Two problems in computational wave dynamics are considered: (i) the use of Klemp-Wilhelmson time splitting at large scales and (ii) analysis of wave-wave instabilities in nonhydrostatic and rotating mountain waves. The use of Klemp-Wilhelmson (KW) time splitting for large-scale and global modeling is assessed through a series of von Neumann accuracy and stability analyses. Two variations of the KW splitting are evaluated in particular: the original acousticmode splitting of Klemp and Wilhelmson (KW78) and a modified splitting due to Skamarock and Klemp (SK92) in which the buoyancy and vertical stratification terms are treated as fast-mode terms. The large-scale cases of interest are the problem of Rossby wave propagation on a resting background state and the classic baroclinic Eady problem. The results show that the original KW78 splitting is surprisingly inaccurate when applied to large-scale wave modes. The source of this inaccuracy is traced to the splitting of the hydrostatic balance terms between the small and large time steps. The errors in the KW78 splitting are shown to be largely absent from the SK92 scheme. Resonant wave-wave instability in rotating mountain waves is examined using a linear stability analysis based on steady-state solutions for flow over an isolated ridge. The analysis is performed over a parameter space spanned by the mountain height (Nh/U) and the Rossby number (U/fL). Steady solutions are found using a newly developed solver based on a nonlinear Newton iteration. Results from the steady solver show that the critical heights for wave overturning are smallest for the hydrostatic case and generally increase in the rotating wave regime. Results of the stability analyses show that the wave-wave instability exists at mountain heights even below the critical overturning values. The most unstable cases are found in the nonrotating regime while the range of unstable mountain heights between initial onset and critical overturning is largest for intermediate Rossby number.

Esta tesis está dedicada al desarrollo, aplicación y test de métodos para la simulación numérica de problemas procedentes de la física y de la ingeniería electrónica. La principal herramienta aplicada a lo largo de todo el trabajo es la ecuación de Vlasov (transporte) en la forma de la Boltzmann Transport Equation (BTE) para la descripción del transporte de partículas cargadas en plasmas y dispositivos electrónicos: las cargas se mueven bajo el efecto de un campo de fuerza y sufren scattering debido a otras cargas o fonones (pseudo-partículas que describen de manera efectiva las vibraciones de los iones del retículo cristalino).La BTE ha de ser acoplada con una ecuación o sistema de ecuaciones para calcular el campo de fuerza: para estructuras simples se usa la ecuación de Poisson; para plasmas, donde los efectos magnéticos no se pueden despreciar debido a las altas velocidades de las partículas, se usa la fuerza de Lorentz, por lo cual se han de resolver las ecuaciones de Maxwell; en nanoestructuras, por ejemplo transistores con dimensiones confinadas, la ecuación de Poisson necesita ser acoplada con la ecuación de Schrödinger para la descripción de las dimensiones cuánticas y para la descomposición en sub-bandas, o niveles de energía.Las colisiones son el scattering que las cargas padecen debido a las interacciones con otras cargas o con el retículo cristalino fijo, representado en forma de fonones. En la tesis se emplean diversos operadores de scattering: los más simples son operadores lineales de relajación; se estudia un modelo para la simulación de semiconductores donde se tienen en cuenta colisiones con fonones acústicos, en aproximación elástica, y fonones ópticos.Tras la introducción, en el primer capítulo se desarrollan los métodos numéricos más importantes: primero un método de interpolación no oscilante (PWENO), necesario para evitar las oscilaciones producidas por la reconstrucción por polinomios de Lagrange, que incrementa la variación total cuando aparecen choques: las oscilaciones en el espacio de fases son características del problema, pero si el método añade oscilaciones espúreas (es decir, debidas al método en sí), entonces el resultado numérico no tiene sentido, o simplemente explota. El segundo método numérico fundamental es la técnica de splitting: cuando se resuelve un problema complicado, si se puede dividir en sub-problemas y resolverlos por separado, entonces se puede reconstruir una aproximación para el problema completo; esta técnica se usa para el time splitting (separación de la parte