The root condition for polynomial of the second order and a spectral stability of finite-difference schemes for Kuramoto-Tsuzuki equation

Abstract

This paper deals with a root condition for polynomial of the second order. We prove the root criterion for such polynomial with complex coefficients. The criterion coincides with well-known Hurwitz criterion in the case of real coefficients. We apply this root criterion for several three-layer finite-difference schemes for Kuramoto-Tsuzuki equation. We investigate polynomials for symmetrical and DuFort-Frankel finite-difference schemes and polynomial for odd-even scheme. We establish spectral (conditional or uncoditional) stability for these schemes.