La validation des systèmes Mécatroniques tels que la supervision d'une chaînede traction hybride utilise de plus en plus la simulation Hardware-in-the-Loop. Cela consiste à interconnecter des composants réels du système et des composantssimulés. On parle alors de simulation temps réel car les composants simulés doivent avoir le même comportement temporel que les réels. En d'autres termes, la simulation temps réel d'un modèle nécessite le maillage de l'évolution du temps simulé sur celle du temps réel. Sur les outils existants, l'intégration de modèles physiques représentatifs se heurte à des modèles de calculs et des contraintes temporelles pessimistes. Cette thèse propose des solutions, analytiques ou tirées d'expérimentations au sein d'IFP Energies nouvelles, pour l'implantation adéquate de la simulation temps réel de modèles physiques. Des métriques ont été introduites pourqualifier et quantifier la validité d'une simulation temps réel. Une définition des contraintes temporelles propres à la simulation temps réel a été proposée, accompagnée des règles régissant leur propagation aux calculs sous-jacents. Ces méthodes ont ensuite été déclinées en étude d'ordonnançabilité pour deux systèmes au comportement pseudo périodique : un simulateur de moteur à combustion et un contrôle moteur. Des expérimentations sur la simulation temps réel distribuée d'un moteur, intégrant des modèles phénoménologiques de combustion, ont permis de justifier et de validerles méthodes proposées. Les dégradations dues à la simulation distribuée ont été corrigées par un mécanisme d'extrapolation paramétrable dont le coût d'exécution a été étudié / Validation of Mechatronics systems such as hybrid automotive powertrains isincreasingly relying on Harware-in-the-Loop simulation. It consists in interconnecting real components to the real-time simulation of physical models, involving their timely behavior to match their real counterpart. In other words, the evolution of simulated and real time have to be meshed together. Involving representative physical models is currently hindered by both pessimistic models of computation and temporal constraints.This thesis proposes several analytical and experimental answers, carried out at IFP Energies nouvelles, toward the proper implantation of real-time simulation of physical models. Several metrics able to qualify and quantify the success of real-time simulation were proposed, as well as the definition of its dedicated timed constraints, along with the rules for their propagation toward the underlying computations involved.Then, we showed how to take advantage of the pseudo periodic behavior of two systems to reach better schedulability bounds for the real-time simulation of : a combustion engine and an engine control. The methods discussed were then accounted for and validated by several experiments, involving the distributed real-time simulation of an engine including phenomenological combustion models. Also, the perturbations induced by the distributed simulation were addressed by proposing a configurable extrapolation mechanism, taking into account its execution time

<p>The next generation satellite communication networks will provide multimedia services supporting high bit rate, mobility, ATM, and TCP/IP. In these cases, the satellite technology will act as the internetwork infrastructure of future global systems and assuming a global wireless system, no distinctions will exist between terrestrial and satellite communications systems, as well as between fixed and 3G mobile networks. In order for satellites to be successful, they must handle bursty traffic from users and provide services compatible with existing ISDN infrastructure, narrowcasting/multicasting services not offered by terrestrial ISDN, TCP/IP-compatible services for data applications, and point-to-point or point-to-multipoint on-demand compressed video services. This calls for onboard processing payloads capable of frequency multiplexing and demultiplexing and interference suppression.</p><p>This thesis introduces a new class of oversampled complex modulated filter banks capable of providing frequency multiplexing and demultiplexing. Under certain system constraints, the system can handle all possible shifts of different user signals and provide variable bandwidths to users. Furthermore, the aliasing signals are attenuated by the stopband attenuation of the channel filter thus ensuring the approximation of the perfect reconstruction property as close as desired. Study of the system efficient implementation and its mathematical representation shows that the proposed system has superiority over the existing approaches for Bentpipe payloads from the flexibility, complexity, and perfect reconstruction points of view. The system is analyzed in both SISO and MIMO cases. For the MIMO case, two different scenarios for frequency multiplexing and demultiplexing are discussed.</p><p>To verify the results of the mathematical analysis, simulation results for SISO, two scenarios of MIMO, and effects of the finite word length on the system performance are illustrated. Simulation results show that the system can perform frequency multiplexing and demultiplexing and the stopband attenuation of the prototype filter controls the aliasing signals since the filter coefficients resolution plays the major role on the system performance. Hence, the system can approximate perfect reconstruction property by proper choice of resolution.</p>