de transporte y de colisión) y el splitting dimensional (dividir el espacio de fases en posición y velocidad). La tercera herramienta fundamental es un sólver para advección lineal: se usan dos métodos, uno basado en trazar hacia atrás las características a nivel puntual y otro basado en reconstruir valores integrales en segmentos en lugar de puntos; el primero controla mejor las oscilaciones, el segundo fuerza la conservación de masa.En el capítulo 2 estos métodos se aplican a algunos tests conocidos para averiguar su solidez.En el capítulo 3 estos métodos se aplican a la simulación de un diodo, y los resultados se comparan con resultados anteriores obtenidos por esquemas Runge-Kutta basados en diferencias finitas para aproximar las derivadas parciales.El capítulo 4 está dedicado a la construcción y simulación de modelos intermedios entre una ecuación cinética, con operador de colisión de tipo relajación, y su aproximación más grosera, ésta última siendo la ecuación del calor. Para obtener modelos intermedios, se busca un cierre de las ecuaciones de los momentos de orden cero y uno. Se proponen esquemas "asymptotic-preserving" para la ecuación cinética, que evitan la stiffness de la parte de advección a través de una descomposición de la función de distribución en su media más fluctuaciones. En cuanto a las clausuras de las ecuaciones de los momentos, se proponen esquemas de relajación para aislar las no-linealidades. Estos métodos son aplicados a un test conocido, el Su-Olson test.El último capítulo está dedicado a la simulación de un MOSFET (Metal Oxide Semiconductor Field Effect Transistor) 2D de dimensión nanométrica en el que los electrones se comportan como partículas en una dimensión y como ondas en las dimensiones confinadas. La descomposición en sub-bandas se realiza a través de una ecuación de Schrödinger 1D en estado estacionario. Las dimensiones, así como las sub-bandas, están acopladas por la ecuación de Poisson en la expresión de la densidad, y por el operador de colisión. Se propone un sólver microscópico para estados transitorios, basado en técnicas de splitting para las BTEs (una para cada nivel de energía), métodos de características para el transporte y una iteración de tipo Newton para resolver el problema acoplado Schrödinger-Poisson para el cálculo del campo de fuerza. / This thesis is dedicated to the development, application and test of numerical methods for the numerical simulation of problems arising from physics and electronic engineering. The main tool which is used all along the work is the Vlasov (transport) equation in the form of the Boltzmann Transport Equation (BTE) for the description of the transport and collisions of charged particles in plasmas and electronic devices: charge carriers are driven by a force field and scattered by other carriers or phonons (pseudo-particles giving an effective representation of the oscillating field produced by the vibrating ions).The BTE must be coupled to an equation or a system of equations for the computation of the force field: for simple structures the Poisson equation is used; for plasmas, where the magnetic phenomena cannot be neglected due to the high velocities of the particles, the Lorentz force is used, so the Maxwell equations have to be solved; for nanostructures, e.g. transistors with confined dimensions, the Poisson equation needs coupling with Schrödinger equation for the description of the quantum dimensions and the decomposition into subbands, or energy levels.Collisions mean the scattering the carriers suffer due to the interactions with other carriers or the fixed lattice, in form of phonons. All along the thesis several scattering operator are used: the simplest ones are linear relaxation-time operators; a model for the simulation of a semiconductor is studied in which collisions are taken into account with acoustic phonons, in the elastic approximation, and optical phonons.After the introduction, in the first chapter the most important numerical methods are developed: first of all a pointwise non-oscillatory interpolation method (PWENO) needed to avoid the simple Lagrange polynomial reconstruction, which increases the total variation when shocks appear: oscillations are part of the physics of the problem, but if the method adds spurious, non-physical oscillations, then the numerical result is meaningless, or it simply blows up. The second fundamental numerical method is the splitting technique: when solving a complicated problem, if we are able to subdivide it into sub-problem and solve them for separate, then