## 2017 |

Matthias Sonntag 2017, ISBN: 978-3-8439-3281-3. @phdthesis{sonntagPHD, title = {Shape derivatives and shock capturing for the Navier-Stokes equations in discontinuous Galerkin methods}, author = {Matthias Sonntag}, url = {http://dx.doi.org/10.18419/opus-9342}, doi = {10.18419/opus-9342}, isbn = {978-3-8439-3281-3}, year = {2017}, date = {2017-08-10}, abstract = {This work addresses two different topics, the shape derivatives for the compressible Navier-Stokes equations on the one hand and, on the other hand, the treatment of shocks or other flow discontinuities in Discontinuous Galerkin methods. There is a strong demand for very efficient methods for shape optimization in the aerospace industry, for example drag reduction or lift maximization of an aircraft. The use of gradient based optimization schemes requires derivatives of the cost function with respect to the shape of an object. With the shape derivatives presented in this work, these derivatives can be calculated independent of the parametrization of the object's shape, and, since the derivation takes place in the continuous space, they can be applied to almost any discretization. Nevertheless, one has to take the numerical scheme, which is later applied, into account. For methods based on the variational formulation a difference in the shape derivative, compared to the pointwise approach, arises, which cannot be neglected. Hence, one objective of this work is to derive the shape derivatives of the drag- and lift-coefficient for the Navier-Stokes equations in variational formulation and compare it with the pointwise approach both analytically and numerically. A discrepancy has to be expected, especially for flow phenomena with high gradients or discontinuities which do not fulfill the strong form of the governing equations. These flow phenomena require a special treatment in numerical methods of high order. In the second part of this work, a shock capturing for the Discontinuous Galerkin method is developed which prevents the oscillations originating from the approximation of discontinuities with high order polynomials. Therefore a hybrid approach is presented, where the original DG scheme is coupled with a second order Finite Volume method. In all elements containing shocks or discontinuities the operator of the DG method is replaced by the Finite Volume scheme. This scheme is, due to the use of slope limiters, well known for its strengths in handling shocks. However, in regions where the flow is smooth the Finite Volume method requires a finer resolution for the same accuracy than the Discontinuous Galerkin scheme. Using the same mesh for the FV method as for the DG scheme would lead to a big reduction in resolution. Hence, to compensate this loss the original elements of the mesh are divided into logical sub-cells. By associating exactly one Finite Volume sub-cell to each degree of freedom of a DG element, the same data structures can be used. This enables an efficient implementation of the outlined shock capturing designated for high performance computations. Therefore, not only the basic properties of this hybrid DG/FV sub-cell approach are investigated with several examples, but also studies regarding the parallel efficiency are performed.}, keywords = {}, pubstate = {published}, tppubtype = {phdthesis} } This work addresses two different topics, the shape derivatives for the compressible Navier-Stokes equations on the one hand and, on the other hand, the treatment of shocks or other flow discontinuities in Discontinuous Galerkin methods. There is a strong demand for very efficient methods for shape optimization in the aerospace industry, for example drag reduction or lift maximization of an aircraft. The use of gradient based optimization schemes requires derivatives of the cost function with respect to the shape of an object. With the shape derivatives presented in this work, these derivatives can be calculated independent of the parametrization of the object's shape, and, since the derivation takes place in the continuous space, they can be applied to almost any discretization. Nevertheless, one has to take the numerical scheme, which is later applied, into account. For methods based on the variational formulation a difference in the shape derivative, compared to the pointwise approach, arises, which cannot be neglected. Hence, one objective of this work is to derive the shape derivatives of the drag- and lift-coefficient for the Navier-Stokes equations in variational formulation and compare it with the pointwise approach both analytically and numerically. A discrepancy has to be expected, especially for flow phenomena with high gradients or discontinuities which do not fulfill the strong form of the governing equations. These flow phenomena require a special treatment in numerical methods of high order. In the second part of this work, a shock capturing for the Discontinuous Galerkin method is developed which prevents the oscillations originating from the approximation of discontinuities with high order polynomials. Therefore a hybrid approach is presented, where the original DG scheme is coupled with a second order Finite Volume method. In all elements containing shocks or discontinuities the operator of the DG method is replaced by the Finite Volume scheme. This scheme is, due to the use of slope limiters, well known for its strengths in handling shocks. However, in regions where the flow is smooth the Finite Volume method requires a finer resolution for the same accuracy than the Discontinuous Galerkin scheme. Using the same mesh for the FV method as for the DG scheme would lead to a big reduction in resolution. Hence, to compensate this loss the original elements of the mesh are divided into logical sub-cells. By associating exactly one Finite Volume sub-cell to each degree of freedom of a DG element, the same data structures can be used. This enables an efficient implementation of the outlined shock capturing designated for high performance computations. Therefore, not only the basic properties of this hybrid DG/FV sub-cell approach are investigated with several examples, but also studies regarding the parallel efficiency are performed. |