Networked control systems (NCSs) are a kind of distributed control systems in which the data between control components are exchanged via communication networks. Because of the attractive advantages of NCSs such as reduced system wiring, low weight, and ease of system diagnosis and maintenance, the research on NCSs has received much attention in recent years. The first part (Chapter 2 - Chapter 4) of the thesis is devoted to designing new controllers for NCSs by incorporating the network-induced delays. The thesis also conducts research on filtering of multirate systems and identification of Hammerstein systems in the second part (Chapter 5 - Chapter 6).<br /><br /> Network-induced delays exist in both sensor-to-controller (S-C) and controller-to-actuator (C-A) links. A novel two-mode-dependent control scheme is proposed, in which the to-be-designed controller depends on both S-C and C-A delays. The resulting closed-loop system is a special jump linear system. Then, the conditions for stochastic stability are obtained in terms of a set of linear matrix inequalities (LMIs) with nonconvex constraints, which can be efficiently solved by a sequential LMI optimization algorithm. Further, the control synthesis problem for the NCSs is considered. The definitions of <em>H₂</em> and <em>H_∞</em> norms for the special system are first proposed. Also, the plant uncertainties are considered in the design. Finally, the robust mixed <em>H₂/H_&infin;</em> control problem is solved under the framework of LMIs. <br /><br /> To compensate for both S-C and C-A delays modeled by Markov chains, the generalized predictive control method is modified to choose certain predicted future control signal as the current control effort on the actuator node, whenever the control signal is delayed. Further, stability criteria in terms of LMIs are provided to check the system stability. The proposed method is also tested on an experimental hydraulic position control system. <br /><br /> Multirate systems exist in many practical applications where different sampling rates co-exist in the same system. The <em>l₂-l_&infin;</em> filtering problem for multirate systems is considered in the thesis. By using the lifting technique, the system is first transformed to a linear time-invariant one, and then the filter design is formulated as an optimization problem which can be solved by using LMI techniques. <br /><br /> Hammerstein model consists of a static nonlinear block followed in series by a linear dynamic system, which can find many applications in different areas. New switching sequences to handle the two-segment nonlinearities are proposed in this thesis. This leads to less parameters to be estimated and thus reduces the computational cost. Further, a stochastic gradient algorithm based on the idea of replacing the unmeasurable terms with their estimates is developed to identify the Hammerstein model with two-segment nonlinearities. <br /><br /> Finally, several open problems are listed as the future research directions.

The next generation satellite communication networks will provide multimedia services supporting high bit rate, mobility, ATM, and TCP/IP. In these cases, the satellite technology will act as the internetwork infrastructure of future global systems and assuming a global wireless system, no distinctions will exist between terrestrial and satellite communications systems, as well as between fixed and 3G mobile networks. In order for satellites to be successful, they must handle bursty traffic from users and provide services compatible with existing ISDN infrastructure, narrowcasting/multicasting services not offered by terrestrial ISDN, TCP/IP-compatible services for data applications, and point-to-point or point-to-multipoint on-demand compressed video services. This calls for onboard processing payloads capable of frequency multiplexing and demultiplexing and interference suppression. This thesis introduces a new class of oversampled complex modulated filter banks capable of providing frequency multiplexing and demultiplexing. Under certain system constraints, the system can handle all possible shifts of different user signals and provide variable bandwidths to users. Furthermore, the aliasing signals are attenuated by the stopband attenuation of the channel filter thus ensuring the approximation of the perfect reconstruction property as close as desired. Study of the system efficient implementation and its mathematical representation shows that the proposed system has superiority over the existing approaches for Bentpipe payloads from the flexibility, complexity, and perfect reconstruction points of view. The system is analyzed in both SISO and MIMO cases. For the MIMO case, two different scenarios for frequency multiplexing and demultiplexing are discussed. To verify the results of the mathematical analysis, simulation results for SISO, two scenarios of MIMO, and effects of the finite word length on the system performance are illustrated. Simulation results show that the system can perform frequency multiplexing and demultiplexing and the stopband attenuation of the prototype filter controls the aliasing signals since the filter coefficients resolution plays the major role on the system performance. Hence, the system can approximate perfect reconstruction property by proper choice of resolution.

