A Posteriori Error Estimates for Higher Order Godunov Finite Volume Methods on Unstructured Meshes

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 u0 and cellwise p-th order reconstructed functions R 0 pu0. 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 pu0. We then devise computable error estimates that exploit the structure of Godunov finite volume methods. The present theory applies directly to a wide range of finite volume methods in current use including MUSCL, TVD, UNO, and ENO methods [LEE 79, HAR 83, HAR 87, HAR 89, SHU 88, BAR 89, BAR 90, DUR 90, BAR 98, ABG 94, VAN 93]. Issues such as the treatment of nonlinearity and post-processing of dual (adjoint) problem data are discussed. Numerical results for linear advection and nonlinear scalar conservation laws at steady-state are presented to validate the analysis.