Chapter 4
Part II – Riemann Solvers
In **finite** **volume** **methods**, integrating conservation laws over a control **volume** leads to a formulation which requires the evaluation of local Riemann problems at each cell interface, see Chapter 2 for more details. The initial states for these problems are typically given by the left and right adjacent cell values. Since these local Riemann problems have to be solved many times in order to find the numerical solution, the Riemann solver is a building block of the **finite** **volume** method. Over the last decades, many different Riemann solvers were developed, see e.g. [61] for a broad overview. The main challenges are the need for com- putational efficiency and easy implementation, while at the same time, accurate results without artificial oscillations need to be obtained.

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approximating the value function, and exhibits the curse of dimensionality. The Monte Carlo simulation will subsequently inherit and amplify the curse of dimensionality from the dynamic programming. In comparison, the **finite** **volume** method only needs to incur the cost once.
This interpolation difference exists even within the function approximation framework, and explains some of the speed differences between continuous time and discrete time **methods**. In continuous time, the search problem only depends on the grids, but not on the specific parameter values. However, under discrete time **methods**, the search problem depends on both the grids and parameter values, and the search problem needs to be resolved for new parameter values. Of course, in practice, the interpolation locations will not change too much for a new set of parameters, and an implementation can and should leverage previous search results to reduce the scale dependence from log(n knots ). However, this is problem-dependent and requires

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Ausgangssituation: The thermal comfort of passengers is the essential design objective in the ca- bin air ventilation layout for modern civil passenger aircraft [1]. The heat flux in an aircraft cabin depends on many physical factors, for example solar radiation, electrical equipment, heat release of the passengers and the interior. Our focal point is the influence of the monuments, for example seats, which are heated up and, if the room temperature drops, returning their heat to the environment. This effect is of major importance for the control of environmental control systems (ECS), since the reac- tion to a changed heat inflow will be delayed. Usually, only the heat flux in the cabin air is simulated with stationary boundary conditions for the ECS and radiation models for the monuments. Solving the Reynolds averaged Navier Stokes equations with a buoyancy term for the cabin air and additio- nally the heat transfer inside the monuments both with **finite** **volume** **methods** would be very time consuming. A solver which solves the heat transfer inside the monuments in linear time reduces the runtime significantly.

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Over the last about ten years the development of the discontinuous Galerkin (DG) **methods** has attracted more and more research groups all over the world, significantly increasing the pace of the development of these **methods**, to work on DG [5, 6, 9, 10, 16, 17, 18, 23, 30, 31, 32, 38, 41, 42]. In fact, it can be observed that to an increasing extent discontinuous Galerkin **methods** are now applied to problems which traditionally where solved using **finite** **volume** **methods**. The reason for this trend can be identified in several advantages of the discontinuous Galerkin **methods** over **finite** **volume** **methods**. Second order **finite** **volume** **methods** are achieved by employing a second order accurate reconstruction. The extension of a second order **finite** **volume** scheme to a (theoretically) third order scheme requires a third order accurate reconstruction which on unstructured meshes is very cumbersome and which in practice shows deterioration of order. On unstructured meshes **finite** **volume** **methods** of even higher order are virtually impossible. These difficulties bound the order of numerical computations in industrial applications to second order. In contrast to this, the order of discontinuous Galerkin **methods**, applied to problems with regular solutions, depends on the degree of the approximating polynomials only which can easily be increased, dramatically simplifying the use of higher order **methods** on unstructured meshes. Furthermore, the stencil of most discontinuous Galerkin schemes is minimal in the sense that each element communicates only with its direct neighbors. In contrast to the increasing number of elements or mesh points communicating for increasing accuracy of **finite** **volume** **methods**, the inter-element communication of discontinuous Galerkin **methods** is the same for any order. The compactness of the discontinuous Galerkin method has clear advantages in parallelization, which does not require additional element layers at partition boundaries. Also due to simple communication at element interfaces, elements with so- called ‘hanging nodes’ can be treated just as easily as elements without hanging nodes, a fact that simplifies local mesh refinement (h-refinement). In addition to this, the communication at element interfaces is identical for any order of the method which simplifies the use of **methods** of differing orders in adjacent elements. This allows for the variation of the order of the numerical scheme over the computational domain, which in combination with h-refinement leads to the so-called hp-refinement algorithms, where p-refinement denotes the variation of the polynomial degree p.