## 2015 |

Jonatan Núñez-de la Rosa High-Order Methods for Computational Astrophysics PhD Thesis Universität Stuttgart, 2015, ISBN: 978-3-8439-2293-7. @phdthesis{nunez2015phd, title = { High-Order Methods for Computational Astrophysics }, author = {Jonatan Núñez-de la Rosa}, editor = {Claus-Dieter Munz, Michael Dumbser}, url = {http://elib.uni-stuttgart.de/opus/volltexte/2015/10355/pdf/NUNEZ_Dissertation.pdf http://elib.uni-stuttgart.de/opus/volltexte/2015/10355/}, isbn = {978-3-8439-2293-7}, year = {2015}, date = {2015-12-07}, school = {Universität Stuttgart}, abstract = {In computational fluid dynamics, high-order numerical methods have gained quite popularity in the last years due to the need of high fidelity predictions in the simulations. High-order methods are suitable for unsteady flow problems and long-term simulations because they are more efficient when obtaining higher accuracy than low-order methods, and because of their outstanding dissipation and dispersion properties. In the present work, the development and application of three high-order numerical methods, namely, the conservative finite difference (FD) method, the finite volume (FV) method, and the discontinuous Galerkin spectral element method (DGSEM), is presented. These methods are used here for solving three equations systems arising in computational astrophysics on flat spacetimes, specifically, the ideal magnetohydrodynamics (MHD), relativistic hydrodynamics (SRHD) and relativistic magnetohydrodynamics (SRMHD). Our computational framework has been subject to the standard testbench in computational astrophysics. Numerical results of problems having smooth flows, and problems with shock-dominated flows, are also reported. Finite volume methods are numerical methods based on the weak solution of conservation laws in integral form. Unlike finite volume methods, where cell averages of the solution are evolved in time, in the conservative finite difference schemes only the solution at specific nodal points are considered. This difference offers a high efficiency of finite difference over finite volume methods in two and three dimensional high-order calculations because of the form of the utilized stencils in the reconstruction step. Recently, a lot of effort has been put into the development of efficient high-order accurate reconstruction procedures on structured and unstructured meshes. The most widely used procedure to achieve high-order spatial accuracy in finite volume and conservative finite difference methods is the WENO reconstruction. The basic idea of the WENO schemes is based on an adaptive reconstruction procedure to obtain a higher-order approximation on smooth regions while the scheme remains non-oscillatory near discontinuities. For this reason, the WENO formulation is particularly effective when solving conservation laws containing discontinuities. In this work, the FD and FV methods are extended to very high-order accuracy on regular Cartesian meshes by making use of the arbitrary high-order reconstruction WENO operator. The time discretization is carried out with a strong stability-preserving Runge-Kutta (SSPRK) method. The MHD, SRHD and SRMHD equations are then solved with these two methods for problems having strong shock configurations. The discontinuous Galerkin (DG) methods combine the ideas of the finite element (FE) and the finite volume methods. From the FE methods, the solution and test functions in the variational formulation of the conservation law are locally represented by polynomials, allowing to be discontinuous at element faces. In order to stabilize the scheme, from the FV methods are borrowed the ideas of using Riemann solvers, which permit to connect a given element with its direct neighboring ones. One special case in the family of DG methods is the DGSEM. In these methods, the domain is decomposed into quadrilateral/hexahedral elements, and the solution and the fluxes are represented by tensor-product basis functions (high-order Lagrangian interpolants). The integrals are approximated by quadrature, and the nodal points, where the solution is computed, are the Gauss-Legendre quadrature points. With these choices, the DG operator has a dimension-by-dimension splitting form, which yields more efficiency due to less operations and less memory consumption. In this work, the DGSEM has been also extended to the equations of computational astrophysics on flat spacetimes, but restricted only to the MHD and SRHD equations. Because discontinuous solutions form part of the nature of the hyperbolic conservation laws, shock capturing strategies have to be devised, especially for the discontinuous Galerkin method. For the DGSEM, a hybrid DG/FV shock capturing approach is used as the main building block for stabilization of the solution when shocks take place. The hybrid DGSEM/FV is constructed in such a way that, in regions of smooth flows, the DGSEM method is employed, and those parts of the flow having shocks, the DGSEM elements are interpreted as quadrilateral/hexahedral subdomains. In each of these subdomains, the nodal DG solution values are used to build a new local domain composed now of finite volume subcells, which are evolved with a robust finite volume method with third order WENO reconstruction. This new numerical framework for computational astrophysics based on the hybridization of high-order methods brings very promising results.