Characteristic finite-difference schemes are attractive for convection-dominated problems because they treat advection along Lagrangian trajectories rather than through an Eulerian stencil. The Hopmoc method combines this characteristic treatment with an odd-even Hopscotch decomposition of diffusion, so every update is local and matrix-free. These methods have been analysed mainly for linear advection-diffusion; no nonlinear stability or convergence theory has been available. We establish a conditional maximum-norm Lipschitz-stability and convergence result for Hopmoc applied to the one-dimensional viscous Burgers equation. Under an explicit monotonicity condition on the time step, the proof uses a discrete Grönwall argument without freezing the advection coefficient; in the convergence estimate the only Lipschitz constant is that of the exact solution, so no bound on the gradient of the discrete solution is required. Supporting results include truncation-error estimates along Lagrangian trajectories, a discrete maximum principle under the same condition, and a two-colour von Neumann analysis of the Hopscotch cycle. It shows that the von Neumann condition holds for Hopmoc for every diffusion number r but fails for its backward-differentiation extension, BDF-Hopmoc, when r>5/2, and that the Hopscotch step is non-normal, which motivates the maximum norm. We also show that BDF-Hopmoc, as formulated, is first-order accurate in time. A modified-equation analysis predicts the numerical broadening of the computed shock. The theory is tested against the exact Cole–Hopf travelling wave, with spatial and space-time refinement studies and separate shock- and smooth-region errors. A comparison with Crank-Nicolson separates the effect of local, matrix-free updates from that of a conventional implicit solve.