Networked control systems (NCSs) are a kind of distributed control systems in which the data between control components are exchanged via communication networks. Because of the attractive advantages of NCSs such as reduced system wiring, low weight, and ease of system diagnosis and maintenance, the research on NCSs has received much attention in recent years. The first part (Chapter 2 - Chapter 4) of the thesis is devoted to designing new controllers for NCSs by incorporating the network-induced delays. The thesis also conducts research on filtering of multirate systems and identification of Hammerstein systems in the second part (Chapter 5 - Chapter 6).<br /><br /> Network-induced delays exist in both sensor-to-controller (S-C) and controller-to-actuator (C-A) links. A novel two-mode-dependent control scheme is proposed, in which the to-be-designed controller depends on both S-C and C-A delays. The resulting closed-loop system is a special jump linear system. Then, the conditions for stochastic stability are obtained in terms of a set of linear matrix inequalities (LMIs) with nonconvex constraints, which can be efficiently solved by a sequential LMI optimization algorithm. Further, the control synthesis problem for the NCSs is considered. The definitions of <em>H₂</em> and <em>H_∞</em> norms for the special system are first proposed. Also, the plant uncertainties are considered in the design. Finally, the robust mixed <em>H₂/H_&infin;</em> control problem is solved under the framework of LMIs. <br /><br /> To compensate for both S-C and C-A delays modeled by Markov chains, the generalized predictive control method is modified to choose certain predicted future control signal as the current control effort on the actuator node, whenever the control signal is delayed. Further, stability criteria in terms of LMIs are provided to check the system stability. The proposed method is also tested on an experimental hydraulic position control system. <br /><br /> Multirate systems exist in many practical applications where different sampling rates co-exist in the same system. The <em>l₂-l_&infin;</em> filtering problem for multirate systems is considered in the thesis. By using the lifting technique, the system is first transformed to a linear time-invariant one, and then the filter design is formulated as an optimization problem which can be solved by using LMI techniques. <br /><br /> Hammerstein model consists of a static nonlinear block followed in series by a linear dynamic system, which can find many applications in different areas. New switching sequences to handle the two-segment nonlinearities are proposed in this thesis. This leads to less parameters to be estimated and thus reduces the computational cost. Further, a stochastic gradient algorithm based on the idea of replacing the unmeasurable terms with their estimates is developed to identify the Hammerstein model with two-segment nonlinearities. <br /><br /> Finally, several open problems are listed as the future research directions.

This thesis considers the application of the type II third order WENO finite volume reconstruction for unstructured tetrahedral meshes proposed by Zhang and Shu in (CCP, 2009) and the third order multirate Runge-Kutta time-stepping to the solution of Maxwell's equations. The dependance of accuracy of the third order WENO scheme on the small parameter in the definition of non-linear weights is studied in detail for one-dimensional uniform meshes and numerical results confirming the theoretical analysis are presented for the linear advection equation. This analysis is found to be crucial in the design of the efficient three-dimensional WENO scheme, full details of which are presented. Several multirate Runge-Kutta (MRK) schemes which advance the solution with local time-steps assigned to different multirate groups are studied. Analysis of accuracy of three different MRK approaches for linear problems based on classic order-conditions is presented. The most flexible and efficient multirate schemes based on works by Tang and Warnecke (JCM, 2006) and Liu, Li and Hu (JCP, 2010) are implemented in three-dimensional finite volume time-domain (FVTD) method. The main characteristics of chosen MRK schemes are flexibility in defining the time-step ratios between multirate groups and consistency of the scheme. Various approaches to partition the three-dimensional computational domain into multirate groups to maximize the achievable speedup are discussed. Numerical experiments with three-dimensional electromagnetic problems are presented to validate the performance of the proposed FVTD method. Three-dimensional results agree with theoretical and numerical accuracy analysis performed for the one-dimensional case. The proposed implementation of multirate schemes demonstrates greater speedup than previously reported in literature. / February 2016

