In the last half a century, Computational Fluid Dynamics (CFD) has been established as an important complementary part and some times a significant alternative to Experimental and Theoretical Fluid Dynamics. Development of efficient computational algorithms for digital simulation of fluid flows has been an ongoing research effort in CFD. An accurate numerical simulation of compressible Euler equations, which are the gov-erning equations of high speed flows, is important in many engineering applications like designing of aerospace vehicles and their components. Due to nonlinear nature of governing equations, such flows admit solutions involving discontinuities like shock waves and contact discontinuities. Hence, it is nontrivial to capture all these essential features of the flows numerically. There are various numerical methods available in the literature, the popular ones among them being the Finite Volume Method (FVM), Finite Difference Method (FDM), Finite Element Method (FEM) and Spectral method. Kinetic theory based algorithms for solving Euler equations are quite popular in finite volume framework due to their ability to connect Boltzmann equation with Euler equations. In kinetic framework, instead of dealing directly with nonlinear partial differential equations one needs to deal with a simple linear partial differential equation. Recently, FEM has emerged as a significant alternative to FVM because it can handle complex geometries with ease and unlike in FVM, achieving higher order accuracy is easier. High speed flows governed by compressible Euler equations are hyperbolic partial differential equations which are characterized by preferred directions for information propagation. Such flows can not be solved using traditional FEM methods and hence, stabilized methods are typically introduced. Various stabilized finite element methods are available in the literature like Streamlined-Upwind Petrov Galerkin (SUPG) method, Galerkin-Least Squares (GLS) method, Taylor-Galerkin method, Characteristic Galerkin method and Discontinuous Galerkin Method. In this thesis a novel stabilized finite element method called as Kinetic Streamlined-Upwind Petrov Galerkin (KSUPG) method is formulated. Both explicit and implicit versions of KSUPG scheme are presented. Spectral stability analysis is done for explicit KSUPG scheme to obtain the stable time step. The advantage of proposed scheme is, unlike in SUPG scheme, diffusion vectors are obtained directly from weak KSUPG formulation. The expression for intrinsic time scale is directly obtained in KSUPG framework. The accuracy and robustness of the proposed scheme is demonstrated by solving various test cases for hyperbolic partial differential equations like Euler equations and inviscid Burgers equation. In the KSUPG scheme, diffusion terms involve computationally expensive error and exponential functions. To decrease the computational cost, two variants of KSUPG scheme, namely, Peculiar Velocity based KSUPG (PV-KSUPG) scheme and Circular distribution based KSUPG (C-KSUPG) scheme are formulated. The PV-KSUPG scheme is based on peculiar velocity based splitting which, upon taking moments, recovers a convection-pressure splitting type algorithm at the macroscopic level. Both explicit and implicit versions of PV-KSUPG scheme are presented. Unlike KSUPG and PV-KUPG schemes where Maxwellian distribution function is used, the C-KUSPG scheme uses a simpler circular distribution function instead of a Maxwellian distribution function. Apart from being computationally less expensive it is less diffusive than KSUPG scheme.

In dieser Arbeit verwende ich den Formalismus der endoreversiblen Thermodynamik um ein hydraulisches Rekuperationssystem für Nutzfahrzeuge zu modellieren und zu untersuchen. Dafür führe ich verlustbehaftete Übergänge extensiver Größen zwischen Teilsystemen eines Systems ein. Diese können einerseits der Modellierung von Leckagen und Reibungsverlusten, welche als Partikel- oder Drehmomentverluste dargestellt würden, dienen. Andererseits ermöglichen sie die Modellierung einer endoreversiblen Maschine, welche – durch Definition eines solchen verlustbehafteten, internen Überganges – ein gegebenes Wirkungsgradkennfeld und daraus resultierende Entropieproduktion inne hat. Diese wird infolge zur Modellierung der Hydraulikeinheit des Rekuperationssystems verwendet. Desweiteren basiert die Beschreibung des Rekuperationssystems auf der Modellierung der Hydraulikflüssigkeit als Van-der-Waals-Fluid, sodass Druckverluste im endoreversiblen Sinne konsistent berücksichtigt werden können. Von gegebenen Materialparamtern werden die dafür notwendigen Van-der-Waals-Parameter hergeleitet. Weitere Aspekte sind Wärmeverluste an die Umgebung sowie Wärmeübergänge zwischen Teilsystemen. Auf Grundlage realer Fahrdaten der Nutzfahrzeuge werden verschiedene dynamische und thermodynamische Effekte im Rekuperationssystem analysiert. Ihr Einfluss auf die resultierenden energetischen Einsparungen beim Abbremsen und Beschleunigen wird durch Variation zugehöriger Parameter aufgezeigt. Zuletzt wird mit einem vereinfachten Modell ohne Druck- und Wärmeverluste, aber unter Einbeziehung des Verbrennungsmotors des Fahrzeuges, eine Optimierung der Steuerung des hydraulischen Rekuperationssystems mit Hinblick auf minimalen Kraftstoffverbrauch durchgeführt. Hier zeigt sich eine erhebliche Verbesserung durch die Leistungsaufteilung zwischen Verbrennungsmotor und Rekuperationssystem nach deren Betriebsbereichen mit maximalem Wirkungsgrad. / In this work I use the formalism of endoreversible thermodynamics to model and investigate a hydraulic recuperation system for commercial vehicles. For that, I introduce lossy transfers of extensive quantities between subsystems of an endoreversible system. On the one hand, these allow modeling of leakages and friction losses, which can be represented as particle or torque losses. On the other hand, they can be used as internal extensity transfers in endoreversible engines which, as a result, have a given efficiency or efficiency map and among other things give an expression for their entropy production. Such an engine is used to model the hydraulic unit of the recuperation system. Furthermore, the description of the recuperation system is based on the modeling of the hydraulic fluid as a van der Waals fluid, so that pressure losses can be taken into account in a consistent endoreversible fashion. From given material parameters the necessary van der Waals parameters are derived. Other aspects of the modeling include heat losses to the environment and heat transfers between subsystems. On the basis of real driving data, various dynamic and thermodynamic effects within the recuperation system are observed and their influence as well as the influence of selected parameters on the resulting energy savings for both acceleration and deceleration are shown. Finally, using a simplified model neglecting pressure and heat losses, but including the internal combustion engine of the vehicle, an optimization of the control strategy for the hydraulic recuperation system with regard to minimum fuel consumption is performed. Here, a significant improvement due to a power distribution between combustion engine and recuperation system according to their high efficiency operating ranges can be achieved.