STUDIES ON ERROR PROPAGATION FOR CERTAIN NONLINEAR APPROXIMATIONS TO HYPERBOLIC-EQUATIONS - DISCONTINUITIES IN DERIVATIVES

The accuracy of numerical approximations to piecewise smooth solutions of hyperbolic partial differential equations is greatly influenced by the presence of singularities in the solution. In the presence of cou pling (through lower-order terms or variable coefficients), high-order numerical approximations can lose accuracy in large regions, where th e analytical solution is known to be smooth, due to the spreading of t he errors that the singularities introduce in the computation. This ph enomenon, which has been analyzed in the past fifteen years for a numb er of classical linear methods, is studied here for numerical approxim ations obtained with nonlinear essentially non oscillatory (ENO) schem es. The study of the local rate of convergence allows for the identifi cation of the necessary techniques to reduce the spread of errors and to avoid the accuracy loss of the computed approximations. The techniq ues developed can be applied to nonlinear hyperbolic partial different ial equations and systems to sharpen the resolution of corners of rare faction waves.