,c\ge 0$ and $a+b+c\le1$, then the explicit method is stable.

And for implicit, the scheme can be written as $$aV(n+1,i-1)+bV(n+1,i)+cV(n+1,i+1)=V(n,i)$$ and if $a,c \le 0, b\ge 0$, and $a+b+c>0$, then the scheme is stable (This condition can be found in Smith book on ths stability of Thomas algorithm).

However, all the finite difference books typically talk about Matrix stability or Fourier stability. But what I have outlined above is not PDE dependent, so why could not I just check the test on coefficients as above and never learn about other stability methods? Does this rule has limitations?

• The Fourier stability can tell you the necessary and sufficient condition for the stability of a scheme. However, the rule you list above is only one sufficient condition. Aug 4, 2016 at 18:50
• @Michael: I see, so it is basically a stronger condition. But does it mean if I have a scheme, check the above, and if all the conditions hold, I am done? If not that I have to move to other methods. Would it always be my first starting point in analyzing stability? Aug 4, 2016 at 18:54
• Yes! I think so. Aug 4, 2016 at 22:55
• What you're using here is not merely positivity but rather convexity (even more precisely, sublinearity). This is elementary but typically only holds for (some) first order methods. Aug 5, 2016 at 18:10
• @DavidKetcheson: I tried this with explicit method for all types of parabolic equation and it seems to always match general matrix analysis results. Does it mean that for the explicit scheme this is necessary and sufficient for parabolic pdes? Sep 23, 2016 at 13:26