Recent advances in medical science regarding the interaction and functional role of fluid compartments in the central nervous system have attracted the attention of many researchers across various disciplines. Neurotoxins are constantly cleared from the brain parenchyma through the intramural periarterial drainage system, glymphatic system and meningeal lymphatic system. Impairment of these systems can potentially contribute to the onset of neurological disorders. The goal of this thesis is to contribute to the understanding of brain fluid dynamics and to the role of vascular pathologies in the context of neurological disorders. To achieve this goal, we designed the first multi-scale, closed-loop mathematical model of the murine fluid system, incorporating: heart dynamics, major arteries and veins, microcirculation, pulmonary circulation, venous valves, cerebrospinal fluid (CSF), brain interstitial fluid (ISF), Starling resistors, Monro-Kellie hypothesis, brain lymphatic drainage and the modern concept of CSF/ISF drainage and absorption based on the {\em Bulat-Klarica-Orešković} hypothesis. The mathematical model relies on one-dimensional Partial Differential Equations (PDEs) for blood vessels and on Ordinary Differential Equations (ODEs) for lumped parameter models. The systems of PDEs and ODEs are solved through a high-order finite volume ADER method and through an implicit Euler method. The computational results are validated against literature values and magnetic resonance flow measurements. Furthermore, the model is validated against {\em in-vivo} intracranial pressure waveforms acquired in healthy mice and in mice with impairment of the intracranial venous outflow. Through a systematic use of our computational model in healthy and pathological cases, we provide a complete and holistic neurovascular view of the main murine fluid dynamics. We propose a hypothesis on the working principles of the glymphatic system, opening a new door towards a comprehensive understanding of the mechanisms which link vascular and neurological disorders. In particular, we show how impairment of the cerebral venous outflow might potentially lead to accumulation of solutes in the parenchyma, by altering CSF and ISF dynamics. This thesis also concerns the development of a high-order ADER-type numerical method for systems of hyperbolic balance laws in networks, based on a new implicit solver for the junction-generalized Riemann problem. The resulting ADER scheme can deal with stiff source terms and can be applied to non-linear systems of hyperbolic balance laws in domains consisting of networks of one-dimensional sub-domains. Also, we design a novel one-dimensional mathematical model for collecting lymphatics coupled with a Electro-Fluid-Mechanical Contraction (EFMC) model for dynamical contractions. The resulting mathematical model gives each lymphangion the autonomous capability to trigger action potentials based on local fluid-dynamical factors.

This PhD thesis presents a novel second order accurate direct Arbitrary-Lagrangian-Eulerian (ALE) Finite Volume scheme for nonlinear hyperbolic systems, written both in conservative and non-conservative form, whose peculiarity is the nonconforming motion of interfaces. Moreover it has been coupled together with specifically designed path-conservative well balanced (WB) techniques and angular momentum preserving (AMC) strategies. The obtained result is a method able to preserve many of the physical properties of the system: besides being conservative for mass, momentum and total energy, also any known steady equilibrium of the studied system can be exactly maintained up to machine precision. Perturbations around such equilibrium solutions are resolved with high accuracy and minimal dissipation on moving contact discontinuities even for very long computational times. The core of our ALE scheme is the use of a space-time conservation formulation in the construction of the final Finite Volume scheme: the governing PDE system is rewritten at the aid of the space-time divergence operator and then a fully discrete one-step discretization is obtained by integrating over a set of closed space-time control volumes. In order to avoid the typical mesh distortion caused by shear flows in Lagrangian-type methods, we adopt a nonconforming treatment of sliding interfaces, which requires the dynamical insertion or deletion of nodes and edges, and produces hanging nodes and space-time faces shared between more than two cells. In this way, the elements on both sides of the shear wave can move with a different velocity, without producing highly distorted elements, the mesh quality remains high and, as a direct consequence, also the time step remains almost constant in time, even for highly sheared vortex flows. Moreover, due to the space-time conservation formulation, the geometric conservation law (GCL) is automatically satisfied by construction, even on moving nonconforming meshes. Our nonconforming ALE scheme is especially well suited for modeling in polar coordinates vortical flows affected by strong differential rotation: in particular, the novel combination with the well balancing make it possible to obtain great results for challenging astronomical phenomena as the rotating Keplerian disk. Indeed, we have formulated a new HLL-type and a novel Osher-type flux that are both able to guarantee the well balancing in a gas cloud rotating around a central object, maintaining up to machine precision the equilibrium between pressure gradient, centrifugal force and gravity force that characterizes the stationary solutions of the Euler equations with gravity. To the best knowledge of the author this work is original for various reasons: it is the first time that the little dissipative Osher scheme is modified in order to be well balanced for non trivial equilibria, and it is the first time that WB is coupled with ALE for the Euler equations with gravity; moreover the use of a well balanced Osher scheme joint with the Lagrangian framework allows, for the first time within a Finite Volume method, to maintain exactly even moving equilibria. In addition, the introduced techniques demonstrate a wide range of applicability from steady vortex flows in shallow water equations to complex free surface flows in two-phase models. In the last case, studied on fixed Cartesian grids, the new well balanced methods have been implemented in parallel exploiting a GPU-based platform and reaching the very high efficiency of ten million of volumes processed per seconds. Finally, in the case of vortical flows we propose a preliminary analysis on how to increase the accuracy of the method by exploiting the redundant conservation law that can be written for the angular momentum, as proposed in Després et al. JCP 2015. Indeed, an easy manipulation of the Euler equations allows to write its additional conservation law: clearly it does not add any supplementary information from the analytical point of view, but from a numerical point of view it provides extra information in particular in the case of rotating systems. We present both a master-slave approach, to deduce a posteriori a more precise approximation of the velocity, and some coupled approaches to investigate how the entire process can take advantage from considering directly the angular momentum during the computation within a strong coupling with other variables. A large set of different numerical tests has been carried out in order to check the accuracy and the robustness of the new methods for both smooth and discontinuous problems, close and far away from the equilibrium, in one and two space dimensions. Many of the presented results show a great enhancement with respect to the state of the art.