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Many dynamical processes can be described by partial differential equations, and convection dominated phenomena as occuring in fluid dynamics are often modeled by hyperbolic conservation laws. As an example, the Euler equations of gas dynamics, the shallow water equations or the Savage-Hutter equations are used to characterize different forms of inviscid fluid flow. Since the structure of hyperbolic conservation laws is different from that of elliptic or parabolic equations, there is a need for specially designed numerical **methods** that take into account characteristics of the solutions, like discontinuities or the transport direction. Classically, partial differential equations are handled using **Finite** Element **Methods** (FEM), **Finite** Difference **Methods** (FDM) or **Finite** **Volume** **Methods** (FVM) on structured or unstructured meshes [BS02], [CL91], [GR96], [EGH00], [Kr¨ o97], [LeV92], [LeV02]. In these **methods**, a mesh is a covering of the computational domain consisting of pairwise disjoint polygons – the cells – which have to satisfy several geometrical and conformity constraints in order to ensure a good approximation of the solution. A main drawback of these **methods** when applied to problems with complicated or even time-dependent computational domains is the time consuming construction of such a mesh.

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Most problems from mathematics, nature, and engineering can be classified to be part of continuum mechanics and can be modeled as boundary value problems using partial differential equations. These problems are usually for- mulated on a subset of the three-dimensional continuous space and can often not be solved analytically, so that we need find a solution with numerical ap- proximations. Numerical **methods** require the discretization of the continuous space which will be divided into smaller entities that couple with neighbor- ing ones. A large variety of these **methods** exist, of which we briefly describe the most commonly used ones. With **finite** difference **methods** (FDM), dif- ferential operators are evaluated as difference quotients on a **finite** number of grid points. The idea of **finite** **volume** **methods** (FVM) is the preservation of conserved quantities on small volumes by applying the Gauss-Ostrogradski theorem, which results in balancing volumetric averages with fluxes on inter- faces of neighboring volumes. In **finite** element **methods** (FEM), we specify a function space of piecewise polynomials in which we find the function that satisfies the partial differential equation of the investigated problem as a best approximation. The residual is projected orthogonal to the space of piecewise polynomials, which is equivalent to minimizing the energy for elliptic equations (Brenner and Scott 2008).

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Several boundary conditions are explained, followed by exemplification of construction of the system matrix, the iteration matrix and the curl curl matrix for assorted **finite** **volume** **methods**. The eigenvalue distribution of the iteration matrix of a given **finite** **volume** time domain method foretells about the sta- bility of the method for a given time step and time update scheme. Various time marching schemes in addition to Runge–Kutta 1 (first order) and Runge–Kutta 2 (second order) are examined for temporal approximation. It is found in practice that first order **methods** have at least one stable time marching scheme for both types of flux approximation, central flux and upwind flux, whereas this is not the case for second order **finite** **volume** **methods**. Second order central flux **finite** **volume** method is not stable for the time marching schemes (Runge–Kutta and Leap–Frog) under consideration, where as there are multiple time marching schemes that can be employed for second order upwind flux **finite** **volume** method. As the higher order **methods** induce more computational cost, only first and second order approximations in time are considered for investigation.

