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978-3-8439-0995-2, Reihe Ingenieurwissenschaften

Alexander Filimon High-Order Schemes for the Solution of Steady State Problems Based on the Iterated Defect Correction Approach

175 Seiten, Dissertation Universität Stuttgart (2012), Softcover, A5

In this work the approach of the iterated defect correction (IDeC) is investigated as a high-order accurate scheme for solving steady state problems. It is applied to the well known conservation laws describing the characteristics of fluid motion, which are the compressible nonlinear Euler and the Navier-Stokes equations.

The present work has been motivated by the German joint venture project MUNA, which had the goal to analyze and, if possible, to minimize the errors in numerical simulations of aerodynamics. This is important in industry as with todays copmutational capacity, the numerical simulation is used far beyond the field of research and development and has found its place in the product development process. The IDeC method has been applied in two ways: as an accurate error estimator without the need of tedious mesh convergence studies as they are used e.g. for the Richardson extrapolation and as a high-order scheme for solving steady state equations. The IDeC method starts from a steady solution obtained by a low-order accurate numerical scheme, which is easy to invert and is characterized by good convergence behaviors. The local defect is then measured by a polynomial reconstruction of the steady solution, which in the present work has been done using WENO reconstruction on unstructured two- and three-dimensional grids. The information of the local defect is then used to formulate a modified problem, which is solved using the low-order scheme. On the one hand the solution of the modified problem provides a good approximation of the global error of the original problem. On the other hand it allows a correction of the starting solution. Applied iteratively one can reach a high-order accurate solution by solving the modified problem with the low-order scheme.

Several other use cases of the IDeC method have been provided in this work, where one of them is the non-intrusive correction of an existing CFD code that can be achieved by solving the local defect completely independently of the code used to solve the low-order scheme. The implementation effort of an iterated defect correction coupling in an existing code is negligible compared to the realization of a high-order polynomial reconstruction in three dimensions on unstructured meshes.