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 wh ere each type is linearly stable. We observe numerically the formation of ternary oscillations, to the left of Riemann shock initial data wi th u(r) = 0, while binary oscillations form to the right of Riemann ra refaction data having u(1) = 0. We derive modulation equations for bot h wave patterns, using them to show that binary oscillations generated by the scheme are numerical artifacts, while computing an explicit so lution for the ternary modulation equations. For positive initial data , the ternary modulation equations remain hyperbolic, while the soluti ons 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. (C) 1998 Elsevier Sci ence B.V.