The last decade has seen an increasing interest in bio-mathematical modelling and scientific computing, resulting in new applications to relevant physiological phenomena and to a better understanding of the origin of various diseases. A topic of great interest to several degenerative diseases is filtration across microvessel walls. The role of the microvessel wall is to let oxygen and nutrients contained in the blood stream to reach the interstitium, and ultimately the surrounding cells, while blocking macromolecules. An understanding of these processes is important in preventing and curing neuro-degenerative diseases, as well as for exploring possible mechanisms to make drug delivery more efficient. This work presents a one-dimensional, time dependent mathematical model describing transport of blood plasma and macromolecules across blood vessel walls. The model takes into account the heterogeneous microvessel wall composition, in order to accurately describe trans-vascular flow. This results in a multi-layered domain, accounting for variable physical properties across the layers forming the micro-vascular wall. In particular, the glycocalyx and endothelium, accounted for in many biological studies, are represented in our model. This micro-structural, yet simplified description of the vascular wall, allows us to simulate the effect of glycocalyx damage and of other pathologies, such as hypertension, hemorrhage and hypovolemia, both in steady and time-dependent states. Due to the simplicity, and thus efficiency of the proposed model, simulations are fast and provide results which are in line with published experimental studies. Furthermore, the simulation tool may be useful for practical applications in physiological and medical studies, by evaluating the possible consequences of pathological conditions.

This thesis is about the development and testing of a novel two-dimensional numerical model (the GIAMT2D model) able to address the hydro-morphodynamic evolution of gravel-bed rivers. The model solves the two-dimensional hyperbolic system of partial differential equations (PDEs) arising from the shallow water-Exner model, describing free surface shallow flows over erodible bed, with suitable closure relations for bedload transport. A coupled formulation of the mathematical problem, which is needed in order to correctly handle sediment transport in Froude trans-critical flow conditions, is implemented, resulting in a non-conservative hyperbolic problem, which requires the adoption of a path-conservative scheme. A drawback of the fully-coupled shallow water-Exner model is that in general the solution of the Riemann problem is not easily available, at least if complex empirical sediment transport formulae are applied, which makes the upwind approach inadequate for designing numerical approximations to the solutions. Adoption of the more general, Riemann solver-free centred approach is thus required, the drawback being that centred schemes are significantly less accurate than upwind schemes in some specific cases, namely for intermediate waves and computations at low CFL number. In GIAMT2D an original centred upwind-biased scheme (UPRICE2-C delta) is applied, recovering accuracy typical of upwind methods, still being able to include any bedload transport formula. The proposed scheme results from original studies in applied mathematics, presented in the first part of the thesis, concerning the development of upwind-biased variations of the centred FORCE scheme for the solution of hyperbolic systems of PDEs, in conservative and non-conservative form. The performance of these schemes is thoroughly assessed in a suite of tests for the shallow water equations. The GIAMT2D model embeds the UPRICE2-Cd scheme extended to second-order accuracy in the ADER framework, inserted in a robust second-order preserving splitting technique for the treatment of frictional source terms, and includes an original wetting-and-drying procedure. The model performance is checked in well-established classical test cases with fixed and movable bed. These applications highlight the capability of the model in correctly and accurately solving the equations in various cases, e.g. in computations at low local CFL number, in the solution of wet-dry fronts with fixed and movable bed and in the prediction of sediment transport in Froude trans-critical conditions. The concept of "morphodynamic benchmark" is introduced for the purpose of assessing the model performance in reproducing basic river morphodynamic processes for which established theoretical and experimental knowledge is available. Unit processes with utmost importance for gravel-bed river morphodynamics, like free and forced bar instability and the stability of channel bifurcations, are chosen for this aim. In this novel approach for assessing the model capabilities, the numerical solutions satisfactorily compare with approximate analytical morphodynamic solution and laboratory data. Having proved that the model is able to reproduce the salient features of these classical morphodynamic solutions, an original morphodynamic study is finally carried out, concerning the non-linear interaction of free and forced bars in straight channels, for which a mature analytical theory is not available at present. The numerical runs of GIAMT2D are used to validate the research hypotheses developed on the basis of existing analytical theories and satisfactorily compare with field observations.

