NASA Ames

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.