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An exact quantitative comparison of run-times is not possible, since the two codes are research codes written in different languages and by different programmers. However, it is fair to say that for the very smooth test prob- lems 10.1 (see Table 10.2 and Table 10.3) and 10.2 the higher order **finite** **volume** scheme is asymptotically more efficient. For the more realistic, and less smooth, test problems Section 10.4, both the FD and the FV code give qualitatively the same results on the same grid, but the FD scheme is much faster than the FV scheme. It would be desirable to run the FV scheme on a coarser grid to reduce the runtime, but then the inflow data for the jet would be resolved by less then 10 cells, and the flow is not sufficiently resolved any more. This might be different for broader currents. Note that the FV scheme could resolve small gravity waves which were completely smeared by the FD scheme. However, these waves quickly leave the computational domain and do not seem to have a noticeable impact on the major currents.

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The ambition and originality of this study concern the modifi cation of a GKS in order to include the effect of turbulence in the numerical fl uxes. A standard linear two-equation turbulence model is used to model turbulence. It is important to note that the use of turbulence models with gas-kinetic schemes is still relatively unexplored, whereas they are commonly used with Lattice Boltzmann **methods** (3) . The scheme used is the GKS developed by Xu in 2001 (6) , also investigated by other researchers (7,8) . It has shown remarkable reliability and accuracy in a number of cases, ranging from viscous subsonic fl ows to hypersonics.

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Several dedicated preconditioners have been developed for immersed **finite** element **methods**. It is demonstrated in [ 47 ] that a diagonal preconditioner in combination with the constraining of very small basis functions results in an effec- tive treatment for systems with linear bases. With certain restrictions to the cut-element-geometry, [ 48 ] derives that a scalable preconditioner for linear bases is obtained by com- bining diagonal scaling of basis functions on cut elements with standard multigrid techniques for the remaining basis functions. In [ 49 ] the scaling of a Balancing Domain Decom- position by Constraints (BDDC) is tailored to cut elements, and this is demonstrated to be effective with linear basis functions. An algebraic preconditioning technique is pre- sented in [ 35 ], which results in an effective treatment for smooth function spaces. References [ 50 ] and [ 51 ] establish that additive Schwarz preconditioners can effectively resolve the conditioning problems of immersed **finite** element meth- ods with higher-order discretizations, for both isogeometric and hp-**finite** element function spaces and for both symmetric positive definite (SPD) and non-SPD problems. Furthermore, the numerical investigation in [ 50 ] conveys that the condi- tioning of immersed systems treated by an additive Schwarz preconditioner is very similar to that of mesh-fitting systems. In particular, the condition number of an additive Schwarz preconditioned immersed system exhibits the same mesh- size dependence as mesh-fitting approaches [ 52 , 53 ], which opens the doors to the application of established concepts of multigrid preconditioning. It should be mentioned that sim- ilar conditioning problems as in immersed **methods** occur in XFEM and GFEM. Dedicated preconditioners have been developed for these problems as well, a survey of which can be found in [ 50 ].

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A stress-based mixed **finite** element method with reduced symmetry for Linear Elas- ticity featuring Raviart-Thomas elements is extended first to the frictionless Signorini problem and then to contact problems obeying Coulomb’s friction law. Different possible discretizations of the contact constraints are examined. The resulting (quasi-)variational inequalities with nonsmooth constraints are solved using Lagrange multipliers, a semi- smooth Newton method and, in case of Coulomb friction, a fixed point iteration. A reconstruction-based a-posteriori error estimator is derived. Its reliabilty is shown under certain regularity assumptions on the solution that correspond to a uniqueness crite- rion for the solution of Coulomb’s problem. The efficiency of the resulting adaptive refinement strategy is tested by a number of numerical experiments in two and three dimensions.