The following work presents a optimization proposal for a digital reverberation architecture applying multirate processing techniques over a reconfigurable platform. Reverberation is one of the acoustic effects that most occur in our lives. Although very common, this phenomenon is often imperceptible. It is noteworthy that the presence of reverberation has a paramount importance, particularly in the musical environment, since it adds sense of space to the recordings (or executions) of a particular song, making it sounds more natural. Due to this importance, the first artificial reverbs came much time before digital computers. These simulators were electro-acoustic devices that simulated the reverberation making use of springs or steel plates equipped with transducers. With the appearance of the first digital computers, digital signal processing techniques began to be used, leading to the first digital reverbs that simulate the reverberation using linear filters in discrete time. Considering the recent developments experienced in the configurable computing field, there is a natural tendency to research and develop acoustic systems based on such a platform. The multirate signal processing is characterized by changing the signal sampling frequency from the removal or addition of samples in the original input sequence. Depending on the application, changing the frequency of sampling can greatly reduce the algorithms and hardware complexity. As the reverb effect is based on digital delay lines which size is proportional to the sampling frequency, and, since multirate processing techniques allow the frequency reduction, is possible visualise the reduction in memory needed to implement the effect in question. In this sense, the architecture of digital reverb proposed by James A. Moorer was chosen as the basis for development and comparison. From the results of this work, it is highlighted the reduction in memory consumption by 50% compared to the reference architecture. Regarding the results compatibility, the proposed architecture presented a satisfactory response, being imperceptible the differences between the reference architecture and the proposed one. At last, it is emphasized that the proposed architecture can be used to build other audio effects based on time delays, which will benefit from the reduction in memory consumption afforded by the proposal. This considerable reduction in memory enables the proposed architecture utilization on a single low-cost chip and presents a new way to manage computational resources required by digital reverberators. / Este trabalho apresenta uma proposta de otimização para uma arquitetura de reverberação digital, empregando técnicas de processamento multitaxa sobre uma plataforma reconfigurável. A reverberação é um dos efeitos acústicos de maior ocorrência em nossa vida. Porém, apesar de muito comum, este fenômeno é muitas vezes imperceptível. Destaca-se que a presença de reverberação é de extrema importância particularmente no meio musical, pois ela adiciona senso de espaço às gravações (ou execuções) de determinada música, proporcionando assim uma maior naturalidade. Devido a esta importância, os primeiros reverberadores artificiais surgiram muito antes dos computadores digitais. Estes simuladores eram dispositivos eletro-acústicos que simulavam a reverberação fazendo uso de molas ou chapas de aço equipadas com transdutores. Com o surgimento dos primeiros computadores digitais, técnicas de processamento digital de sinais começaram a ser utilizadas, dando origem aos primeiros reverberadores digitais que simulavam a reverberação através do uso de filtros lineares em tempo discreto. Tendo em vista a recente evolução experimentada na área da computação configurável, surge uma tendência natural à pesquisa e desenvolvimento de sistemas acústicos baseados em tal plataforma. O processamento de sinais multitaxa se caracteriza pela mudança da freqüência de amostragem de um sinal, a partir da remoção ou adição de amostras na seqüência de entrada original. Dependendo da aplicação, a mudança da freqüência de amostragem pode reduzir consideravelmente a complexidade dos algoritmos e do hardware. Como o efeito de reverberação digital se baseia em linhas de atraso cujo tamanho é proporcional à freqüência de amostragem, e as técnicas de processamento multitaxa possibilitam a redução desta freqüência, visualiza-se então a redução da quantidade de memória necessária para a implementação do efeito em questão. Neste sentido, a arquitetura de reverberação digital proposta por James A. Moorer foi escolhida como base de desenvolvimento e comparação. Dos resultados obtidos neste trabalho, destaca-se a redução do consumo de memória em 50% em relação à arquitetura de referência. No tocante a compatibilidade de resultados, a arquitetura proposta apresentou uma resposta satisfatória, sendo imperceptíveis as diferenças entre a arquitetura de referência e a arquitetura proposta. Por fim, destaca-se que a arquitetura proposta pode ser utilizada na construção de outros efeitos de áudio baseados em atrasos de tempo, que se beneficiarão com a redução do consumo de memória proporcionada pela proposta em questão. Essa redução considerável de memória possibilita o emprego da arquitetura proposta em um chip único (single-chip) de baixo custo, e apresenta uma nova maneira de gerenciar os recursos computacionais exigidos pelos reverberadores digitais.

