<p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small;"><span style="font-family: Times New Roman;">This thesis consists of two different parts, related to two different fields in mathematical physics: wave propagation and kinetic theory of gases. Various mathematical and computational problems for equations from these areas are treated.</span></span></span></p><p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small; font-family: Times New Roman;"> </span></span></p><p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small;"><span style="font-family: Times New Roman;">The first part is devoted to high order finite difference methods for the Helmholtz equation and the wave equation. Compact schemes with high order accuracy are obtained from an investigation of the function derivatives in the truncation error. With the help of the equation itself, it is possible to transfer high order derivatives to lower order or to transfer time derivatives to space derivatives. For the Helmholtz equation, a compact scheme based on this principle is compared to standard schemes and to deferred correction schemes, and the characteristics of the errors for the different methods are demonstrated and discussed. For the wave equation, a finite difference scheme with fourth order accuracy in both space and time is constructed and applied to a problem in discontinuous media.</span></span></span></p><p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small; font-family: Times New Roman;"> </span></span></p><p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small;"><span style="font-family: Times New Roman;">The second part addresses some problems related to kinetic equations. A direct simulation Monte-Carlo method is constructed for the Landau-Fokker-Planck equation, and numerical tests are performed to verify the accuracy of the algorithm. A formal derivation of the method from the Boltzmann equation with grazing collisions is performed. The linear and linearized Boltzmann collision operators for the hard sphere molecular model are studied using exact reduction of integral equations to ordinary differential equations. It is demonstrated how the eigenvalues of the operators are found from these equations, and numerical values are computed. A proof of existence of non-zero discrete eigenvalues is given. The ordinary diffential equations are also used for investigation of the Chapman-Enskog distribution function with respect to its asymptotic behavior.</span></span></span></p><p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small; font-family: Times New Roman;"> </span></span></p>

These thesis review some recent results on the construction of very high order multidimensional upwind schemes for the solution of steady and unsteady conservation laws on unstructured triangular grids. We also consider the extension to the approximation of solutions to conservation laws containing second order dissipative terms. To build this high order schemes we use a sub-triangulation of the triangular Pk elements where we apply the distribution used for a P1 element. This manuscript is divided in two parts. The first part is dedicated to the design of the high order schemes for scalar equations and focus more on the theoretical design of the schemes. The second part deals with the extension to system of equations, in particular we will compare the performances of 2nd, 3rd and 4th order schemes. The first part is subdivided in four chapters: The aim of the second chapter is to present the multidimensional upwind residual distributive schmes and to explain what was the status of their development at the beginning of this work. The third chapter is dedicated to the first contribution: the design of 3rd and 4th order quasi non-oscillatory schemes. The fourth chapter is composed of two parts: We start by understanding the non-uniformity of the accuracy of the 2nd order schemes for advection-diffusion problem. To solve this issue we use a Finite Element hybridisation. This deep study of the 2nd order scheme is used as a basis to design a 3rd order scheme for advection-diffusion. Finally, in the fifth chapter we extend the high order quasi non-oscillatory schemes to unsteady problems. In the second part, we extend the schemes of the first part to systems of equations as follows: The sixth chapter deals with the extension to steady systems of hyperbolic equations. In particular, we discuss how to solve some issues such as boundary conditions and the discretisation of curved geometries. Then, we look at the performance of 2nd and 3rd order schemes on viscous flow. Finally, we test the space-time schemes on several test cases. In particular, we will test the monotonicity of the space-time non-oscillatory schemes and we apply residual distributive schemes to acoustic problems.

