The efficient evaluation of the arising integrals in NURBS-based boundary element methods is numerically challenging. First, the boundary is parametrized by arbitrary NURBS curves, there- fore the use of analytic formulae for the numerical integration is not possible. Hence, the arising integrals have to be evaluated approximately, which induces a consistency error. Second, besides regular integrals also singular and nearly singular integrals have to be evaluated accurately for **high**-**order** basis functions. Theorem 1.6.7 states that the Galerkin error decays exponentially with respect to the degrees of freedom on geometrically graded meshes. The numerical results in Chapter 5 show a similar decay for collocation methods. In **order** to obtain algorithms for the numerical integration with algebraic complexity, that preserve the exponential convergence of the collocation and the Galerkin errors, the quadrature error has to decay exponentially for all integrals, too.

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In the literature several methods have been discussed to solve the PDEs arising in financial option pricing problems. The most common approach is to apply central finite differences to discretize the spatial domain with second-**order** accuracy, e.g. in [37, 38, 51, 60]. In a low dimensional setting Crank-Nicolson time marching is frequently used [4, 60, 90]. Even for a moderate number of spatial dimensions the resulting system of linear equa- tions becomes expensive to solve. In’t Hout et al. [37, 38, 51] applied dimensional splitting techniques, such as Alternating Direction Implicit (ADI) schemes, to derive efficient meth- ods for the Heston and Heston-Hull-White PDEs. In addition to second-**order** accurate schemes higher-**order** discretizations were introduced and discussed by various researchers as well: Leentvaar and Oosterlee [66, 67] used standard fourth-**order** finite difference ap- proximations, while Linde [69] employed broad sixth-**order** finite difference stencils. These schemes are generally more expensive from a computational point of view, since the dis- cretization matrix is broadly banded. With the help of so called **high**-**order**-compact (HOC) schemes one can derive a fourth-**order** accurate approximation on the compact stencil [3, 21, 22, 47, 89]. Düring et al. [21, 22] constructed HOC schemes for stochas- tic volatility models with one underlying asset and one risk-factor as well as for basket options. In the time domain they applied Crank-Nicolson time stepping. Recently ADI splitting in combination with HOC discretizations has been introduced for convection- diffusion equations with mixed derivatives and constant coefficients by Düring et al. [23]. Their work was extended to PDEs arising in stochastic volatility models [25] and to the multivariate Black-Scholes model in [45].

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Challenging engineering problems often involve weakly or strongly coupled fields, where mechanical parts have to account for changes in temperature, electric fields, phase separation, chemical potentials, and solid-fluid interaction. Numerical simulations of such problems often include partial differential equations (PDEs) of higher-**order**, e.g., mechanical deformations in combination with Cahn-Hilliard diffusion (Stein et al., 2017), Kuramoto-Sivashinsky flow interfaces (Anders et al., 2012a), or **high**-**order** topology-optimization and phase-field fracture (Borden et al., 2014; Dede et al., 2012; Hesch et al., 2016). From a numerical point of view, the solution of such PDEs can be achieved in different ways. The simplest way is the finite difference method with all their pros and cons, particularly their restriction to rectangular domains. This geometrical limitation can be overcome by the finite element method (FEM) whereby standard techniques require mixed formulations of the **high**-**order** terms which come with additional effort and, in part, unclear boundary constraints, cf. (Anders et al., 2012b). An efficient way to fulfill all continuity requirements of the FEM is the choice of sophisticated basis functions, e.g., non-rational basis spline functions (B-splines) or non-uniform rational B-spline functions (NURBS). These functions have become popular in the context of the Isogeometric Analysis (IGA), cf. (Cottrell et al., 2009). However, the focus of the IGA is on the geometry description, which should comply with shapes generated by computer-aided design. NURBS typically describe these models’ geometry and may go along with a local C 0 -continuity, which is sufficient to solve classical second-**order** PDEs. In opposite to this concept, we refer here to the

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One of the main theoretical tools in understanding HG is the strong-field approximation (SFA), also known as the Lewenstein model. Originally proposed to study HG in atoms [8], it was later extended to molecules. It is the quantum-mechanical formulation of the three-step model [9], which ascribes HG to a sequence of (i) ionization, (ii) accel- eration of the continuum electron, and (iii) recombination. The three-step model has had great success in describing qualitatively the dynamics of harmonic generation. It also predicts correctly the value of the cutoff energy of the emit- ted harmonic radiation. Regarding the quantitative predic- tive power of the Lewenstein model for HG in molecules, we note that the model ignores the Coulomb forces act- ing on the active electron in the continuum. This affects most significantly the region of low harmonics, which is thus not accurately described. For **high**-**order** harmonics, the absolute value of the harmonic intensity is usually lower than the value obtained by numerically integrating the time- dependent Schr¨odinger equation (TDSE) (for atoms, see [10] where such a comparison is made). Nevertheless, in the case of atoms, it was shown that the qualitative behavior of **high**- **order** harmonics usually agrees well with the TDSE result [10]. For molecules, the extra degrees of freedom can hinder the agreement with the exact results. One needs to consider different possible formulations and gauges to decide which one fits better the analysis of a given process. For an ex- tended discussion about the choice of gauge and formulation in the context of molecules, see [11, 12]. For example, to describe the two-center interference effects [13], it is advan-