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Besonderer Dank gilt Alexander Schwarz f¨ ur seine Unterst¨ utzung bereits in meiner Mas- terarbeit zu gemischten **Finite** Elemente Methoden. F¨ ur die inhaltliche Ausrichtung und die thematische Eingrenzung meiner Dissertation hat er einen wichtigen Beitrag geleis- tet. Durch sein großes Engagement, die fachlichen Diskussionen und seine unersch¨opfliche Begeisterung f¨ ur die Mechanik hat er wesentlich zum erfolgreichen Abschluss der Arbeit beigetragen. Im Besonderen m¨ochte ich meinen B¨ uronachbarn und Weggef¨ahrten Vero- nica Lemke und Simon Fausten danken. Sie sind in der Zeit des Zusammenarbeitens zu wichtigen Freunden geworden, die zu jeder Zeit Freude und Mut, aber auch Frust und Entt¨auschung mit mir teilten und einfach immer da sind. Ich danke der gesamten Ar- beitsgruppe f¨ ur die freundschaftliche Arbeitsatmosph¨are, den wertvollen fachlichen Aus- tausch und die stete Hilfsbereitschaft. Aufs Herzlichste m¨ochte ich meinen derzeitigen und ehemaligen Kolleginnen und Kollegen am Institut danken, zu denen Solveigh Averweg, Daniel Balzani, Julia Bergmann, Moritz Bloßfeld, Dominik, Brands, Sarah Brinkhues, Vera Ebbing, Bernhard Eidel, Simon Fausten, Ashutosh Gandhi, Markus von Hoegen, Maximilian Igelb¨ uscher, Veronika Jorisch, Marc-Andre Keip, Simon Kugai, Matthias Labusch, Veronica Lemke, Petra Lindner-Roull´e, Sascha Maassen, Simon Maike, Rainer Niekamp, Paulo Nigro, Yasemin ¨ Ozmen, Mangesh Pise, Sabine Ressel, Mohammad Sarhil, Lisa Scheunemann, Thomas Schmidt, Serdar Serdas, Steffen Specht, Karl Steeger, Masato Tanaka, Huy Ngoc Thai, Sonja Uebing und Nils Viebahn geh¨oren.

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In recent years, several high-order **methods** for the shallow water equa- tions have been proposed that can handle these problems, like [BoM10, GPC07, CGL08, NPP06, XiS11]. However, most of them either only treat the one-dimensional case or are of at most second order. Schemes for the two-dimensional shallow water equations that are of at least third order are from Gallardo et al.,[GPC07], and Xing and Shu, [XiS11]. The rst is of third order in wet, smooth regions and is restricted to grids with quadrilat- eral cells. The latter can theoretically be extended to arbitrary high order, but is restricted to grids with rectangular cells. The scheme presented in this work is of theoretically arbitrary high order and the requirement of a grid with triangular cells stems from the organization of the computation, but it is not relevant for the scheme itself.

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In order to make this work self-consistent, in this chapter we present a short outline of the mathematical framework needed in the subsequent chapters. First, we summarize some results from functional analysis like partial diﬀerential operators, linear functionals, sesquilin- ear ( bilinear) forms or Sobolev spaces. We brieﬂy discuss some facts from matrix theory and introduce the standard algebraic eigenvalue problem. In Section 2.2 we deﬁne an elliptic second-order PDE eigenvalue problem in classical and variational formulation. We describe the Galerkin method as well as the **Finite** Element Method (FEM) for approximating the solution of PDE eigenvalue problem. Afterwards, we discuss the quality of approximate solutions based on a priori and a posteriori error analysis and introduce the Adaptive Fi- nite Element Method (AFEM) which reduces the computational complexity of standard FEM, while retaining the overall accuracy. In Section 2.3 we consider Generalized Algebraic Eigenvalue Problems. We introduce iterative **methods** for large, sparse eigenvalue problems, namely the Arnoldi/Lanczos method. Moreover, we discuss homotopy **methods**. We con- clude this section by recalling perturbation results for the algebraic eigenvalue problems. Last but not least, we expose some relations between continuous and discrete inner products and norms on Sobolev spaces.