Optical parametric oscillators (OPOs) and optical parametricamplifiers (OPAs) constitute a class of optical frequencyconverting devices that have many possible applications, e.g.in range finding, molecular spectroscopy and medicine. They canconvert the frequency of the incident pump field with highefficiency, and generate two waves at new frequencies that willbe continuously tuneable over a wide spectral range. Virtuallyany wavelengths within the transparency region of the nonlinearmaterial can be generated if the material can bequasi-phasematched (QPM). In addition, QPM gives thepossibility to utilise the largest nonlinear tensor element ofthe material and allows walk-off free interaction between thewaves. The aims of this thesis have been to investigate thepossibility to use QPM KTiOPO4crystals as nonlinear material in nanosecond OPOsand OPAs operating at room-temperature, and to explore theadvantages and shortcomings of these devices. The technique ofelectric field poling has been employed to implement the QPMstructure in flux grown KTiOPO4(KTP). The main conclusion is that periodically poled KTP (PPKTP)is a suitable material to use in nanosecond OPOs and OPAs. Thematerial properties that foremost make KTP into an attractivenonlinear material are: The large value of the nonlinearcoefficient d33, the high resistance to optically inducedbreakdown, the low susceptibility to grey-track formation, theinsensitivity to the photorefractive effect, the widetransparency and the low coercive field. The thesis shows that it is possible to pole large volumesof KTP with a high quality of the QPM structure. Highlyefficient nanosecond OPOs have been constructed during thisproject. Maximum conversion efficiencies have reached 45 % inthe case of a singly resonant OPO (SRO) built around a 3 mmthick PPKTP crystal. Total pulse energies for both the signal(1.72 µm) and the idler (2.8 µm) of up to 18 mJ wasreached and an average output power of 2 W was obtained forthis sample. However, up to 24 W was produced in a doublyresonant OPO operating close to degeneracy. The efficiencyreached 48 % for that case. Truly continuous and very widespectral tuning has also been demonstrated, as well as a narrowbandwidth OPO operating on one single longitudinal mode. <b>Keywords:</b>optical parametric oscillators, opticalparametric amplifiers, quasi-phasematching, KTiOPO4, nonlinear optics, frequency conversion, periodicelectric field poling, ferroelectrics, high-order secondharmonic generation, electro-optic effect.

Residual-distribution methods offer several potential benefits over classical methods, such as a means of applying upwinding in a multi-dimensional manner and a multi-dimensional positivity property. While it is apparent that residual-distribution methods also offer higher accuracy than finite-volume methods on similar meshes, few studies have directly compared the performance of the two approaches in a systematic and quantitative manner. In this study, comparisons between residual distribution and finite volume are made for steady-state smooth and discontinuous flows of gas dynamics, governed by hyperbolic conservation laws, to illustrate the strengths and deficiencies of the residual-distribution method. Deficiencies which reduce the accuracy are analyzed and a new nonlinear scheme is proposed that closely reproduces or surpasses the accuracy of the best linear residual-distribution scheme. The accuracy is further improved by extending the scheme to fourth order using established finite-element techniques. Finally, the compact stencil, arithmetic workload, and data parallelism of the fourth-order residual-distribution scheme are exploited to accelerate parallel computations on an architecture consisting of both CPU cores and a graphics processing unit. Numerical experiments are used to assess the gains to efficiency and possible monetary savings that may be provided by accelerated architectures.

Residual-distribution methods offer several potential benefits over classical methods, such as a means of applying upwinding in a multi-dimensional manner and a multi-dimensional positivity property. While it is apparent that residual-distribution methods also offer higher accuracy than finite-volume methods on similar meshes, few studies have directly compared the performance of the two approaches in a systematic and quantitative manner. In this study, comparisons between residual distribution and finite volume are made for steady-state smooth and discontinuous flows of gas dynamics, governed by hyperbolic conservation laws, to illustrate the strengths and deficiencies of the residual-distribution method. Deficiencies which reduce the accuracy are analyzed and a new nonlinear scheme is proposed that closely reproduces or surpasses the accuracy of the best linear residual-distribution scheme. The accuracy is further improved by extending the scheme to fourth order using established finite-element techniques. Finally, the compact stencil, arithmetic workload, and data parallelism of the fourth-order residual-distribution scheme are exploited to accelerate parallel computations on an architecture consisting of both CPU cores and a graphics processing unit. Numerical experiments are used to assess the gains to efficiency and possible monetary savings that may be provided by accelerated architectures.