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We proposed several strategies to generate a curved three-dimensional **high** or- der mesh. The question of which one to choose depends on the initial data provided. Our starting point is always a linear mesh, but there is a huge dif- ference between a simple mesh file as is, with a list of discrete points and element connections, and the possibility to load the mesh with the mesh gener- ator, where it was built, providing a connection to the geometry data. To our knowledge, there are no standardized mesh file formats for storing the mesh and the geometry together. We know that the accuracy of the surface repre- sentation is of utmost importance for a **high** **order** mesh, especially for wall boundary conditions in fluid dynamics or reflective boundaries in wave prop- agation problems. It is clear that for a linear mesh as is, the curved surface information has to be the generated or reconstructed a-posteriori, and the ac- curacy will simply depend on the linear mesh resolution. If the mesh generator is available, the additional information can be provided a-priori, and we are able to improve the accuracy by increasing the surface resolution.

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intensity **high**-**order** harmonic generation source. The coherence properties are analyzed and several reconstructions show the shot-to-shot fluctuations of the incident beam wavefront. The method is based on a multi-step approach. First, the spectrum is extracted from double-slit diffraction data. The spectrum is used as input to extract the monochromatic sample diffraction pattern, then phase retrieval is performed on the quasi-monochromatic data to obtain the sample’s exit surface wave. Reconstructions based on guided error reduction (ER) and alternating direction method of multipliers (ADMM) are compared. ADMM allows additional penalty terms to be included in the cost functional to promote sparsity within the reconstruction.

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In this thesis, we present a **high**-**order** Discontinuous Galerkin Method (DGM) for simulating incompressible and low-Mach number flows. Let ∆x be a characteristic length of the numerical mesh. Then, the error e of a numerical method of **order** k behaves like e ∝ ∆x k . In 2007, a survey within the computational fluid dynamics (CFD) community revealed that an **order** of three and above is considered to be **high**-**order**, cf. (Wang, Fidkowski, Abgrall, Bassi, Caraeni, Cary, Deconinck, Hartmann, Hillewaert, Huynh, Kroll, May, Persson, van Leer & Visbal 2013). State of the art CFD codes used in industry are based on the Finite Volume Method (FVM), which are usually of second **order**. In the field of research the **high**-**order** DGM has gained considerable interest for simulating flow problems, see the reviews for incompressible flows in Section 4.1 and for variable density flows at low-Mach numbers in Section 6.1.2. In general, the DGM can be of any **order** by choosing the **order** of the local polynomials, which are used to approximate the solution. The main motivation for using a **high**-**order** method is the capability to reach the same accuracy with less degrees of freedom (DOF) compared to a low-**order** method, or, in other words, to get a result of higher accuracy with the same number of DOF.

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Because DTV/LRO of hot judder are temporary and dynamic, they will disappear when brake disc cools down and the positions of hot spots in both circumferential and radial directions of the disc surfaces alternate in the subsequent braking applications 152 . Be- sides, it is difficult to calculate the transfer functions with two or even four inputs of hot judder excitations from the two or four wheel brakes of a vehicle. Moreover, as inspect- ed by the identification test with production passenger cars that it is difficult to find the hot judder with one single dominant **high** **order** in the production brakes (see the inves- tigation in 4.2.4), or even in the brakes with specifically produced brake pads that show evident **high**-**order** hot judder in the brake dynamometer tests (see the comparison test in 4.2). Therefore, in **order** to identify the transfer functions of **high**-**order** hot judder with a higher signal-to-noise ratio and higher quality, brake discs that are artificially modified by using the numerical control machine into desired surface shapes simulating the **high**- **order** DTV/LRO are taken as the solution. As **high**-**order** hot judder around 10 th **order** is the research objective established in this work, the 10 th DTV or LRO have mainly been imitated. For example, Figure 4-10 shows two discs with 60 µm “10 th **order**” DTV (grooves at corresponding positions on both sides of the disc) and 100 µm “10 th **order**” LRO (grooves at alternating positions on both sides of the disc), respectively.