}, keywords = {}, pubstate = {published}, tppubtype = {phdthesis} } In computational fluid dynamics, high-order numerical methods have gained quite popularity in the last years due to the need of high fidelity predictions in the simulations. High-order methods are suitable for unsteady flow problems and long-term simulations because they are more efficient when obtaining higher accuracy than low-order methods, and because of their outstanding dissipation and dispersion properties. In the present work, the development and application of three high-order numerical methods, namely, the conservative finite difference (FD) method, the finite volume (FV) method, and the discontinuous Galerkin spectral element method (DGSEM), is presented. These methods are used here for solving three equations systems arising in computational astrophysics on flat spacetimes, specifically, the ideal magnetohydrodynamics (MHD), relativistic hydrodynamics (SRHD) and relativistic magnetohydrodynamics (SRMHD). Our computational framework has been subject to the standard testbench in computational astrophysics. Numerical results of problems having smooth flows, and problems with shock-dominated flows, are also reported. Finite volume methods are numerical methods based on the weak solution of conservation laws in integral form. Unlike finite volume methods, where cell averages of the solution are evolved in time, in the conservative finite difference schemes only the solution at specific nodal points are considered. This difference offers a high efficiency of finite difference over finite volume methods in two and three dimensional high-order calculations because of the form of the utilized stencils in the reconstruction step. Recently, a lot of effort has been put into the development of efficient high-order accurate reconstruction procedures on structured and unstructured meshes. The most widely used procedure to achieve high-order spatial accuracy in finite volume and conservative finite difference methods is the WENO reconstruction. The basic idea of the WENO schemes is based on an adaptive reconstruction procedure to obtain a higher-order approximation on smooth regions while the scheme remains non-oscillatory near discontinuities. For this reason, the WENO formulation is particularly effective when solving conservation laws containing discontinuities. In this work, the FD and FV methods are extended to very high-order accuracy on regular Cartesian meshes by making use of the arbitrary high-order reconstruction WENO operator. The time discretization is carried out with a strong stability-preserving Runge-Kutta (SSPRK) method. The MHD, SRHD and SRMHD equations are then solved with these two methods for problems having strong shock configurations. The discontinuous Galerkin (DG) methods combine the ideas of the finite element (FE) and the finite volume methods. From the FE methods, the solution and test functions in the variational formulation of the conservation law are locally represented by polynomials, allowing to be discontinuous at element faces. In order to stabilize the scheme, from the FV methods are borrowed the ideas of using Riemann solvers, which permit to connect a given element with its direct neighboring ones. One special case in the family of DG methods is the DGSEM. In these methods, the domain is decomposed into quadrilateral/hexahedral elements, and the solution and the fluxes are represented by tensor-product basis functions (high-order Lagrangian interpolants). The integrals are approximated by quadrature, and the nodal points, where the solution is computed, are the Gauss-Legendre quadrature points. With these choices, the DG operator has a dimension-by-dimension splitting form, which yields more efficiency due to less operations and less memory consumption. In this work, the DGSEM has been also extended to the equations of computational astrophysics on flat spacetimes, but restricted only to the MHD and SRHD equations. Because discontinuous solutions form part of the nature of the hyperbolic conservation laws, shock capturing strategies have to be devised, especially for the discontinuous Galerkin method. For the DGSEM, a hybrid DG/FV shock capturing approach is used as the main building block for stabilization of the solution when shocks take place. The hybrid DGSEM/FV is constructed in such a way that, in regions of smooth flows, the DGSEM method is employed, and those parts of the flow having shocks, the DGSEM elements are interpreted as quadrilateral/hexahedral subdomains. In each of these subdomains, the nodal DG solution values are used to build a new local domain composed now of finite volume subcells, which are evolved with a robust finite volume method with third order WENO reconstruction. This new numerical framework for computational astrophysics based on the hybridization of high-order methods brings very promising results. |

