R. Donat, STUDIES ON ERROR PROPAGATION FOR CERTAIN NONLINEAR APPROXIMATIONS TO HYPERBOLIC-EQUATIONS - DISCONTINUITIES IN DERIVATIVES, SIAM journal on numerical analysis, 31(3), 1994, pp. 655-679
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.