[pt] Muitas das atuais técnicas de simulação de redes elétricas de potência são baseadas em plataformas computacionais do tipo EMTP. Basicamente, este método de simulação transforma os componentes elétricos a parâmetros concentrados e a topologia da rede em um procedimento matemático recursivo. Para cada nova iteração, a solução da rede é estimada como uma função de estados prévios e valores de entrada atuais. Passos de integração pequenos geram soluções precisas por estender a banda de freqüências representada. No entanto, isto também aumenta a carga computacional limitando o tamanho da rede, a faixa de freqüência ou afetando o tempo de resposta do processo. Esta situação conflitante tem sido atacada por vários pesquisadores, muitos procurando formas alternativas de aumentar o passo de integração. Este trabalho introduz um novo método para simulação de redes elétricas lineares. Mesmo baseando nas técnicas utilizadas no EMTP, o método leva a um conjunto diferente de equações convenientes para conjugar com técnicas de filtragem digital multitaxa. O resultado final é a decomposição de sinais e da rede em sub-bandas de freqüências. Cada uma das simulações em sub-bandas da rede elétrica é feita com um passo de integração máximo, sempre representando carga computacional mínima. Além disso, o procedimento permite uma estimativa on-line se uma saída de sub-banda tem contribuição irrelevante ao resultado final da simulação, suspendendo o correspondente módulo de operação e, consequentemente, reduzindo a carga computacional. Como resultado, tem-se um simulador capaz de adaptar a complexidade do modelo de acordo com os dados da simulação.Modelos de linhas de transmissão a parâmetros distribuídos são introduzidos, conectados ao modelo da rede a parâmetros concentrados. Um método de computação específica é desenvolvido para uma operação global do sistema.Casos ilustrativos são incluídos mostrando a eficiência do simulador proposto em comparação com técnicas tradicionais. / [en] Most of present day electric power network simulation techniques are based on computational platforms of the EMTP type. Basically, following this approach, the simulation method transforms the electric lumped components and the network topology into a recursive mathematical procedure. For each new integration step the network solution is evaluated as a function of previous state and present input values. Small integration steps increase solution accuracy, by extending the frequency band. However, this also increases computational requirements, limiting network size, frequency bandwidth or affecting the procedure response time. This conflicting situation hes been attacked by many researchers, mostly looking for different forms of increasing the integration step.This work introduces a new approach to linear electric network simulation. Although closely following the EMTP basic techniques, the method leads to a different set of equations, more convenient to conjugate with multirate digital filtering techniques. The final result is the decomposition of signal and network models into subbands of frequencies. Each of the network subband simulations is performed with maximum integration time step, always representing minimum computational burden. Moreover, the procedure allows to on-line estimate if a particular subband model output has negligible contribution to the final simulation result. While this situation remains, the corresponding module operation is suspended, reducing the computational load. As a result, the simulator is able to adapt its model complexity, on line, according to the simulation requirements. Distributed transmission line models are introduced, connected to the lumped network parameter models. A specific computational procedure is shown to operate the overall system.Illustrative cases are included, supporting the simulator proposed efficiency when compared to conventional procedures.