## 2014 |

Florian Hindenlang Mesh Curving Techniques for High Order Parallel Simulations on Unstructured Meshes PhD Thesis Universität Stuttgart, 2014, ISBN: 978-3-8439-1771-1. @phdthesis{HindenlangPhD2014, title = {Mesh Curving Techniques for High Order Parallel Simulations on Unstructured Meshes}, author = {Florian Hindenlang}, editor = {Claus-Dieter Munz and David Kopriva and Gregor Gassner}, url = {https://nrg.iag.uni-stuttgart.de/wp-content/uploads/2016/04/Dissertation.pdf http://dx.doi.org/10.18419/opus-3957}, isbn = {978-3-8439-1771-1}, year = {2014}, date = {2014-12-15}, school = {Universität Stuttgart}, abstract = {In this work, the generation of high order curved three-dimensional hybrid meshes and its application are presented. Meshes with linear edges are the standard of today's state-of-the-art meshing software. Industrial applications typically imply geometrically complex domains, mostly described by curved domain boundaries. To apply high order methods in this context, the geometry - in contrast to classical low order methods - has to be represented with a high order approximation, too. Therefore, a high order element mapping has to be used for the discretization of curved domain boundaries. The main idea here is to rely on existing linear mesh generators and provide additional information to produce high order curved elements. A very promising candidate for future numerical solvers in computational fluid dynamics is the family of high order discontinuous Galerkin (DG) schemes. They are locally conservative schemes, with a continuous polynomial representation within each element and allow a discontinuous solution across element faces. Elements couple only to direct face neighbors, and the discontinuity is resolved via numerical flux functions. As the main focus of this work are curved elements, the different formulations and possible implementations of the DG scheme with non-linear element mappings are discussed in detail. Especially, a highly efficient variant of the DG scheme for hexahedra, namely the discontinuous Galerkin spectral element method (DG-SEM), is presented. The main focus of this thesis is the generation of high order meshes. Several techniques to generate curved elements are described and their applicability to complex geometries is demonstrated. Starting from a linear mesh, the first step curves the element faces representing the curved geometry. Two approaches are presented, the first based on continuity conditions using surface normal vectors and the second based on interpolation of additionally generated surface points. The high order mapping of the volumetric element is computed as a blending of the curved element faces. In the case of boundary layer meshes, the blending may lead to inverted elements. As a remedy to this problem, an additional mesh deformation approach is proposed and validated. Independent thereof, another approach is presented, allowing one to directly generate curved volume mappings from the agglomeration of block-structured meshes. One of the reasons making high order DG schemes attractive for the simulation of fluid dynamics is their parallel efficiency. As future applications in fluid dynamics comprise the resolution of three-dimensional unsteady effects and are increasingly complex, the simulations require more and more computing resources, and weak and strong scalability of the numerical method becomes extremely important. In the last part of this thesis, the parallelization concept of the DG-SEM code Flexi is described in detail. A new domain decomposition strategy based on space filling curves is introduced, and is shown to be simple and flexible. A thorough parallel performance analysis confirms that the overall implementation scales perfectly. Ideal speed-up is maintained for high polynomial degrees, up to the limit of one element per core. As the DG scheme only communicates with direct neighbors, the same parallel efficiency is found on both cartesian meshes as well as fully unstructured meshes. The findings underline that the proposed Discontinuous Galerkin scheme exhibit a great potential for highly resolved simulations on current and future large scale parallel computer systems.}, keywords = {}, pubstate = {published}, tppubtype = {phdthesis} } In this work, the generation of high order curved three-dimensional hybrid meshes and its application are presented. Meshes with linear edges are the standard of today's state-of-the-art meshing software. Industrial applications typically imply geometrically complex domains, mostly described by curved domain boundaries. To apply high order methods in this context, the geometry - in contrast to classical low order methods - has to be represented with a high order approximation, too. Therefore, a high order element mapping has to be used for the discretization of curved domain boundaries. The main idea here is to rely on existing linear mesh generators and provide additional information to produce high order curved elements. A very promising candidate for future numerical solvers in computational fluid dynamics is the family of high order discontinuous Galerkin (DG) schemes. They are locally conservative schemes, with a continuous polynomial representation within each element and allow a discontinuous solution across element faces. Elements couple only to direct face neighbors, and the discontinuity is resolved via numerical flux functions. As the main focus of this work are curved elements, the different formulations and possible implementations of the DG scheme with non-linear element mappings are discussed in detail. Especially, a highly efficient variant of the DG scheme for hexahedra, namely the discontinuous Galerkin spectral element method (DG-SEM), is presented. The main focus of this thesis is the generation of high order meshes. Several techniques to generate curved elements are described and their applicability to complex geometries is demonstrated. Starting from a linear mesh, the first step curves the element faces representing the curved geometry. Two approaches are presented, the first based on continuity conditions using surface normal vectors and the second based on interpolation of additionally generated surface points. The high order mapping of the volumetric element is computed as a blending of the curved element faces. In the case of boundary layer meshes, the blending may lead to inverted elements. As a remedy to this problem, an additional mesh deformation approach is proposed and validated. Independent thereof, another approach is presented, allowing one to directly generate curved volume mappings from the agglomeration of block-structured meshes. One of the reasons making high order DG schemes attractive for the simulation of fluid dynamics is their parallel efficiency. As future applications in fluid dynamics comprise the resolution of three-dimensional unsteady effects and are increasingly complex, the simulations require more and more computing resources, and weak and strong scalability of the numerical method becomes extremely important. In the last part of this thesis, the parallelization concept of the DG-SEM code Flexi is described in detail. A new domain decomposition strategy based on space filling curves is introduced, and is shown to be simple and flexible. A thorough parallel performance analysis confirms that the overall implementation scales perfectly. Ideal speed-up is maintained for high polynomial degrees, up to the limit of one element per core. As the DG scheme only communicates with direct neighbors, the same parallel efficiency is found on both cartesian meshes as well as fully unstructured meshes. The findings underline that the proposed Discontinuous Galerkin scheme exhibit a great potential for highly resolved simulations on current and future large scale parallel computer systems. |