Aquesta tesi proposa una millora del clàssic mètode dels elements finits (finite element method, FEM) per a un tractament eficient de dominis amb contorns corbs: el denominat NURBS-enhanced finite element method (NEFEM). Aquesta millora permet descriure de manera exacta la geometría mitjançant la seva representació del contorn CAD amb non-uniform rational B-splines (NURBS), mentre que la solució s'aproxima amb la interpolació polinòmica estàndard. Per tant, en la major part del domini, la interpolació i la integració numèrica són estàndard, retenint les propietats de convergència clàssiques del FEM i facilitant l'acoblament amb els elements interiors. Només es requereixen estratègies específiques per realitzar la interpolació i la integració numèrica en elements afectats per la descripció del contorn mitjançant NURBS.La implementació i aplicació de NEFEM a problemes que requereixen una descripció acurada del contorn són, també, objectius prioritaris d'aquesta tesi. Per exemple, la solució numèrica de les equacions de Maxwell és molt sensible a la descripció geomètrica. Es presenta l'aplicació de NEFEM a problemes d'scattering d'ones electromagnètiques amb una formulació de Galerkin discontinu. S'investiga l'habilitat de NEFEM per obtenir solucions precises amb malles grolleres i aproximacions d'alt ordre, i s'exploren les possibilitats de les anomenades malles NEFEM, amb elements que contenen singularitats dintre d'una cara o aresta d'un element. Utilitzant NEFEM, la mida de la malla no està controlada per la complexitat de la geometria. Això implica una dràstica diferència en la mida dels elements i, per tant, suposa un gran estalvi tant des del punt de vista de requeriments de memòria com de cost computacional. Per tant, NEFEM és una eina poderosa per la simulació de problemes tridimensionals a gran escala amb geometries complexes. D'altra banda, la simulació de problemes d'scattering d'ones electromagnètiques requereix mecanismes per aconseguir una absorció eficient de les ones scattered. En aquesta tesi es discuteixen, optimitzen i comparen dues tècniques en el context de mètodes de Galerkin discontinu amb aproximacions d'alt ordre.La resolució numèrica de les equacions d'Euler de la dinàmica de gasos és també molt sensible a la representació geomètrica. Quan es considera una formulació de Galerkin discontinu i elements isoparamètrics lineals, una producció espúria d'entropia pot evitar la convergència cap a la solució correcta. Amb NEFEM, l'acurada imposició de la condició de contorn en contorns impenetrables proporciona resultats precisos inclús amb una aproximació lineal de la solució. A més, la representació exacta del contorn permet una imposició adequada de les condicions de contorn amb malles grolleres i graus d'interpolació alts. Una propietat atractiva de la implementació proposada és que moltes de les rutines usuals en un codi d'elements finits poden ser aprofitades, per exemple rutines per realitzar el càlcul de les matrius elementals, assemblatge, etc. Només és necessari implementar noves rutines per calcular les quadratures numèriques en elements corbs i emmagatzemar el valor de les funciones de forma en els punts d'integració. S'han proposat vàries tècniques d'elements finits corbs a la literatura. En aquesta tesi, es compara NEFEM amb altres tècniques populars d'elements finits corbs (isoparamètics, cartesians i p-FEM), des de tres punts de vista diferents: aspectes teòrics, implementació i eficiència numèrica. En els exemples numèrics, NEFEM és, com a mínim, un ordre de magnitud més precís comparat amb altres tècniques. A més, per una precisió desitjada NEFEM és també més eficient: necessita un 50% dels graus de llibertat que fan servir els elements isoparamètrics o p-FEM per aconseguir la mateixa precisió. Per tant, l'ús de NEFEM és altament recomanable en presència de contorns corbs i/o quan el contorn té detalls geomètrics complexes. / This thesis proposes an improvement of the classical finite element method (FEM) for an efficient treatment of curved boundaries: the NURBSenhanced FEM (NEFEM). It is able to exactly represent the geometry by means of the usual CAD boundary representation with non-uniform rational Bsplines (NURBS), while the solution is approximated with a standard piecewise polynomial interpolation. Therefore, in the vast majority of the domain, interpolation and numerical integration are standard, preserving the classical finite element (FE) convergence properties, and allowing a seamless coupling with standard FEs on the domain interior. Specifically designed polynomial interpolation and numerical integration are designed only for those elements affected by the NURBS boundary representation.The implementation and application of NEFEM to problems demanding an accurate boundary representation are also primary goals of this thesis. For instance, the numerical solution of Maxwell's equations is highly sensitive to geometry description. The application of NEFEM to electromagnetic scattering problems using a discontinuous Galerkin formulation is presented. The ability of NEFEM to compute an accurate solution with coarse meshes and high-order approximations is investigated, and the possibilities of NEFEM meshes, with elements containing edge or corner singularities, are explored. With NEFEM, the mesh size is no longer subsidiary to geometry complexity, and depends only on the accuracy requirements on the solution, whereas standard FEs require mesh refinement to properly capture the geometry. This implies a drastic difference in mesh size that results in drastic memory savings, and also important savings in computational cost. Thus, NEFEM is a powerful tool for large-scale scattering simulations with complex geometries in three dimensions. Another key issue in the numerical solution of electromagnetic scattering problems is using a mechanism to perform the absorption of outgoing waves. Two perfectly matched layers are discussed, optimized and compared in a high-order discontinuous Galerkin framework.The numerical solution of Euler equations of gas dynamics is also very sensitive to geometry description. Using a discontinuous Galerkin formulation and linear isoparametric elements, a spurious entropy production may prevent convergence to the correct solution. With NEFEM, the exact imposition of the solid wall boundary condition provides accurate results even with a linear approximation of the solution. Furthermore, the exact boundary representation allows using coarse meshes, but ensuring the proper implementation of the solid wall boundary condition. An attractive feature of the proposed implementation is that the usual routines of a standard FE code can be directly used, namely routines for the computation of elemental matrices and vectors, assembly, etc. It is only necessary to implement new routines for the computation of numerical quadratures in curved elements and to store the value of shape functions at integration points. Several curved FE techniques have been proposed in the literature. In this thesis, NEFEM is compared with some popular curved FE techniques (namely isoparametric FEs, cartesian FEs and p-FEM), from three different perspectives: theoretical aspects, implementation and performance. In every example shown, NEFEM is at least one order of magnitude more accurate compared to other techniques. Moreover, for a desired accuracy NEFEM is also computationally more efficient. In some examples, NEFEM needs only 50% of the number of degrees of freedom required by isoparametric FEs or p-FEM. Thus, the use of NEFEM is strongly recommended in the presence of curved boundaries and/or when the boundary of the domain has complex geometric details.