In this thesis we investigate variational problems involving 1-dimensional sets (e.g., curves, networks) and variational inequalities related to obstacle-type dynamics from a twofold prospective. On one side, we provide variational approximations and convex relaxations of the relevant energies and dynamics, moving mainly within the framework of Gamma-convergence and of convex analysis. On the other side, we thoroughly investigate the numerical optimization of the corresponding approximating energies, both to recover optimal 1-dimensional structures and to accurately simulate the actual dynamics.

As recently shown in laboratory bench scale experiments, chemotaxis, i.e.the movement of microorganisms toward or away from the concentration gradient of a chemical species, could have a fundamental role in the transport of bacteria through saturated porous media. Chemotactic bacteria could enhance bioremediation by directing their own motions to residual contaminants in less conductive zones in aquifers. The aim of the present work is to develop a proper numerical scheme to define and to quantify the magnitude and the role of chemotaxis in the complex groundwater system framework. We present a new class of meshless Lagrangian particle methods based on the Smooth Particle Hydrodinamics (SPH) formulation of Vila & Ben Moussa, combined with a new Weighted Essentially Non-Oscillatory (WENO) reconstruction technique on moving point clouds in multiple space dimensions. The purpose of this new scheme is to fully exploit the advantages of SPH among traditional meshbased and meshfree schemes and to overcome its inapplicability for modeling chemotaxis in porous media. The key idea is to produce for each particle first a set of high order accurate Moving Least Squares (MLS) reconstructions on a set of different reconstruction stencils. Then, these reconstructions are combined with each other using a nonlinear WENO technique in order to capture at the same time discontinuities and to maintain accuracy and low numerical dissipation in smooth regions. The numerical fluxes between interacting particles are subsequently evaluated using this MLS-WENO reconstruction at the midpoint between two particles, in combination with a Riemann solver that provides the necessary stabilization of the scheme based on the underlying physics of the governing equations. We propose the use of two different Riemann solvers: the Rusanov flux and an Osher-type flux. The use of monotone fluxes together with a WENO reconstruction ensures accuracy, stability, robustness and an essentially non oscillatory solution without the artificial viscosity term usually employed in conventional SPH schemes. To our knowledge, this is the first time that the WENO method, which has originally been developed for mesh-based schemes in the Eulerian framework on fixed grids, is extended to meshfree Lagrangian particle methods like SPH in multiple space dimensions. In the first part, we test the new algorithm on two dimensional blast wave problems and on the classical one-dimensional Sod shock tube problem for the Euler equations of compressible gas dynamics. We obtain a good agreement with the exact or numerical reference solution in all cases and an improved accuracy and robustness compared to existing standard SPH schemes. In the second part, the new SPH scheme is applied to advection-diffusion equation in heterogeneous porous media with anisotropic diffusion tensor. Several numerical test case shows that the new scheme is accurate. Unlike standard SPH, it reduces the occurrence of negative concentration. In the third part, we show the applicability of the new scheme for modeling chemotaxis in porous media. We test the new scheme against analytical reference solutions. Under the assumption of complete mixing at the Darcy scale, we perform different two-dimensional conservative solute transport simulations under steady-state conditions with instant injection showing that chemotaxis significantly affect the quantification of field-scale mixing processes.

