High-order methods for convection dominated nonlinear problems using multilevel techniques
For industrial aerodynamic applications to compressible flow simulation, finite volume methods with low order of consistency are a mature technology, usually combined with shock capturing techniques. However, high accuracy is needed for those problems which involve a wide range of spatial and temporal scales. High-order methods are potentially able to deliver high accuracy while at the same time avoiding excessive grid resolution. In spite of that, these methods have not made their way into industrial flow solvers for aircraft design yet. One of the reasons for this is the lack of tailor-suited, best-practice solution techniques that compare favorably to highly-tuned loworder methods. Currently, reliable, efficient, high-order numerical solutions can be guaranteed only in a limited number of cases. One issue to address in order to make these methods competitive is efficiency, which involves different aspects of the solution strategy. This thesis contributes to the development of efficient strategies for solving compressible flows via high-order discretizations. Systems of nonlinear hyperbolic conservation laws govern the dynamics of these flows. When discretizing steady problems, one obtains a large system of nonlinear algebraic equations. We focus on convergence accelerators for the solution of steady problems. Nonetheless, since implicit time discretization produces a very similar set of equations, the techniques presented can be applied to time-dependent problems as well. A combination of implicit Newton’s relaxation and explicit multilevel methods is proposed. The resulting technique is reliable and easy to tune, exceptionally simple compared to other ‘expert systems’. In the framework of implicit relaxation, an important issue is the storage of the system matrix. Strategies alternative to explicit storage of the matrix are introduced and analyzed. We also investigate the underlying high-order spatial discretization by considering and directly comparing the performance of three different methods, all employing a local, discontinuous solution space. Finally, a novel method for multiwavelet-based mesh adaptivity is extended to systems of conservation laws so as to adapt the mesh to the flow features. On the one hand, high-order approximations on rather large-size cells are used in regions where the solution is smooth. On the other hand, low-order approximations together with very fine cells are applied in regions where shocks or large gradients appear. Such an adaptive grid can allocate the resources efficiently, in that cells are concentrated in areas where they are needed, as opposed to uniform mesh refinement.
info:eu-repo/classification/ddc/510, Eulersche Formel, Implizites Euler-Verfahren, Numerische Strömungssimulation, Kompressible Strömung, Erhaltungssatz, Mathematik, high-order methods, computational fluid dynamics, compressible flows, Euler equations, multilevel techniques
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