On the behavior of the total variation in CWENO methods for conservation laws

We consider a family of high-order, weighted essentially non-oscillatory central schemes (CWENO) for approximating solutions of one-dimensional hyperbolic systems of conservation laws. We are interested in the behavior of the total variation (TV) of the approximate solution obtained with these methods. Our numerical results suggest that even though CWENO methods are not total variation diminishing (TVD), they do have bounded total variation (TVB). Moreover. the TV of the approximate solution seems to never increase above the theoretical value, and it approaches it as the mesh is refined. These results are hopefully a first step in the quest for proving the convergence of such high-order methods. (C) 2000 IMACS. Published by Elsevier Science B.V. All rights reserved.