Effective pollution mitigation is the key to successful air quality management. Although states invest millions of dollars to predict future air quality, the regulatory modeling and analysis process to inform pollution control strategy remains uncertain. Traditionally deterministic ‘bright-line’ tests are applied to evaluate the sufficiency of a control strategy to attain an air quality standard. A critical part of regulatory attainment demonstration is the prediction of future pollutant levels using photochemical air quality models. However, because models are uncertain, they yield a false sense of precision that pollutant response to emission controls is perfectly known and may eventually mislead the selection of control policies. These uncertainties in turn affect the health impact assessment of air pollution control strategies. This thesis explores beyond the conventional practice of deterministic attainment demonstration and presents novel approaches to yield probabilistic representations of pollutant response to emission controls by accounting for uncertainties in regional air quality planning. Computationally-efficient methods are developed and validated to characterize uncertainty in the prediction of secondary pollutant (ozone and particulate matter) sensitivities to precursor emissions in the presence of uncertainties in model assumptions and input parameters. We also introduce impact factors that enable identification of model inputs and scenarios that strongly influence pollutant concentrations and sensitivity to precursor emissions. We demonstrate how these probabilistic approaches could be applied to determine the likelihood that any control measure will yield regulatory attainment, or could be extended to evaluate probabilistic health benefits of emission controls, considering uncertainties in both air quality models and epidemiological concentration–response relationships. Finally, ground-level observations for pollutant (ozone) and precursor concentrations (oxides of nitrogen) have been used to adjust probabilistic estimates of pollutant sensitivities based on the performance of simulations in reliably reproducing ambient measurements. Various observational metrics have been explored for better scientific understanding of how sensitivity estimates vary with measurement constraints. Future work could extend these methods to incorporate additional modeling uncertainties and alternate observational metrics, and explore the responsiveness of future air quality to project trends in emissions and climate change.

