Residual Methods for Discretization Error Estimation

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The largest and most difficult numerical approximation error to estimate is discretization error. This study investigates the accuracy of two different residual-based discretization error estimators: discretization error transport equations and defect correction methods. Residual methods are a category of error estimators that use a discrete solution and information about the problem being solved (either the differential or the discrete equations) to calculate an error estimate using only one grid level. As few as two grid levels are needed if the reliability of the error estimate is to be assessed. This is advantageous because the reliability of all discretization error estimators require that the numerical solution, or solutions, be solved on sufficiently fine grids, which is often difficult to achieve for complex scientific computing applications. Two different linearization methods for the discretization error transport equation will also be studied. The estimated discretization error for these residual-based methods will be compared to Richardson extrapolation using exact solutions to 1D Burgers’ equation and the 2D Euler equations. The exact solutions to the Euler equations include two manufactured solutions, Ringleb's flow, and a supersonic vortex flow.