we can reconstruct an approximation for the complete problem; this technique is used for both time splitting (separate transport from collisions) and dimensional splitting (split the phase space into either dimensions). The third fundamental instrument is the solver for linear advections: two methods are used, one based on pointwise following backwards the characteristics and another one based on reconstructing integral values along segments instead of point values; the first one controls better oscillations, the second one forces mass conservation.These methods are applied in chapter 2 to some well-known benchmark tests to control their robustness.In chapter 3 these methods are applied to the simulation of a diode, and the results compared to previous results obtained by Runge-Kutta schemes based on finite differences schemes for the approximation of the partial derivatives.Chapter 4 is dedicated to the construction and simulation of intermediate models between a kinetic equation, with relaxation-time collision operator, and its coarsest approximation, this one being the heat equations. In order to obtain intermediate models, the moment equations are closed at zeroth and first order. Asymptotic-preserving schemes are proposed for the kinetic equation, which avoid the stiffness of the advection part by decomposing the distribution function into its average plus fluctuations. As for the moment closures, relaxation schemes are proposed in order to confine the non-linearities in the right hand side. These methods are then applied to a known benchmark, the Su-Olson test.The last chapter is dedicated to the simulation of a nanoscaled 2D MOSFET (Metal Oxide Field Effect Transistor) in which electrons behave as particles in one dimension and as waves in the confined dimensions. The subband decomposition is realized through a stationary-state 1D Schrödinger equation. The dimensions as well as the subbands are coupled by the Poisson equation in the expression of the density and by the collision operator. A transient-state microscopic solver is proposed, based on splitting techniques for the BTE's (one for each energy level), characteristics methods for the transport and a Newton iteration for the solution of the coupled Schrödinger-Poisson system for computing the force field.

La simulation d’écoulements incompressibles pose de nombreuses difficultés. Une première est la question de savoir comment traiter la contrainte d’incompressibilité et le couplage vitesse/pression afin d’obtenir une solution précise à moindre coût. Pour cela, nous nous intéressons en particulier à deux méthodes de time splitting : la correction de pression et la correction de vitesse. Une seconde difficulté porte sur des conditions limites de sortie. Nous nous intéressons ici à deux d’entre elles : la condition limite de traction et la condition limite d’Orlanski. Après avoir détaillé les difficultés pouvant apparaître lors de l’implémentation des méthodes de time-splitting, nous proposons une nouvelle implémentation de la condition limite de traction qui permet d’améliorer les ordres de convergence obtenus. Nous nous intéressons ensuite à la condition limite d’Orlanski qui nécessite une certaine vitesse d’advection C dans la direction normale à la limite dont nous proposons ici une nouvelle définition. Nos propositions sont confrontées à de multiples écoulements physiques afin de valider leurs comportements : l’écoulement en aval d’une marche descendante, l’écoulement au niveau d’une bifurcation,l’écoulement autour d’un obstacle et des écoulements de Poiseuille-Rayleigh-Bénard. / One of the understudied difficulties in the simulation of incompressible flows is how to treat the incompressibilityconstraint and the velocity/pressure coupling in order to obtain an accurate solution at low computationnalcost. In this context, we develop two methods: pressure-correction and velocity-correction. An anotherdifficulty is due to the boundary conditions. We study here two of them : the traction boundary condition andthe Orlanski boundary condition. After having developed the difficulties that appears when implementing timesplittingmethods, we propose a new way to enforce the traction boundary condition which improves the orderof convergence. Then we propose a new definition of the advective velocity C which is needed for the Orlanskiboundary condition. Our propositions are validated against multiple physical flows: flow over a backward facingstep, flow around a biffurcation, flow around an obstacle and several Poiseuille-Rayleigh-Bénard flows.

