New rational methods for the numerical solution of first order initial value problem

Exponentially-fitted numerical methods are appealing because L-stability is guaranteed when solving initial value problems of the form Y¹= ʎy,y(a)=ɳ, ʎ є C , Rc(λ)<0. Such numerical methods also yield the exact solution when solving the above-mentioned problem. Whilst rational methods have been well established in the past decades, most of them are not ‘completely’ exponentially fitted.Recently, a class of one-step exponential-rational methods (ERMs) were discovered.Analyses showed that all ERMs are exponentially-fitted, hence implying L-stability.Several numerical experiments showed that ERMs is more accurate than existing rational methods in solving general initial value problem. However, ERMs have several weaknesses: i) every ERM is non-uniquely defined; ii) may return complex values; and iii) less accurate numerical solution when solving problem whose solution possesses singularity.Therefore, the first purpose of this study is to modify the original ERMs so that the first two weaknesses will be overcomed. Theoretical analyses such as consistency, stability and convergence of the modified ERMs are presented.Numerical experiments showed that the modified ERMs and the original ERMs are found to have comparable accuracy; hence modified ERMs are preferable to original ERMs.The second purpose of this study is to overcome the third weakness of the original ERMs where a variable step-size strategy is proposed to improve the accuracy ERMs.The procedures of the strategy are detailed out in this report.Numerical experiments have revealed that the affects from the implementation of the strategy is less obvious

Efficient Reduction Techniques for the Simulation and Optimization of Parametrized Systems:Analysis and Applications

This thesis is concerned with the development, analysis and implementation of efficient reduced order models (ROMs) for the simulation and optimization of parametrized partial differential equations (PDEs). Indeed, since the high-fidelity approximation of many complex models easily leads to solve large-scale problems, the need to perform multiple simulations to explore different scenarios, as well as to achieve rapid responses, often requires unaffordable computational resources. Alleviating this extreme computational effort represents the main motivation for developing ROMs, i.e. low-dimensional approximations of the underlying high-fidelity problem. Among a wide range of model order reduction approaches, here we focus on the so-called projection-based methods, in particular Galerkin and Petrov-Galerkin reduced basis methods. In this context, the goal is to generate low cost and fast, but still sufficiently accurate ROMs which characterize the system response for the whole range of input parameters we are interested in. In particular, several challenges have to be faced to ensure reliability and computational efficiency. As regards the former, this thesis presents some heuristic approaches to approximate the stability factor of parameterized nonlinear PDEs, a key ingredient of any a posteriori error estimate. Concerning computational efficiency, we propose different strategies to combine the `Matrix Discrete Empirical Interpolation Method' (MDEIM) with a state approximation resulting either from a proper orthogonal decomposition or a greedy approach. Specifically, we exploit the MDEIM to develop fast and efficient ROMs for nonaffinely parametrized elliptic and parabolic PDEs, as well as for the time-dependent Navier-Stokes equations. The efficacy of the proposed methods is demonstrated on a variety of computationally-intensive applications, such as the shape optimization of an acoustic device, the simulation of blood flow in cerebral aneurysms and the simulation of solute dynamics in blood flow and arterial walls. %and coupled blood flow and mass transport in human arteries. Furthermore, the above-mentioned techniques have been exploited to develop a model order reduction framework for parametrized optimization problems constrained by either linear or nonlinear stationary PDEs. In particular, among this wide class of problems, here we focus on those featuring high-dimensional control variables. To cope with this high dimensionality and complexity, we propose an all-at-once optimize-then-reduce paradigm, where a simultaneous state and control reduction is performed. This methodology is applied first to a data reconstruction problem arising in hemodynamics, and then to several optimal flow control problems

Effiziente numerische Methoden für fraktionale Differentialgleichungen und ihr analystischer Hintergrund

Efficient numerical methods for fractional differential equations and their theoretical background are presented. A historical review introduces and motivates the field of fractional calculus. Analytical results on classical calculus as well as special functions and integral transforms are repeated for completeness. Known analytical results on non-integer order differentiation and integrations are presented and corrected and extended where needed. On those results several numerical methods for the solution of fractional differential equations are based. These methods are described and compared to each other in detail. Special attention is paid to the question of applicability of higher oder methods and in connection the practical implementation of such methods is analyzed. Different ways of improvements of the presented numerical methods are given. Numerical calculations confirm the results which were deduced theoretically. Moreover, some of the presented methods are generalized to deal with partial differential equations of fractional order. Finally a problem of physics/chemistry is presented and some of the presented numerical methods are applied.Effiziente numerische Methoden für fraktionale Differentialgleichungen und ihr theoretischer Hintergrund werden betrachtet. Ein historischer Rückblick liefert eine Einführung und Motivation in das Gebiet der fraktionalen Differential- und Integralrechnung. Analytische Ergebnisse der klassischen Differential- und Integralrechnung, sowie spezielle Funtktionen und Integraltransformationen werden zur Vollständigkeit wiederholt. Bekannte analytische Ergebnisse nicht-ganzzahliger Differentiation und Integration werden dargelegt und berichtigt und erweitert falls nötig. Auf diesen Ergebnissen beruhen mehrere numerische Methoden für die Lösung fraktionaler Differentialgleichungen. Diese Methoden werden detailliert beschrieben und untereinander verglichen. Besonderer Wert wird auf die Frage nach der Anwendbarkeit der Methoden höherer Ordnung gelegt und in diesem Zusammenhang die praktische Implementierung solcher Methoden untersucht. Verschiedene Möglichkeiten zur Verbesserung der beschriebenen Methoden werden vorgestellt. Numerische Berechnungen bestätigen die theoretisch hergeleiteten Ergebnisse. Des Weiteren werden einige der vorgestellten Methoden verallgemeinert, um auf partielle Differentialgleichungen fraktionaler Ordnung angewendet werden zu können. Letzlich wird ein Problem aus der Physik/Chemie vorgestellt und einige der dargestellten numerischen Methoden darauf angewendet