Remarks on the Consistency of Upwind Source at Interface Schemes on Nonuniform Grids

This note presents a preliminary study of the supra-convergence of well-balanced (finite volume) schemes for conservation laws with a (nonlinear) geometrical source term. In particular, we consider scalar (linear) advection equations in one dimension, for which smooth analytical solutions are available, and upwind interfacial discretizations for the numerical simulation on nonuniform grids (also generated by adaptive procedures). We point out the inconsistent characteristics of the (local) truncation error, and then we illustrate through a simple example the consistency condition formulated by Wendroff and White, which turns out to be effectively compatible with a (strong) convergence theory at optimal rates.