Magister Scientiae - MSc / This thesis deals with the design and implementation of some novel numerical methods for non-linear singular pertubations problems (NSPPs). It provide a survey of asymptotic and numerical methods for some NSPPs in the past decade. By considering two test problems, rigorous asymptotic analysis is carried out. Based on this analysis, suitable numerical methods are designed, analyzed and implemented in order to have some relevant results of physical importance. Since the asymptotic analysis provides only qualitative information, the focus is more on the numerical analysis of the problem which provides the quantitative information. / South Africa

>Magister Scientiae - MSc / This thesis deals with the design and implementation of some novel numerical methods for nonlinear singular perturbations problems (NSPPs). We provide a survey of asymptotic and numerical methods for some NSPPs in past decade. By considering two test problems, rigorous asymptotic analysis is carried out. Based on this analysis, suitable numerical methods are designed, analyzed and implemented in order to have some relevant results of physical importance. Since the asymptotic analysis provides only qualitative information, the focus is more on the numerical analysis of the problem which provides the quantitative information.

The aim of this thesis was to make a practical tool for low frequency analysis in room acoustics.The need arises from Acad’s experience that their results from simulations using raytracing software deviate in the lower frequencies when compared to field measurements inrooms. The tool was programmed in Matlab and utilizes the Finite Difference Time Domain (FDTD) method, which is a form of rapid finite element analysis in the time domain.A number of tests have been made to investigate the practical limitations of the FDTD method, such as numerical errors caused by sound sources, discretization and simulation time. Boundary conditions, with and without frequency dependence, have been analysed bycomparing results from simulations of a virtual impedance tube and reverberation room to analytical solutions. These tests show that the use of the FDTD method appears well suited for the purpose of the tool.A field test was made to verify that the tool enables easy and relatively quick simulations of real rooms, with results well in line with measured acoustic parameters. Comparisons of the results from using the FDTD method, ray-tracing and finite elements (FEM) showed goodcorrelation. This indicates that the deviations Acad experience between simulated results and field measurements are most likely caused by uncertainties in the sound absorption data used for low frequencies rather than by limitations in the ray-tracing software. The FDTDtool might still come in handy for more complex models, where edge diffraction is a more important factor, or simply as a means for a “second opinion” to ray-tracing - in general FEM is too time consuming a method to be used on a daily basis.Auxiliary tools made for importing models, providing output data in the of room acoustic parameters, graphs and audio files are not covered in detail here, as these lay outside the scope of this thesis. / Målet för detta examensarbete var att undersöka möjligheten att programmera ett praktisktanvändbart verktyg för lågfrekvensanalys inom rumsakustik. Behovet uppstår från Acadserfarenhet att resultat från simuleringar med hjälp av strålgångsmjukvara avviker i lågfrekvensområdeti jämförelse med fältmätningar i färdigställda rum. Verktyget är programmerati Matlab och använder Finite Difference Time Domain (FDTD) metoden, vilket är en typav snabb finita elementanalys i tidsdomänen.En rad tester har genomförts för att se metodens praktiska begräsningar orsakade av numeriskafel vid val av ljudkälla, diskretisering och simuleringstid. Randvillkor, med och utanfrekvensberoende, har analyserats genom jämförelser av simulerade resultat i virtuella impedansröroch efterklangsrum mot analytiska beräkningar. Testerna visar att FDTD-metodentycks fungerar väl för verktygets tilltänkta användningsområde.Ett fälttest genomfördes för att verifiera att det med verktyget är möjligt att enkelt och relativtsnabbt simulera resultat som väl matcher uppmätta rumsakustiska parametrar. Jämförelsermellan FDTD-metoden och resultat beräknade med strålgångsanalys och finita elementmetoden(FEM) visade även på god korrelation. Detta indikerar att de avvikelser Acaderfar mellan simulerade resultat och fältmätningar troligen orsakas av osäkerheter i den ingåendeljudabsorptionsdata som används för låga frekvenser, snarare än av begränsningar istrålgångsmjukvaran. Verktyget kan fortfarande komma till användning för mer komplexamodeller, där kantdiffraktion är en viktigare faktor, eller helt enkelt som ett sätt att få ett”andra utlåtande” till resultaten från strålgångsmjukvaran då FEM-analys generellt är en förtidskrävande metod för att användas på daglig basis.Kringverktyg skapade för t.ex. import av modeller, utdata i form av rumsakustiska parametrar,grafer och ljudfiler redovisas inte i detalj i denna rapport eftersom dessa ligger utanförexamensarbetet.