Die zu erwartenden Temperaturen und Regenmengen der folgenden Tage bis Stunden sind heutzutage eine der wichtigsten Informationen. Diese Kenntnis ist nicht nur von allgemeinem Interesse. Insbesondere Bereiche wie die Landwirtschaft und Forstwirtschaft sind die zu erwartenden Regenmengen selbst über einen langen Zeitraum von Wochen von besonderen Interesse um zum Beispiel die Ernte oder den Schutz von Pflanzen zu planen. Daher ist die Fähigkeit, das Wetter zuverlässig und schnell für ausreichend lange Zeiträume vorher zu sagen, wesentlich. Die Zuverlässigkeit der Wettervorhersage, oder genau genommen der numerischen Wettervorhersage, hängt von mehreren Faktoren ab. Einer dieser Faktoren ist die Detailliertheit der Atmosphärenmodelle. Während die ersten numerischen Experimente die Atmosphäre als eine Schicht trockenen idealen Gases betrachteten, beinhalten aktuelle Modelle die Feuchte, Wolken, Niederschlag und Strahlung. Mit jedem zusätzlichen Detail steigt natürlich der Simulationsaufwand. Daher müssen parallel zur verbesserten Modellierung auch die numerischen Verfahren erweitert werden. Im allgemeinen sind die Atmosphärenmodelle Systeme nichtlinearer hyperbolischer Differentialgleichungen (PDEs). Insbesondere beinhalten die Modelle Wellen unterschiedlicher Ausbreitungsgeschwindigkeit, welche nahezu nicht gedämpft werden. Diese unterschiedlichen Geschwindigkeiten sind die Grundlage für den Mehrskalencharakter der Atmosphärenmodelle. Eine effektive numerische Methode muss daher die unterschiedlichen Skalen adäquat behandeln. Die Entwicklung und Analyse numerischer Mehrskalenverfahren zur Lösung von Systemen hyperbolischer Differentialgleichungen ist herausfordernd. Beispiele für hyperbolische Systeme beginnen bei der einfachen skalaren linearen Advektionsgleichung, der Wellengleichung und enden bei nichtlinearen Systemen wie den Flachwassergleichungen oder den (reibungsfreien) Eulergleichungen. Letztere sind die Grundlage für alle Atmosphärenmodelle. Viele hyperbolische PDEs besitzen eine additive Struktur, wobei die Aufteilung gerade den Zeitskalen entsprechen. Wir gehen von einer angepassten Diskretisierung im Raum, in der Regel eine Finite-Volumen Diskretisierung, aus. Diese Diskretisierung erhält die additive Struktur des kontinuierlichen Problems in der (ortsdiskretisierten) gewöhnlichen Differentialgleichung (ODE). Daher entwickeln wir eine neue numerische Methode zur Lösung gewöhnlicher Differentialgleichungen, welche die additive Struktur und gleichzeitig die zugehörigen Zeitskalen ausnutzt. Die Analyse von Splittingverfahren ist herausfordernd sowohl in der Entwicklung der Ordnungsbedingungen als auch der Stabilitätskriterien. Jeder Mehrskalenansatz kombiniert die unterschiedlichen Zeitskalen auf unterschiedliche Art und Weise. Daher gibt es keine einheitliche Ordnungs- und Stabilitätstheorie. Wir entwickeln die Ordnungsbedingungen auf klassischem Wege, durch Differentiation der numerischen Lösung. Die Aufteilung der rechten Seite in schnelle und langsame Terme führt auf zusätzliche Koeffizienten und Kombinationen der elementaren Differentiale. Im Vergleich zu klassischen Verfahren hat jedes elementare Differential unterschiedliche nicht-klassische Koeffizienten, ohne erkennbare Struktur. Dieser Strukturverlust erschwert die numerische Lösung zusätzlich. Analytische Lösungen gibt es nur in Sonderfällen. Wir entwickeln und analysieren eine neue Klasse von Mehrskalen methoden, welche mit den Integrator der schnellen Skale parametriert ist. Dieser neue Ansatz erlaubt die Verallgemeinerung der Ausgangsmethode und vereinfacht etliche Schritte in der Herleitung der Ordnungsbedinungen. Zusätzlich hat die Verallgemeinerung auch den Vorteil, die Ordnungsbedingungen des Gesamtverfahrens und die Struktur des darunter liegenden Lösers der schnellen Zeitskale zu assoziieren. Wir untersuchen ebenfalls die lineare Stabilität der neuen Methoden. Aufgrund der Aufteilung in langsame und schnelle Terme gibt es viele verschiedene Modellprobleme. Wir leiten ein Modellproblem auf Basis eines vereinfachten hyperbolischer PDEs her. Auf Basis dieses Stabilitsproblems konstruieren wir die neuen Methoden und untersuchen ihre Effizienz anhand zweier nichtlineare Benchmarkprobleme. Analog zur Herleitung der Ordnungsbedingungen vereinheitlichen wir die Konstruktion der Stabilitätsfunktionen und heben im nachhinein die Unterschiede aufgrund des fast-scale integrators hervor. Gute numerische Methoden führen nicht nur zu einem kleinen Fehler, sondern haben auch ein großes Stabilitätsgebiet. Daher optimizieren wir die Methodenparameter im Hinblick auf die Größe des Stabilitätsgebiets. Unsere neuen Methoden besitzen sowohl reelle, als auch rationale Parameter. Die Lösung des gemischten ganzzahligen-reellen Optimierungsproblem vereinfachen wir durch die Auswahl einzelner rationaler Parameter. Dadurch erhalten