2 May 2002

14:00

Prof Tim Barth

Abstract

A-Posteriori Error estimates for high order Godunov finite
volume methods are presented which exploit the two solution
representations inherent in the method, viz. as piecewise
constants $u_0$ and cell-wise $q$-th order reconstructed
functions $R^0_q u_0$. The analysis provided here applies
directly to schemes such as MUSCL, TVD, UNO, ENO, WENO or any
other scheme that is a faithful extension of Godunov's method
to high order accuracy in a sense that will be made precise.
Using standard duality arguments, we construct exact error
representation formulas for derived functionals that are
tailored to the class of high order Godunov finite volume
methods with data reconstruction, $R^0_q u_0$. We then consider
computable error estimates that exploit the structure of higher
order Godunov finite volume methods. The analysis technique used
in this work exploits a certain relationship between higher
order Godunov methods and the discontinuous Galerkin method.
Issues such as the treatment of nonlinearity and the optional
post-processing of numerical dual data are also discussed.
Numerical results for linear and nonlinear scalar conservation
laws are presented to verify the analysis. Complete details can
be found in a paper appearing in the proceedings of FVCA3,
Porquerolles, France, June 24-28, 2002.