Binary and ternary oscillations in a cubic numerical scheme

We study a central difference semi-discretization of the cubic scalar conservation law. Both spatial period-2 (binary) and period-3 (ternary) oscillations are stationary solutions of this scheme, and we find where each type is linearly stable. We observe numerically the formation of ternary oscillations, to the left of Riemann shock initial data with u_r = 0, while binary oscillations form to the right of Riemann rarefaction data having u_l = 0. We derive modulation equations for both wave patterns, using them to show that binary oscillations generated by the scheme are numerical artifacts, while computing an explicit solution for the ternary modulation equations. For positive initial data, the ternary modulation equations remain hyperbolic, while the solutions enter an elliptic region for data of both signs. Conditions under which solutions of the ternary modulation equations can be inverted to yield period-3 oscillations are also discussed.