The Vanna-Volga (VV) method has been in wide use as one of the major tools for several years among foreign exchange (FX) trading desks. Despite its popularity, the properties of the VV method are not well studied and understood. This thesis attempts to understand better why and when the VV method makes sense, and how to use it better. Often under practical circumstances the state of calibration can be described as being frequent but imperfect. To take advantage of this level of calibration, we studied the properties and benefits of the Karasinski method, and extended this method to a few useful applications. We have found that the Karasinski method, if used with a reasonably calibrated model, can provide significant performance improvement over the VV method.The VV and Karasinski chapters contain most of the original research in this thesis; there are a wealth of discoveries made in these chapters. Novel methods and applications related to the VV and Karasinski methods are proposed, and some of which can be readily applied to the practical trading environment. To make the VV and Karasinski methods work well in practice, the numerical issues for computing the price and Greeks have been carefully addressed with finite difference schemes that are second-order convergent and fast to compute. As an example of easy-to-compute but difficult-to-calibrate model candidates for the Karasinski method, the Multi-Heston model has been discussed too. A sound computational preparation enables the VV and in particular Karasinski methods to enjoy high viability as being fast, efficient and practical. This thesis is tailored to the purpose of making a detailed study on these useful methods whose great potential has not been adequately understood and fully realised.

This thesis concerns the analytical and practical aspects of applying the Closest Point Method to solve elliptic partial differential equations (PDEs) on smooth surfaces and domains with smooth boundaries. A new numerical scheme is proposed to solve surface elliptic PDEs and a novel geometric multigrid solver is constructed to solve the resulting linear system. The method is also applied to coupled bulk-surface problems. A new embedding equation in a narrow band surrounding the surface is formulated so that it agrees with the original surface PDE on the surface and has a unique solution which is constant along the normals to the surface. The embedding equation is then discretized using standard finite difference scheme and barycentric Lagrange interpolation. The resulting scheme has 2nd-order accuracy in practice and is provably 2nd-order convergent for curves without boundary embedded in ℝ². To apply the method to solve elliptic equations on surfaces and domains with boundaries, the "ghost" point approach is adopted to handle Dirichlet, Neumann and Robin boundary conditions. A systematic method is proposed to represent values of ghost points by values of interior points according to boundary conditions. A novel geometric multigrid method based on the closest point representation of the surface is constructed to solve the resulting large sparse linear systems. Multigrid solvers are designed for surfaces with or without boundaries and domains with smooth boundaries. Numerical results indicate that the convergence rate of the multigrid solver stays roughly the same as we refine the mesh, as is desired of a multigrid algorithm. Finally the above methods are combined to solve coupled bulk-surface PDEs with some applications to biology.

Approved for public release; distribution is unlimited / A FORTRAN 77 computer code employing an adaptation of the finite differencing algorithm proposed by Brian was developed for the solution of transient heat conduction problems in cylindrical geometries. Validation of code was accomplished by comparison with an ana­lytic solution derived for a model with symmetric, linear boundary conditions. Accuracy of results for asymmetric and non-linear boundary conditions was determined by comparison with a similarly vali­dated code employing the explicit method. Code effectiveness was then demonstrated by conducting a transient temperature analysis for a simulated earth-orbiting satellite. Brian's method demonstrated unconditional stability with associated significant reductions in execu­tion time compared to the explicit method. The effects of discretization error on the accuracy of results require further investigation. / http://archive.org/details/applicationofbri00hein / Lieutenant Commander, United States Navy

A two-dimensional numerical model has been developed to simulate wave-current induced nearshore circulation patterns in beaches and surf zones. The wave model is based on the parabolic wave equation for mild slope beaches. The parabolic equation method has been chosen because it is a viable means of predicting the characteristics of surface waves in slowly varying domains and in its present form dissipation and wave breaking are also included. The two dimensional parabolic mild slope equation was discretised and solved in a fully implicit manner, so stability did not create a major problem. This wave model was then embedded into the existing numerical model DIVAST. The sediment transport formulae from Van Rijn was used to calculate the nearshore sediment transport rate.

This paper is concerned with finite difference methods for approximating solutions to the partial differential heat equation. The first chapel gives some introductory background into the physical problem, then motivates three finite difference methods. Chapters II through IV provide statements and proofs for the theorems used in the methods of Chapter I. The final Chapter, V, provides conclusions and an indication of future work. An appendix includes the computer codes written by the author with numerical results.

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A provably stable and high-order accurate semi-discrete finite difference scheme modeling the shuttle movement of a seismic airgun is derived using the SBP-SAT method. The one dimensional airgun model studied consists of two pressurized compartments separated by a moving shuttle. The air inside the compartments is modeled by the compressible Euler equations, whereas the shuttle movement is governed by the pressure difference across the shuttle. Well-posedness for the continuous problem and stability for the numerical scheme is proven using the energy method. Numerical studies verify accuracy and convergence.