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5.4 Conversion into a B-Spline **Volume** Mesh
As explained in Section 5.1, a **volume** mesh which is well-suited for adaptation based on a nested mesh hierarchy is needed. So far, it has been shown how to generate a block- structured **volume** mesh with each block representing a discrete regular rectangular mesh. For the conversion of these blocks into B-spline volumes, the B-spline control points have to be defined based on the control points for the inner Catmull-Clark surface and the discrete points away from the inner surface control mesh. Two main goals for that matter are to keep the good approximation of the given B-spline surface representing the body and to achieve the best possible smoothness, especially on the body and close to it, i.e., in and on the offset mesh. Special care has to be taken at each block border where a smooth join with the particular adjacent block has to be ensured if possible. For the purpose of geometric modeling, a method for the creation of smoothly joining bicubic Nurbs starting from a Catmull-Clark surface mesh was proposed by Peters [76]. Another method using bicubic patches for the approximation of Catmull- Clark subdivision surfaces was presented by Loop and Schaefer [77]. These approaches are based on the same concept as the one in this work. For an easy explanation, it is demonstrated by showing how a C 2 join of two uniform cubic B-spline curves can

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recover the optimal order of convergence in the case of non-smooth solutions. We prove an a posteriori error estimator for the coupling of the **finite** **volume** element method and the boundary element method for a T -piecewise constant diffusion coefficient, which has a quasi-monotone distribution, and a possibly dominating convection vector or a possibly dominating reaction function. The constants in the upper and lower bound are robust with respect to the ratio of the piecewise constant α and with respect to a small diffu- sion compared to the convection field b or the reaction term c. This result can also be used for the pure **finite** **volume** element method. Following the ideas of [22] we prove an L 2 -orthogonality property of the residual for our system. Furthermore, we use a ro- bust interpolation operator with respect to the piecewise constant diffusion coefficient, see [63, 47]. Roughly speaking, the analytical idea is to extend an estimate (coming from the energy norm error) by the L 2 -orthogonality property and by this interpolation oper- ator, which further allows estimates similar to the well-known a posteriori error analysis from the context of the **finite** element method and estimates between discrete error terms. With the Galerkin orthogonality and localization techniques for the Sobolev norms on the boundary of [15], we can prove a robust a posteriori estimator with respect to the model data. Besides the usual residual and a normal jump term, where the part of the normal jump on the coupling boundary is replaced by the coupling condition, we additionally have a term from the boundary element method measuring the error of the Cauchy data through the Calderón system. A tangential jump measures the error in the tangential direction on the coupling boundary. If we apply an upwind method, we also derive a quantity, which measures the amount of upwinding. We give an alternative proof to [22], which does not use a continuous bilinear form of the **finite** **volume** element scheme. To get local efficiency and thus a lower bound, we estimate the **finite** **volume** element quantities by the standard arguments of [73, 74], which consist of bubble functions and an edge lifting operator. For the tangential jump on the boundary we use results from the theory of the a posteriori error estimate in the context of the non-conforming **finite** element method. Our proof can also be used to estimate a similar tangential component in [19], where the proof contains a mistake. Collecting all the results we show that the estimator is local and, in case of a quasi-uniform mesh on the boundary, also generically efficient.

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pre-conditioning **methods** are not appropriate for temporal accuracy of the method [WSW02] or if regions of low and high Mach numbers co-exist. They may also suffer from very re- strictive time step restrictions, i.e., ∆t . ε 2 ∆x; see [BM05b, BM05a, Del10]. Furthermore,
[Kle95] initiated the multiple pressure variables (MPV) approach, treating different orders in the asymptotic expansion of the pressure function in a clever way (cf. Remark 3.5.4); see [HP94, KM95] and [MRKG03, PM05, MDR07, Vat13] for further discussions and some ex- tensions. On the other hand, one may extend the incompressible **methods**, like the pressure- correction scheme, for compressible flows; one can name [DLP93, KP96, MD01] for collocated grids, and [BW98, CG84, WSW02, vdHVW03] for staggered grids. An important example is the extension of the celebrated marker-and-cell (MAC) scheme [HW65] to compressible regimes as done in [HA68, HA71], which led to several recent contributions [GGHL08, GHMN17, GHK + 11,

