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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?

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    $\begingroup$ 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. $\endgroup$
    – Michael
    Aug 4 '16 at 18:50
  • $\begingroup$ @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? $\endgroup$
    – Kamil
    Aug 4 '16 at 18:54
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    $\begingroup$ Yes! I think so. $\endgroup$
    – Michael
    Aug 4 '16 at 22:55
  • $\begingroup$ 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. $\endgroup$
    – David Ketcheson
    Aug 5 '16 at 18:10
  • $\begingroup$ @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? $\endgroup$
    – Kamil
    Sep 23 '16 at 13:26
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The fact that positivity of coefficients of explicit and implicit methods usually leads to sufficient conditions for stability is well known, but rarely discussed in detail. (I'm not aware of any textbooks discussing it in any detail, for example.) I suspect that the method does not receive much attention precisely because it only gives sufficient conditions whereas the von-Neumann method (Fourier analysis) usually gives sufficient and necessary conditions. This paper compares results obtained with the two methods:

B. J. Noye, A new third-order finite-difference method for transient one-dimensional advection—diffusion, Communications in Applied Numerical Methods, 6(4):279–288, 1990

A simple case in which the positive coefficients do not lead to sufficient stability restrictions is the leapfrog method. This is a simple consequence of the fact that not all oscillations indicate an instability.

Positivity of course implies that local extrema do not grow in time, which is why requiring positive coefficients is discussed quite extensively in the literature dealing with the computation of flows with discontinuities, see, e.g.,

S. Spekreijse, Multigrid solution of monotone second-order discretizations of hyperbolic conservation laws, Mathematics of Computation, 49(179):135-155, 1987

A. Jameson, Positive schemes and shock modelling for compressible flows, International Journal for Numerical Methods in Fluids, 20(8-9):743–776, 1995

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