Non-Oscillatory Schemes for Scalar Conservation Laws

Bojan Popov
Department of Mathematics, Texas A&M University

In the recent years one of the challenging topic in applied
mathematics has been the development of analytical theory and
numerical approximations of nonlinear hyperbolic conservation
laws. Modern algorithms were developed for the accurate computation of
shock discontinuities, rarefaction waves and other phenomena
associated with such equations.  High order Godunov-type schemes are a
fundamental tool for solving conservation laws numerically. There are
two main steps in such schemes: evolution and projection.  In the
original Godunov scheme, the projection is onto piecewise constant
functions -- the cell averages. In a general Godunov-type method, the
projection is onto piecewise polynomials. Many well known methods are
non-oscillatory, however, non-oscillation is, in general, not
sufficient to prove convergence of such methods to the entropy
solution or derive error estimates. For example, the original MinMod,
UNO, ENO, and many central schemes are known to be numerically robust,
at least for piecewise smooth initial data, but theoretical results
about convergence are still missing. We prove a convergence theorem
and an error estimate for a subclass of Godunov-type schemes which
includes simple modifications of Minmod and UNO. In the case of a
linear flux, we derive a new stability result for the original MinMod
scheme and based on this prove an error estimate. Our numerical
experiments show that this stability result holds for any convex flux
and suggest a different approach in the design and analysis of
numerical methods.