From industrial revolution to the present day, fossil fuels are the main sources for ensuring energy supply. Fossil fuel usages have negative effects on environment that are highlighted by several local or international policy initiatives at support of the big energy transition. The effects urge energy planners to integrate renewable energies into the corresponding energy systems. However, large-scale incorporation of renewable energies into the systems is difficult because of intermittent behaviors, limited availability and economic barriers. It requires intricate balancing among different energy producing resources and the syringes among all the major energy sectors. Although it is possible to evaluate a given energy scenario (complete set of parameters describing a system) by using a simulation model, however, identifying optimal energy scenarios with respect to multiple objectives is a very difficult to accomplished. In addition, no generalized optimization framework is available that can handle all major sectors of an energy system. In this regards, we propose a complete generalized framework for identifying scenarios with respect to multiple objectives. The framework is developed by coupling a multi-objective evolutionary algorithm and EnergyPLAN. The results show that the tool has the capability to handle multiple energy sectors together; moreover, a number of optimized trade-off scenarios are identified. Furthermore, several improvements are proposed to the framework for finding better-optimized scenarios in a computationally efficient way. The framework is applied on two different real-world energy system optimization problems. The results show that the framework is capable to identify optimized scenarios both by considering recent demands and by considering projected demands. The proposed framework and the corresponding improvements make it possible to provide a complete tool for policy makers for designing optimized energy scenarios. The tool can be able to handle all major energy sectors and can be applied in short and long-term energy planning.

This Ph.D. thesis presents a threefold revisitation and reformulation of the linear sampling method (LSM) for the qualitative solution of inverse scattering problems (in the resonance region and in time-harmonic regime): 1) from the viewpoint of its implementation (in a 3D setting), the LSM is recast in appropriate Hilbert spaces, whereby the set of algebraic systems arising from an angular discretization of the far-field equation (written for each sampling point of the numerical grid covering the investigation domain and for each sampling polarization) is replaced by a single functional equation. As a consequence, this 'no-sampling' LSM requires a single regularization procedure, thus resulting in an extremely fast algorithm: complex 3D objects are visualized in around one minute without loss of quality if compared to the traditional implementation; 2) from the viewpoint of its application (in a 2D setting), the LSM is coupled with the reciprocity gap functional in such a way that the influence of scatterers outside the array of receiving antennas is excluded and an inhomogeneous background inside them can be allowed for: then, the resulting 'no-sampling' algorithm proves able to detect tumoural masses inside numerical (but rather realistic) phantoms of the female breast by inverting the data of an appropriate microwave scattering experiment; 3) from the viewpoint of its theoretical foundation, the LSM is physically interpreted as a consequence of the principle of energy conservation (in a lossless background). More precisely, it is shown that the far-field equation at the basis of the LSM (which does not follow from physical laws) can be regarded as a constraint on the power flux of the scattered wave in the far-field region: if the flow lines of the Poynting vector carrying this flux verify some regularity properties (as suggested by numerical simulations), the information contained in the far-field constraint is back-propagated to each point of the background up to the near-field region, and the (approximate) fulfilment of such constraint forces the L^2-norm of any (approximate) solution of the far-field equation to behave as a good indicator function for the unknown scatterer, i.e., to be 'small' inside the scatterer itself and 'large' outside.

The AEgIS experiment pursues the ambitious goal of measuring for the first time the gravitational pull on neutral antimatter. The envisioned method consists in producing a beam of cold anti-hydrogen and measuring the deflection of its free fall by means of a Moiré deflectometer. To do so the pulsed production of abundant cold anti-hydrogen is paramount, therefore the charge exchange production mechanism has been elected as the most promising candidate production method. Performing the charge exchange anti-hydrogen production requires access to an abundant source of cold positronium which can be achieved by the employment of oxide-coated nanochanneled silica plates (NCPs). We spend chapter 1 formulating a classical model of positronium production and thermalisation in NCPs and validating it by testing it against the available experimental data. In chapter 2 we describe the measurement of the energy spectrum of positronium produced by nanochanneled plates using the beam produced by the SURF machine. We then compare the measured energy spectra with the model proposed in chapter 1 showing, in the comparison, the indication of a transition during thermalisation process to a regime where quantum phenomena become significant. We describe in detail in chapter 3 several positronium spectroscopy measurements that we performed during the course of the last three years by employing the positron beam line of the experiment AEgIS. We will the proceed to illustrate an improved version of the detrending technique commonly employed in signal analysis which, applied to the analysis of SSPALS spectra, improves the achievable precision on the experimental results. In chapter 4 we describe an innovative approach that we are currently pursuing to employ the detector FACT, part of the AEgIS apparatus, to confirm the successful production of anti-hydrogen.

The theory of partial differential equations constitutes today one of the most important topics of scientific understanding. A standard approach for solving a time-dependent partial differential equation consists in discretizing the spatial variables by finite differences or finite elements. This results in a huge system of (stiff) ordinary differential equations that has to be integrated in time. Exponential integrators constitute an interesting class of numerical methods for the time integration of stiff systems of differential equations. Their efficient implementation heavily relies on the fast computation of the action of certain matrix functions; among those, the matrix exponential is the most prominent one. In this manuscript, we go through the steps that led to the development of backward error accurate routines for computing the action of the matrix exponential.