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DISCUSSION
4 Discussion
4.1 Summary
The quantification of prostate enlargement is a significant clinical challenge. WB-MRI provides an innovative option, which was obtained in this thesis. Using kernel-based **methods**, the first aim of this thesis research was to fully automate prostate segmentation and **volume** estimation based on WB-MRI in the context of a population-based epidemiological study. This method reduces the need for human resources in image readings and thus allows processing a large number of MR scans with a reasonable amount time and effort. The second aim was to develop a new, accurate, and reliable method without any prostate shape assumption. Although this method requires involvement of an expert, it provides accurate PV estimation. This method was additionally compared to the ellipsoid shape assumption for the prostate [Habes 2013c]. The quantitative evaluation of the prostate shape assumption – similar to the shape assumption used as the standard in clinical diagnostics with the TRUS modality – in the sample of this thesis yielded systematically lower PV compared to method 2 [Habes 2013c] and manual prostate delineation by the expert (8.6 ml underestimation with the ellipsoid formula compared to 0.05 ml overestimation with method 2). A similar comparison was not provided for method 1, since the ellipsoid formula plays no role in epidemiological studies. This cumulative thesis is based on two peer-reviewed journal publications [Habes 2013b, Habes 2013c], which are attached in the Appendix and cover both aims separately.

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Figure 1.2 : Three different kind of gaps inside the screw pump
Previous investigations of multiphase screw pumps have largely been concerned with the general pumping behaviour of these systems, [1]. Moreover these studies have been restricted to relatively medium sized pumps, in which the maximum power consumption and gas concentrations are relatively modest. In these situations the heat capacity and density of the gas-liquid mixture is dominated by the liquid phase so that the pumping process is mainly isothermal and thermodynamic effects can be neglected, [2] and [3]. However, this assumption cannot be justified for larger, more powerful screw pumps, which are capable of conveying two-phase fluids with very high concentrations of the gaseous phase up to 100 %. On the other hand, past investigations used a more simple theory for the flow through the gaps and often neglect the compressibility of the gaseous phase. Another area of research is the application of screw pumps in blood conveyance. With the focus on low hemolysis, a channel flow model was introduced to investigate the velocity field, the flow rate and the shear stress distribution, [4]. The model was solved by a **finite** analytical method, with the assumptions, that the fluid is incompressible and the flow is fully developed in channel direction.

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The numerical prediction of flow induced vortex sound in the low Mach number range including rotating do- mains demands high accuracy and efficiency. Using hybrid **methods** like the hydrodynamic/acoustic splitting (HAS) gives one possibility to apply Computational Aeroacoustics (CAA) in a two-step approach. Starting with an incompressible Computational Fluid Dynamics (CFD) simulation, the acoustics is defined as compress- ible perturbations. As there are various **methods** including different theoretical derivations and assumptions, the application of the most suitable method considering accuracy and efficiency is oftentimes case dependent. In this paper an axial fan benchmark case by Zenger et al. [ 1 ] is investigated numerically, which includes the necessity to discretize and model a rotating domain. Motivated by the Mach number scaling laws from Moon [ 2 ], velocity terms and their influence onto aeroacoustic source and propagation effects is focused by the application of different HAS approaches. First, a short introduction to the benchmark case is given. In addition to three well established HAS approaches, a new modified wave equation formulation is introduced in the theory chapter. The numerical simulation setups summarizing the important simplifications are presented for CFD and CAA each. Based on the experimental data obtained from the benchmark [ 1 ], the numerical flow results are presented and discussed. Finally, the sound fields computed by different HAS approaches are compared and investigated. In this context, the newly introduced wave equation formulation shows consistent and promising results. Summarizing the main features of the investigations, a conclusion and an outlook are given.

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