Discontinuous Galerkin Methods for Solving Acoustic Problems

Parciální diferenciální rovnice hrají důležitou v inženýrských aplikacích. Často je možné tyto rovnice řešit pouze přibližně, tj. numericky. Z toho důvodu vzniklo množství diskretizačních metod pro řešení těchto rovnic. Uvedená nespojitá Galerkinova metoda se zdá jako velmi obecná metoda pro řešení těchto rovnic, především pak pro hyperbolické systémy. Naším cílem je řešit úlohy aeroakustiky, přičemž šíření akustických vln je popsáno pomocí linearizovaných Eulerových rovnic. A jelikož se jedná o hyperbolický systém, byla vybrána právě nespojitá Galerkinova metoda. Mezi nejdůležitější aspekty této metody patří schopnost pracovat s geometricky složitými oblastmi, možnost dosáhnout metody vysokého řádu a dále lokální charakter toho schématu umožnuje efektivní paralelizaci výpočtu. Nejprve uvedeme nespojitou Galerkinovu metodu v obecném pojetí pro jedno- a dvoudimenzionalní úlohy. Algoritmus následně otestujeme pro řešení rovnice advekce, která byla zvolena jako modelový případ hyperbolické rovnice. Metoda nakonec bude testována na řadě verifikačních úloh, které byly formulovány pro testování metod pro výpočetní aeroakustiku, včetně oveření okrajových podmínek, které, stejně jako v případě teorie proudění tekutin, jsou nedílnou součástí výpočetní aeroakustiky.Partial differential equations play an important role in engineering applications. It is often possible to solve these equations only approximately, i.e. numerically. Therefore number of successful discretization techniques has been developed to solve these equations. The presented discontinuous Galerkin method seems to be very general method to solve this type of equations, especially useful for hyperbolic systems. Our aim is to solve aeroacoustic problems, where propagation of acoustic waves is described using linearized Euler equations. This system of equations is indeed hyperbolic and therefore the discontinuous Galerkin method was chosen. The most important aspects of this method is ability to deal with complex geometries, possibility of high-order method and its local character enabling efficient computation parallelization. We first introduce the discontinuous Galerkin method in general for one- and two-dimensional problems. We then test the algorithm to solve advection equation, which was chosen as a model case of hyperbolic equation. The method will be finally tested using number of verification problems, which were formulated to test methods for computational equations, including verification of boundary conditions, which, similarly to computational fluid dynamics, are important part of computational aeroacoustics.

Iteration-free computation of Gauss-Legendre quadrature nodes and weights

Gauss-Legendre quadrature rules are of considerable theoretical and practical interest because of their role in numerical integration and interpolation. In this paper, a series expansion for the zeros of the Legendre polynomials is constructed. In addition, a series expansion useful for the computation of the Gauss-Legendre weights is derived. Together, these two expansions provide a practical and fast iteration-free method to compute individual Gauss-Legendre node-weight pairs in O(1) complexity and with double precision accuracy. An expansion for the barycentric interpolation weights for the Gauss-Legendre nodes is also derived. A C++ implementation is available online

Transition probability of Brownian motion in the octant and its application to default modeling

We derive a semi-analytic formula for the transition probability of three-dimensional Brownian motion in the positive octant with absorption at the boundaries. Separation of variables in spherical coordinates leads to an eigenvalue problem for the resulting boundary value problem in the two angular components. The main theoretical result is a solution to the original problem expressed as an expansion into special functions and an eigenvalue which has to be chosen to allow a matching of the boundary condition. We discuss and test several computational methods to solve a finite-dimensional approximation to this nonlinear eigenvalue problem. Finally, we apply our results to the computation of default probabilities and credit valuation adjustments in a structural credit model with mutual liabilities

Bogoliubov modes of a dipolar condensate in a cylindrical trap

The calculation of properties of Bose-Einstein condensates with dipolar interactions has proven a computationally intensive problem due to the long range nature of the interactions, limiting the scope of applications. In particular, the lowest lying Bogoliubov excitations in three dimensional harmonic trap with cylindrical symmetry were so far computed in an indirect way, by Fourier analysis of time dependent perturbations, or by approximate variational methods. We have developed a very fast and accurate numerical algorithm based on the Hankel transform for calculating properties of dipolar Bose-Einstein condensates in cylindrically symmetric traps. As an application, we are able to compute many excitation modes by directly solving the Bogoliubov-De Gennes equations. We explore the behavior of the excited modes in different trap geometries. We use these results to calculate the quantum depletion of the condensate by a combination of a computation of the exact modes and the use of a local density approximation

Accuracy and Stability of Computing High-Order Derivatives of Analytic Functions by Cauchy Integrals

High-order derivatives of analytic functions are expressible as Cauchy integrals over circular contours, which can very effectively be approximated, e.g., by trapezoidal sums. Whereas analytically each radius r up to the radius of convergence is equal, numerical stability strongly depends on r. We give a comprehensive study of this effect; in particular we show that there is a unique radius that minimizes the loss of accuracy caused by round-off errors. For large classes of functions, though not for all, this radius actually gives about full accuracy; a remarkable fact that we explain by the theory of Hardy spaces, by the Wiman-Valiron and Levin-Pfluger theory of entire functions, and by the saddle-point method of asymptotic analysis. Many examples and non-trivial applications are discussed in detail.Comment: Version 4 has some references and a discussion of other quadrature rules added; 57 pages, 7 figures, 6 tables; to appear in Found. Comput. Mat

Numerical aspects of enriched and high-order boundary element basis functions for Helmholtz problems.

In this thesis several aspects of the Partition of Unity Boundary Element Method (PUBEM) are investigated, with novel results in three main areas: 1. Enriched modelling of wave scattering from polygonal obstacles. The plane waves are augmented by a set of enrichment functions formed from fractional order Bessel functions, as informed by classical asymptotic solutions for wave fields in the vicinity of sharp corners. It is shown that the solution accuracy can be improved markedly by the addition of a very small number of these enrichment functions, with very little effect on the run time. 2. High-order formulations. Plane waves are not the only effective means of introducing oscillatory approximation spaces. High-Order Lagrange polynomials and high-order Non-Uniform Rational B-Splines (NURBS) also exhibit oscillation and these are tested and compared against PUBEM. It is found that these high-order functions significantly outperform the corresponding low-order (typically quadratic) polynomials and NURBS that are commonly used, and that for large problems the highest order tested (11th) has potential to be competitive with PUBEM without the associated ill-conditioning. 3. Integration. The accuracy of PUBEM traditionally comes at the cost of the requirement to evaluate many highly-oscillatory integrals. Several candidate integration strategies are investigated with the aim of find- ing a robust, accurate and efficient approach. Schemes tested include the Filon and asymptotic methods, as well as the Method of Stationary Phase (MSP). Although these schemes are found to be spectacularly successful for many cases, they fail for a sufficient number of situations to cause a complete PUBEM analysis based on these methods to lack robustness. Conclusions are drawn about the effective use of more traditional quadrature for robust implementations