Symmetric solutions of evolutionary partial differential equations

We show that for a large class of evolutionary nonlinear and nonlocal partial differential equations, symmetry of solutions implies very restrictive properties of the solutions and symmetry axes. These restrictions are formulated in terms of three principles, based on the structure of the equations. The first principle covers equations that...

An adjoint-based shape-optimization method for aerodynamic design

This thesis presents a shape optimization framework for problems that are encountered routinely in Aerodynamic Design. The nature of the framework is numerical. Its focus is wide as different aspects of the shape optimization practice are treated, e.g., the solution of the flow equations, the sensitivity analysis and the parameterization of the...

A meshless front tracking method for the Euler equations of fluid dynamics

A second order front tracking method is developed for solving the Euler equations of inviscid fluid dynamics numerically. Front tracking methods are usually limited to first order accuracy, since they are based on a piecewise constant approximation of the solution. Here the second order convergence is achieved by building a piecewise linear...

Hierarchical Methods for Shape Optimization in Aerodynamics: Multilevel parametric shape algorithms and additive preconditioners

This lecture is devoted to the presentation of two particular "hierarchical" approaches to numerical shape optimization in aerodynamics design: 1) multilevel parametric shape optimization algorithms; 2) additive preconditioners. The first approach is inspired from the classical geometrical multigrid method, except that here the geometrical...

Development and application of generalized MUSTA schemes

Numerical methods for solving non-linear systems of hyperbolic conservation laws via finite volume methods or discontinuous Galerkin finite element methods require, as the building block, a monotone numerical flux. The simplest approach for providing a monotone numerical utilizes a symmetric stencil and does not explicitly make use of wave...

Preconditioners for Linearized Discrete Compressible Euler Equations

We consider a Newton-Krylov approach for discretized compressible Euler equations. A good preconditioner in the Krylov subspace method is essential for obtaining an efficient solver in such an approach. In this paper we compare point-block-Gauss-Seidel, point-block-ILU and point-block-SPAI preconditioners. It turns out that the SPAI method is...

A space-time discontinuous Galerkin finite-element discretization of the Euler equations using entropy variables

A method to numerically solve the Euler equations for fluids with general equations of state is presented. It is based on a formulation solving the conservation equations for either pressure primitive variables or entropy variables, instead of the commonly used conservation variables. We use a space-time discontinuous Galerkin finite-element...

A survey of numerical methods for the calculation of inviscid, possibly rotational Euler flows around aeronautical configurations

A survey of numerical methods for the calculation of inviscid Euler flows is presented. This survey has been prepared for AGAKD FDP WGOT: "Test cases for steady inviscid transonic or supersonic flows". The survey is augmented by a short assessment of the usefulness of the methods. For aerodynamic applications, the existing methods are in a...