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The aim of this thesis is to develop a new constitutive model describing the behavior of trans- versely isotropic materials under both mechanical and thermo-mechanical coupled loadings and to use the developed model to perform numerical simulations using **high**-**order** finite elements.
First of all, for the isothermal case, a new constitutive model is formulated based on the mul- tiplicative decomposition of the deformation gradient tensor. The model is an extension of the volumetric/isochoric decomposition, where the isochoric deformation gradient is further decom- posed into two parts: one part describing the deformation in the preferred direction (fiber direc- tion) and another part that contains the remaining deformations. The proposed model fulfills the condition of a stress-free undeformed state, and has advantages over the classical modeling given in Sec. 3.4: firstly, it leads to a clear split of the stress-state in the preferred direction from the remaining stresses; secondly, it overcomes the drawback of applying the volumetric/isochoric de- coupling to the case of anisotropy. The proposed model includes only three material parameters, which can easily be obtained from simple experiments.

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In Section 3.2.1 we have defined different numerical properties [22, 56, 58, 95, 96] which help to decide if a numerical method is suitable in the low Mach context. If all properties are fulfilled a stable **high** **order** method can be obtained which is consistent with the asymptotic limit as ε → 0. Therefore, we check (analytically or numerically) if these properties are fulfilled by the chosen discretization. In a theoretical analysis we are able to assume that the reference solution is given exactly, for numerical computations we need to compute an approximation. Therefore, we derive a proper numerical method for the computation of the reference solution. This method is obtained by computing the ε → 0 limit of the given discretiza- tion and identifying this limit with a fully implicit method. This observation is also motivated by the results of Chapter 4, where we obtained that the RS-IMEX splitting behaves similarly to a fully implicit discretization.

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The discretization of inviscid ﬂuxes is still a challenging part of numerical simulation. Especially in supersonic ﬂow there is a demand for **high** accuracy discretizations which suppress oscillations at shock waves and maintain mono- tonicity. In classical **high** **order** MUSCL (Monotonic Upstream-centered Scheme for Conservation Laws) [2] approaches, TVD (Total Variation Diminishing) lim- iters [3] are used to avoid oscillations at discontinuities. Besides classical TVD limiters like minmod, superbee, van Albada, or the van Leer limiter, there is a number of newly developed limiter functions [4, 5] which are constructed to achieve a sharp and accurate shock capturing while at the same time avoid clipping and squaring eﬀects of classical second **order** limiters [5]. Some lim- iter functions are even able to maintain their formal accuracy at local extrema [5]. Another topic of research is the handling of interface value reconstruc- tion on highly stretched irregular grids. Moreover, there are activities to use multi-dimensional information in the limiter design. Conventional ﬂux vector or ﬂux diﬀerence splittings treat any coordinate direction separately from the remaining ones. It is easy to show, that such one-dimensional limiters fail to achieve a good shock resolution if the shock is located in direction diagonal to the computational grid. This may cause an oscillatory behavior and a stall of convergence. During the last two decades there was some activity in developing multi-dimensional limiting techniques [6, 7, 8] without meeting a wide accep- tance. However, the newly developed MLP approach of Kim and coworkers [9, 1, 10, 11] seems to have a **high** potential to achieve signiﬁcant improvements in this ﬁeld.

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Finally, the numerical computations can be extended by considering more different IMEX Runge-Kutta schemes to see the influence of **order** reduction on the overall convergence behavior. Here, especially the case of very **high** **order** methods is interesting. Unfortunately, there are only a few IMEX Runge-Kutta schemes available which have a classical **order** of convergence larger than three. Therefore additional IMEX Runge-Kutta schemes with a **high** **order** of accuracy and ideally with a large implicit stage **order** are needed. One way to obtain such a scheme is the integral deferred correction procedure, see [37] and the references therein and [27, 36] for an IMEX extension. Integral deferred correction methods apply an IMEX Runge-Kutta scheme in a first step to the ordinary differential equation and then in a second step to a differential equation which describes the error between the classical solution and the numerical approximation. By this, the error can be reduced to obtain a very **high** **order**. Due to this structure the final method can be written as an IMEX Runge-Kutta scheme, see [27], but this scheme has many stages and is therefore not very efficient.

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We present a **high** **order** spherical microphone array design using consumer grade MEMS microphones com- bined with an equalization kernel implemented on an FPGA. A performance analysis of the prototype regarding the usable frequency range and the achievable angular resolution with special regard to the analysis of sound fields in reverberation rooms is given.

In industry and research, Computational Fluid Dynamics (CFD) methods play an essential role in the study of compressible flows which occur, for example, around airplanes or in jet engines, and complement experiments as well as theoretical analysis. In transonic flows, the flow speed may already exceed the speed of sound locally, giving rise to discontinuous flow phenomena, such as shock waves. These phenomena are numerically challenging due to having a size of only a few mean free paths and featuring a large gradient in physical quantities. The application of traditional low-**order** approaches, such as the Finite Element Method (FEM) or the Finite Volume Method (FVM), is usually limited by their immense computational costs for large three- dimensional problems with complex geometries when aiming for highly accurate solutions. By contrast, **high**-**order** methods, such as the discontinuous Galerkin (DG) method, inherently enable a deep insight into complex fluid flows due to their **high**-**order** spatial convergence rate for smooth problems while requiring comparatively few degrees of freedom.

