  Large time step explicit schemes in the form originally proposed by LeVeque
have seen a significant revival in recent years. In this paper we consider a
general framework of local 2k + 1 point schemes containing LeVeque's scheme
(denoted as LTS-Godunov) as a member. A modified equation analysis allows us to
interpret each numerical cell interface coefficient of the framework as a
partial numerical viscosity coefficient.
  We identify the least and most diffusive TVD schemes in this framework. The
most diffusive scheme is the 2k + 1-point Lax-Friedrichs scheme (LTS-LxF). The
least diffusive scheme is the Large Time Step scheme of LeVeque based on Roe
upwinding (LTS-Roe). Herein, we prove a generalization of Harten's lemma: all
partial numerical viscosity coefficients of any local unconditionally TVD
scheme are bounded by the values of the corresponding coefficients of the
LTS-Roe and LTS-LxF schemes.
  We discuss the nature of entropy violations associated with the LTS-Roe
scheme, in particular we extend the notion of transonic rarefactions to the LTS
framework. We provide explicit inequalities relating the numerical viscosities
of LTS-Roe and LTS-Godunov across such generalized transonic rarefactions, and
discuss numerical entropy fixes. Finally, we propose a one-parameter family of
Large Time Step TVD schemes spanning the entire range of the admissible total
numerical viscosity. Extensions to nonlinear systems are obtained through the
Roe linearization. The 1D Burgers equation and the Euler system are used as
numerical illustrations.
