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 coupling (through lower-order terms or variable coefficients), high-order numerical approximations can lose accuracy in large regions, where the analytical solution is known to be smooth, due to the spreading of the errors that the singularities introduce in the computation. This phenomenon, which has been analyzed in the past fifteen years for a number of classical linear methods, is studied here for numerical approximations obtained with nonlinear essentially non oscillatory (ENO) schemes. The study of the local rate of convergence allows for the identification of the necessary techniques to reduce the spread of errors and to avoid the accuracy loss of the computed approximations. The techniques developed can be applied to nonlinear hyperbolic partial differential equations and systems to sharpen the resolution of corners of rarefaction waves.