Cette thèse avait comme but le développement, l'analyse et l'application de schémas numériques pour la simulation de modèles cinétiques basés sur l'équation de Vlasov, notamment de schémas basés sur le splitting de Strang et une méthode d'interpolation essentiellement non oscillatoire (WENO). Les schémas sont testés sur des cas test de plus en plus compliqués, et finalement sur un modèle Boltzmann-Schrödinger-Poisson qui décrit les états transitoires d'un transistor à l'echelle nanométrique.

Nano-scale machining, or Nanomachining is a hybrid process in which the total thermal energy necessary to remove atoms from a work-piece surface is applied from external sources. In the current study, the total thermal energy necessary to remove atoms from a work-piece surface is applied from two sources: (1) localized energy from a laser beam focused to a micron-scale spot to preheat the work-piece, and (2) a high-precision electron-beam emitted from the tips of carbon nano-tubes to remove material via evaporation/sublimation. Macro-to-nano scale heat transfer models are discussed for understanding their capability to capture and its application to predict the transient heat transfer mechanism required for nano-machining. In this case, thermal transport mechanism during nano-scale machining involves both phonons (lattice vibrations) and electrons; it is modeled using a parabolic two-step (PTS) model, which accounts for the time lag between these energy carriers. A numerical algorithm is developed for the solution of the PTS model based on explicit and implicit finite-difference methods. Since numerical solution for simulation of nanomachining involves high computational cost in terms of wall clock time consumed, performance comparison over a wide range of numerical techniques has been done to devise an efficient numerical solution procedure. Gauss-Seidel (GS), successive over relaxation (SOR), conjugate gradient (CG), d -form Douglas-Gunn time splitting, and other methods have been used to compare the computational cost involved in these methods. Use of the Douglas-Gunn time splitting in the solution of 3D time-dependent heat transport equations appears to be optimal especially as problem size (number of spatial grid points and/or required number of time steps) becomes large. Parallel computing is implemented to further reduce the wall clock time required for the complete simulation of nanomachining process. Domain decomposition with inter-processor communication using Message Passing Interface (MPI) libraries is adapted for parallel computing. Performance tuning has been implemented for efficient parallelization by overlapping communication with computation. Numerical solution for laser source and electron-beam source with different Gaussian distribution are presented. Performance of the parallel code is tested on four distinct computer cluster architecture. Results obtained for laser source agree well with available experimental data in the literature. The results for electron-beam source are self-consistent; nevertheless, they need to be validated experimentally.

Time splitting is a numerical procedure used in solution of partial differential equations whose solutions allow multiple time scales. Numerical schemes are split for handling the stiffness in equations, i.e. when there are multiple time scales with a few time scales being smaller than the others. When there are such terms with smaller time scales, due to the Courant number restriction, the computational cost becomes high if these terms are treated explicitly. In the present work a nonlinear advective equation is solved numerically using different techniques based on a generalised framework for splitting methods. The nonlinear advective equation was chosen because it has an analytical solution making comparisons with numerical schemes amenable and also because its nonlinearity mimics the equations encountered in atmospheric modelling. Using the nonlinear advective equation as a test bed, an analysis of the splitting methods and their influence on the split solutions has been made. An understanding of influence of splitting schemes requires knowledge of behaviour of unsplit schemes beforehand. Hence a study on unsplit methods has also been made. In the present work, using the nonlinear advective equation, it shown that the three time level schemes have high phase errors and underestimate energy (even though they have a higher order of accuracy in time). It is also found that the leap-frog method, which is used widely in atmospheric modelling, is the worst among examined unsplit methods. The semi implicit method, again a popular splitting method with atmospheric modellers is the worst among examined split methods. Three time-level schemes also need explicit filtering to remove the computational mode. This filtering can have a significant impact on the obtained numerical solutions, and hence three-time level schemes appear to be unattractive in the context of the nonlinear convective equation. Based on this experience, splitting methods for the two-time level schemes is proposed. These schemes realistically capture the phase and energy of the nonlinear advective equation.

