2 May 2002
Prof Tim Barth
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.
- Computational Mathematics and Applications Seminar