The linear elastodynamic two-dimensional anti-plane stress problem, where deformations occur in only one direction is considered for one sided non-planar faults. Fault dynamics are modeled using purely velocity dependent friction laws, and applied on boundaries with complex geometry. Summation-by-parts operators and energy estimates are used to couple a high-order finite difference method with an unstructured finite volume method. The unstructured finite volume method is used near the fault and the high-order finite difference method further away from the fault where no complex geometry is present. Boundary conditions are imposed weakly on characteristic form using the simultaneous approximation term technique, allowing explicit time integration to be used. Numerical computations are performed to verify the accuracy and time stability, of the method.

<p>Optical parametric oscillators (OPOs) and optical parametricamplifiers (OPAs) constitute a class of optical frequencyconverting devices that have many possible applications, e.g.in range finding, molecular spectroscopy and medicine. They canconvert the frequency of the incident pump field with highefficiency, and generate two waves at new frequencies that willbe continuously tuneable over a wide spectral range. Virtuallyany wavelengths within the transparency region of the nonlinearmaterial can be generated if the material can bequasi-phasematched (QPM). In addition, QPM gives thepossibility to utilise the largest nonlinear tensor element ofthe material and allows walk-off free interaction between thewaves.</p><p>The aims of this thesis have been to investigate thepossibility to use QPM KTiOPO₄crystals as nonlinear material in nanosecond OPOsand OPAs operating at room-temperature, and to explore theadvantages and shortcomings of these devices. The technique ofelectric field poling has been employed to implement the QPMstructure in flux grown KTiOPO₄(KTP).</p><p>The main conclusion is that periodically poled KTP (PPKTP)is a suitable material to use in nanosecond OPOs and OPAs. Thematerial properties that foremost make KTP into an attractivenonlinear material are: The large value of the nonlinearcoefficient d₃₃, the high resistance to optically inducedbreakdown, the low susceptibility to grey-track formation, theinsensitivity to the photorefractive effect, the widetransparency and the low coercive field.</p><p>The thesis shows that it is possible to pole large volumesof KTP with a high quality of the QPM structure. Highlyefficient nanosecond OPOs have been constructed during thisproject. Maximum conversion efficiencies have reached 45 % inthe case of a singly resonant OPO (SRO) built around a 3 mmthick PPKTP crystal. Total pulse energies for both the signal(1.72 µm) and the idler (2.8 µm) of up to 18 mJ wasreached and an average output power of 2 W was obtained forthis sample. However, up to 24 W was produced in a doublyresonant OPO operating close to degeneracy. The efficiencyreached 48 % for that case. Truly continuous and very widespectral tuning has also been demonstrated, as well as a narrowbandwidth OPO operating on one single longitudinal mode.</p><p><b>Keywords:</b>optical parametric oscillators, opticalparametric amplifiers, quasi-phasematching, KTiOPO₄, nonlinear optics, frequency conversion, periodicelectric field poling, ferroelectrics, high-order secondharmonic generation, electro-optic effect.</p>

Coding is an essential technology for efficient fiber-optic communications and secure quantum communications. In particular, low-density parity-check (LDPC) coding is favoured due to its strong error correction capability and high-throughput implementation feasibility. In fiber-optic communications, it has been realized that advanced high-order modulation formats and soft-decision forward error correction (FEC) such as LDPC codes are the key technologies for the next-generation high-speed optical communications. Therefore, energy-efficient LDPC coding in combination with advanced modulation formats is an important topic that needs to be studied for fiber-optic communications. In secure quantum communications, large-alphabet quantum key distribution (QKD) is becoming attractive recently due to its potential in improving the efficiency of key exchange. To recover the carried information bits, efficient information reconciliation is desirable, for which the use of LDPC coding is essential. In this dissertation, we first explore different efficient LDPC coding schemes for optical transmission of polarization-division multiplexed quadrature-amplitude modulation (QAM) signals. We show that high energy efficiency can be achieved without incurring extra overhead and complexity. We then study the transmission performance of LDPC-coded turbo equalization for QAM signals in a realistic fiber link as well as that of pragmatic turbo equalizers. Further, leveraging the polarization freedom of light, we expand the signal constellation into a four-dimensional (4D) space and evaluate the performance of LDPC-coded 4D signals in terms of transmission reach. Lastly, we study the security of a proposed weak-coherent-state large-alphabet QKD protocol and investigate the information reconciliation efficiency based on LDPC coding.