Scientific Computation and Applied & Industrial Mathematics Seminar: Tyrone Phillips

The largest and most difficult numerical approximation error to estimate is discretization error.  Residual-based discretization error estimation methods are a category of error estimators that use an estimate of the source of discretization error and information about the specific application to estimate the discretization error using only one grid level. The higher-order terms are truncated from the discretized equations and are the local source of discretization error. The accuracy of the resulting discretization error estimate depends solely on the accuracy of the estimated truncation error. Residual-based methods require only one grid level compared to the more commonly used Richardson extrapolation which requires at least two. Reducing the required number of grid levels reduces computational expense and, since only one grid level is required, can be applied to unstructured grids where multiple quality grid levels are difficult to produce. The two residual-based discretization error estimators of interest are defect correction and error transport equations.  The focus of this work is the development, improvement, and evaluation of various truncation error estimation methods considering the accuracy of the truncation error estimate and the resulting discretization error estimates. The minimum requirements for accurate truncation error estimation is specified along with proper treatment for several boundary conditions. The single grid methods require that the continuous operator be modified at the boundary to be consistent with the implemented boundary conditions.  The methods are evaluated using various Euler applications. Defect correction showed to be more accurate for areas of larger discretization error; however, the cost was substantial (although cheaper than the primal problem) compared to the cost of solving the ETEs which was essentially free due to the linearization.