wir allerdings einige tausend unabhängige Teilprobleme. Zum Abschluss analysieren wir die Effizienz der neuen Methoden anhand zweier nichtlinearer Benchmarkprobleme und vergleichen die Genauigkeit und Stabilität mit Referenzverfahren. / The expected temperatures and rainfall in the next days to hours is one of the most important information nowadays. This knowledge is not only of general interest. Disciplines like agriculture and forestry the knowledge of the rain is even more important for a time span of weeks to plan the harvest or protect the plants. Therefore, the possibility to forecast the weather reliably and fast is very important nowadays. The reliability of weather forecast, or more accurate the numerical weather forecast, depends on several factors. One factor is the complexity of atmosphere models. Whereas the first numerical experiments treat the atmosphere as dry ideal gas with one layer, recent models incorporate the humidity, clouds, precipitation and radiation. But every higher detail in the model come at higher costs for simulation. Hence the development of finer grained models also require more advanced numerical methods to solve them. The atmosphere models are in general a nonlinear hyperbolic set of partial differential equations (PDEs). In particular the models consist of several waves, traveling with different speeds with nearly no damping. Roughly speaking these varying velocities lead to the multiscale nature of the atmosphere models and a suitable numerical method should respect the different time scales. The development and analysis of multirate methods for hyperbolic systems remains a challenging problem. Examples for class of hyperbolic systems of PDEs range from the scalar and linear advection equation, the wave equation to nonlinear systems like the shallow water equations or the (inviscid) Euler equations, which are the basis for the atmosphere models. The hyperbolic PDEs often have an additive split structure, which in turn account for the different time scales. We assume a suitable, often finite volume, discretization in space. Hence we retain the additive splitting from the continuous problem in the semi-discretized ordinary differential equation (ODE). Hence we develop a new numerical method which accounts for the additive split structure and the multiscale nature. The development of splitting methods is challenging in the analysis of the order conditions and the stability criteria. In particular the interaction between the fast and slow scales render the order conditions often complicated and unstructured. Furthermore every multiscale approach combines the scales in a different way, which is why there is no unified order condition theory. With these challenges in mind we derive the order conditions in a classical way by differentiation of the numerical method. The splitting in a fast and a slow right hand side leads to several combinations of elementary differentials. And every differential has different non-standard coefficients, without any structure between these combinations. This loss in structure renders the numerical solutions of the order conditions quite complicated, and the analytical solutions are only possible in rare cases. We develop a new class of multirate methods, which is parameterized by the fast scale solver. That new approach allows for a better generalization and simplifies several steps by unification. Nevertheless this new type of generalization has the advantage to associate the order conditions of the complete (macro scale) method with the structure of the underlying (micro scale) integrator. The second challenge is the analysis of the (linear) stability of multirate methods. We also analyze the (linear) stability of the newly developed methods. Due to the splitting structure there are many different model problems possible. We deduce a model problem from a simplified system of hyperbolic PDEs. On top of these stability model problems we will construct the new methodss. In analogy to the analysis of the order conditions, we unify the construction of the stability functions and highlight the differences due to the different fast scale integrators afterwards. A good method does not only lead to low errors, but also has a large stability area. Hence we optimize the method parameters with respect to the stability area. In our case, the parameter set contains rational and real parameters. We circumvent the solution of a mixed-integer optimization problem by considering only some rational parameters and optimize for them independently. Nevertheless, we obtain several thousand sub problems. Finally we consider two nonlinear benchmark problems. With these problems we analyze the accuracy and stability again and compare the efficiency with two reference multiscale methods.

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