UTA Department of Mathematics

Applied Mathematics Seminar

Date/Time/Room: Friday (02/14/2003) at 2:00pm in 487 Pickard Hall

Speaker: Bojan Popov, Department of Mathematics, Texas A&M University

"Non-Oscillatory Schemes for Scalar Conservation Laws"

Abstract: A class of Godunov-type schemes for solving scalar conservation laws will be considered. 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 the general Godunov-type method, the projection is onto piecewise polynomials. Many well-known methods are nonoscillatory, however, nonoscillation is, in general, not sufficient to prove convergence of such methods to the entropy solution or derive error estimates. For example, MinMod, UNO, ENO, and WENO methods are known to be numerically robust, at least for piecewise smooth initial data, but theoretical results about convergence are still missing. The notion of weakly nonoscillatory schemes (WNO) will be introduced. For example, any Godunov-type scheme with nonoscillatory evolution and projection is WNO. The main result is a convergence theorem and an error estimate for a subclass of WNO schemes which includes simple modifications of MinMod and UNO. In the case of a linear flux, it will be shown that the modified MinMod scheme coincides with the original MinMod scheme, and the error estimate can be improved in this case.