## 2012 |

Christoph Altmann Explicit Discontinuous Galerkin Methods for Magnetohydrodynamics PhD Thesis Universität Stuttgart, 2012, ISBN: 987-3-8439-0707-2. @phdthesis{AltmannPHD2012, title = {Explicit Discontinuous Galerkin Methods for Magnetohydrodynamics}, author = {Christoph Altmann}, editor = {Claus-Dieter Munz and James Rossmanith}, url = {http://elib.uni-stuttgart.de/opus/frontdoor.php?source_opus=7998}, isbn = {987-3-8439-0707-2}, year = {2012}, date = {2012-09-14}, school = {Universität Stuttgart}, abstract = {In this work, the explicit space-time expansion discontinuous Galerkin (STEDG) method is adapted and applied to unsteady ideal and viscous magnetohydrodynamic (MHD) computations. With a special emphasis on shock-capturing and divergence correction of the magnetic eld, enhancements to the STE-DG method are proposed that are necessary within the MHD context.}, keywords = {}, pubstate = {published}, tppubtype = {phdthesis} } In this work, the explicit space-time expansion discontinuous Galerkin (STEDG) method is adapted and applied to unsteady ideal and viscous magnetohydrodynamic (MHD) computations. With a special emphasis on shock-capturing and divergence correction of the magnetic eld, enhancements to the STE-DG method are proposed that are necessary within the MHD context. |

## 2009 |

G. J. Gassner Discontinuous Galerkin Methods for the Unsteady Compressible Navier-Stokes Equations PhD Thesis Universität Stuttgart, 2009. @phdthesis{GassnerPHD2009, title = {Discontinuous Galerkin Methods for the Unsteady Compressible Navier-Stokes Equations}, author = {G. J. Gassner}, url = {http://elib.uni-stuttgart.de/opus/volltexte/2009/3948/}, year = {2009}, date = {2009-01-01}, school = {Universität Stuttgart}, keywords = {}, pubstate = {published}, tppubtype = {phdthesis} } |

# PhD Thesis

## 2017 |

2017, ISBN: 978-3-8439-3281-3. |

## 2015 |

High-Order Methods for Computational Astrophysics PhD Thesis Universität Stuttgart, 2015, ISBN: 978-3-8439-2293-7. |

## 2014 |

Mesh Curving Techniques for High Order Parallel Simulations on Unstructured Meshes PhD Thesis Universität Stuttgart, 2014, ISBN: 978-3-8439-1771-1. |

## 2012 |

Explicit Discontinuous Galerkin Methods for Magnetohydrodynamics PhD Thesis Universität Stuttgart, 2012, ISBN: 987-3-8439-0707-2. |

## 2009 |

Discontinuous Galerkin Methods for the Unsteady Compressible Navier-Stokes Equations PhD Thesis Universität Stuttgart, 2009. |