On the sign-stability of the finite difference solutions of one-dimensional parabolic problems

In the numerical solutions of partial differential equations, the preservation of the qualitative properties of the original problem is a more and more important requirement. For ID parabolic equations, one of this properties is the so-called sign-stability: the number of sign-changes of the solution cannot increase in time. This property is investigated for the finite difference solutions, and a sufficient condition is given to guarantee the numerical sign-stability. We prove sufficient conditions for the sign-stability and sign-unstability of tridiagonal matrices.