The implicit upwind method for 1-D scalar conservation laws with continuous fluxes

We present the first part of a theory of monotone implicit methods for scalar conservation laws. In this paper, we focus on the implicit upwind scheme. The theoretical investigation of this method is centered around a rigorously verified implicit monotonicity criterion. The relation between the upwind scheme and a discrete entropy inequality is constructed analogously to the classical approach of Crandall and Majda [M. G. Crandall and A. Majda, Math. Comp., 34 (1980), pp. 1-21]. A proof of convergence is given which does not rely on a classical compactness argument. The theoretical results are complemented by a discussion of numerical aspects.