Ultra-fast laser heating of nano-films is investigated using 3-D Dual Phase Lag heat transport equation with laser heating at different locations on the metal film. The energy absorption rate, which is used to model femtosecond laser heating, is modified to accommodate for three-dimensional laser heating. A numerical solution based on an explicit finite-difference method is employed to solve the DPL equation. The stability criterion for selecting a time step size is obtained using von Neumann eigenmode analysis, and grid function convergence tests are performed. DPL results are compared with classical diffusion and hyperbolic heat conduction models and significant differences among these three approaches are demonstrated. We also develop an implicit finite-difference scheme of Crank-Nicolson type for solving 1-D and 3-D DPL equations. The proposed numerical technique solves one equation unlike other techniques available in the literature, which split the DPL equation into a system of two equations and then apply discretization. Stability analysis is performed using a von Neumann stability analysis. In 3-D, the discretized equation is solved using delta-form Douglas and Gunn time splitting. The performance of the proposed numerical technique is compared with the numerical techniques available in the literature.

Une modélisation numérique des ondes poroélastiques, décrites par le modèle de Biot, est proposée dans le domaine temporel. La dissipation visqueuse à l'intérieur des pores est décrite par le modèle de perméabilité dynamique de Johnson-Koplik-Dashen (JKD). Certains coefficients du modèle de Biot-JKD sont proportionnels à la racine carrée de la fréquence, introduisant dans le domaine temporel des dérivées fractionnaires décalées d'ordre 1/2, revenant à un produit de convolution. Basé sur une représentation diffusive, le produit de convolution est remplacé par un nombre fini de variables de mémoire satisfaisant une équation différentielle ordinaire locale en temps, menant au modèle de Biot-DA (diffusive approximation). Les propriétés des deux modèles sont analysées : hyperbolicité, décroissance de l'énergie, dispersion. On montre que la meilleure méthode de détermination des coefficients de l'approximation diffusive - quadratures de Gauss, optimisation linéaire ou non-linéaire sur la plage de fréquence d'intérêt - est l'optimisation non-linéaire. Une méthode de splitting est utilisée numériquement : la partie propagative est discrétisée par un schéma aux différences finies ADER d'ordre 4, et la partie diffusive est intégrée exactement. Les conditions de saut aux interfaces sont discrétisées avec une méthode d'interface immergée. Des simulations numériques sont présentées pour des milieux isotropes et isotropes transverses. Des comparaisons avec des solutions analytiques montrent l'efficacité et la précision de cette approche. Des simulations numériques en milieux complexes sont réalisées : influence de la porosité d'os spongieux, diffusion multiple en milieu aléatoire. / A time-domain numerical modeling of Biot poroelastic waves is proposed. The viscous dissipation in the pores is described using the dynamic permeability model of Johnson-Koplik-Dashen (JKD). Some of the coefficients in the Biot-JKD model are proportional to the square root of the frequency: in the time-domain, these coefficients introduce shifted fractional derivatives of order 1/2, involving a convolution product. Based on a diffusive representation, the convolution product is replaced by a finite number of memory variables that satisfy local-in-time ordinary differential equations, resulting in the Biot-DA (diffusive approximation). The properties of the two models are analyzed: hyperbolicity, decrease of energy, dispersion. To determine the coefficients of the diffusive approximation, different methods of quadrature are analyzed: Gaussian quadratures, linear or nonlinear optimization procedures in the frequency range of interest. The nonlinear optimization is shown to be the best way of determination. A splitting strategy is applied numerically: the propagative part is discretized using a fourth-order ADER scheme on a Cartesian grid, and the diffusive part is solved exactly. An immersed interface method is implemented to discretize the jump conditions at interfaces. Numerical experiments are presented for isotropic and transversely isotropic media. Comparisons with analytical solutions show the efficiency and the accuracy of this approach. Some numerical experiments are performed in complex media: influence of the porosity of a cancellous bone, multiple scattering across a set of random scatterers.