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The vehicle considered in this paper is the SHEFEX-3 (SHarp Edge Flight EXperiment) pro- totype, a vehicle planned by the German Aerospace Center (DLR) [31, 32] for the demonstration of several entry technologies. The proposed tracking law can be used as feedback control scheme together with onboard trajectory-generation algorithms [33, 34], as well as in conjunction with pure optimal trajectory-generation tools [6, 7]. The work is organized as follows. In Sec. II the vehicle and the scenario are briey introduced, while in Sec. III the adaptive **high**-**order** sliding-mode is described in detail, together with a series of simulations coming from a simplied example motivat- ing the current work. In Sec. IV the proposed technique is applied to the longitudinal equations of motion of an unpowered entry vehicle, while Sec. V focuses on the validation of the proposed algorithms, and compares the results with a traditional sliding-mode control algorithm. Finally, in Sec. VI some conclusions on the work are drawn.

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The discretisation is **high**-**order** with a **high** accuracy and less numerical diffusion than in low-**order** discretisation methods such as FV. This causes severe problems solving benchmark problems with sharp corners or **high**, possibly exponential gradients in the stresses within the flow leading to the HWNP. However, since the focus for this implementation is on the simulation of droplets where no such gradients are expected, the software is not optimized for **high** Weissenberg numbers. If a better adaption is needed in future, a possible solution would be the implementation of the log-conformation formulation introduced by Fattal and Kupferman (2004). The use of the LDG method as a discretisation of a hyperbolic first **order** system of equations is the reason that there is no need for elliptic stabilization methods, since the fluxes can handle the hyperbolic equations very well. Compared to the DEVSS methods this means that we have fewer dependent variables, thus a smaller operator matrix and less memory and computational effort when solving the system fully coupled. Actually, the use of the elliptic stabilization methods makes it hard to solve the non-linear system coupled and to solve the linear system directly, even by using HPC in combination with **high** memory nodes. The smaller operator matrix using LDG makes a fully coupled solution possible, which is more robust than a decoupled system. However, for very fine computational meshes combined with **high** polynomial orders, even when using LDG, the memory requirements of direct solvers may eventually become unfeasible. Therefore, in the long run, iterative solvers are the means of choice, and several iterative methods are currently under development within the BoSSS group for solving large linear systems of equations.

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University of Stellenbosch Business School (USB), Bellville, South Africa
Suggested Citation: Meij, J. T. (1982) : Separable programming for aggregate production
planning: A **high**-**order** cost case, South African Journal of Business Management, ISSN 2078-5976, African Online Scientific Information Systems (AOSIS), Cape Town, Vol. 13, Iss. 1, pp. 18-22,

The objective of this study is to develop a method for incorporating LR and ER boundary conditions into **high**- **order**, nodal, time-domain finite element methods, without adding considerable computational load to the already computationally intensive simulation. Firstly, a method for modeling LR frequency dependent impedance bound- ary conditions is presented. In this method, the boundary impedance is mapped to a multipole rational function, and then formulated in differential form. This allows for a convenient incorporation of the boundary condi- tions into the numerical scheme. The accuracy of the LR boundary condition model is assessed by comparing simulations against analytic solutions. Secondly, an approximate method for modeling ER impedance boundary conditions is presented. This method builds upon and extends the LR method. Here, the impedance properties of the boundary are adjusted continuously during the simulation, as a function of the incident wave field angle. The accuracy of the ER boundary model is analyzed by comparisons against measurements.

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Most of the algorithms currently employed for compressible flows are low (first or second) **order** in space. Classical low-**order** techniques, such as finite volume methods combined with shock capturing techniques, have been tuned to industrial applications during the last thirty years and represent a very mature technology. However, the governing laws of fluid dynamics often involve a wide range of spatial and temporal scales. For example, the accurate computation of vortex-dominated flows can play a major role in some aeronautical problems. **High** accuracy can also be of interest in other applications, e.g., computational aeroacoustics. For such applications one may need to resolve the flow at scales of very different length. For this purpose, **high** levels of accuracy are needed, which can be challenging to achieve for low-**order** methods [164, 16]. **High**-**order** methods, namely methods with **order** of consistency higher than two, have become a very active field of research in the last decade. To be mentioned are the final report of the European ADIGMA project [122] and the collection of contributions [198], both focusing on the state of the art of **high**-**order** methods in computational fluid dynamics (CFD) for compressible flow. The expectation is that with these methods **high** accuracy can be achieved, while at the same time avoiding excessive grid resolution.

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