In this thesis, we consider the problem of solving large and sparse linear systems of saddle point type stemming from optimization problems. The focus of the thesis is on iterative methods, and new preconditioning srategies are proposed, along with novel spectral estimtates for the matrices involved.

This Ph.D thesis focuses on iterative regularization methods for regularizing linear and nonlinear ill-posed problems. Regarding linear problems, three new stopping rules for the Conjugate Gradient method applied to the normal equations are proposed and tested in many numerical simulations, including some tomographic images reconstruction problems. Regarding nonlinear problems, convergence and convergence rate results are provided for a Newton-type method with a modified version of Landweber iteration as an inner iteration in a Banach space setting.

In this thesis we present a new and accurate series of computation methods for compressible multi-phase flows with capillary effects based upon the full seven-equation Baer-Nunziato model. For that reason, there are some numerical methods to obtain high accuracy solutions, which will be shown here. First, a high resolution shock capturing Total Variation Diminishing (TVD) finite volume scheme is used on both Cartesian and unstructured triangular grids. Regarding the TVD finite volume scheme on the unstructured grid, time-accurate local time stepping (LTS) is applied to compute the solutions of the governing PDE system, in which the results are also compared with time-accurate global time stepping. Second, we propose a novel high order accurate numerical method for the solution of the seven equation Baer-Nunziato model based on ADER discontinuous Galerkin (DG) finite element schemes combined with a posteriori subcell finite volume limiting and adaptive mesh refinement (AMR). In multi-phase flows, the difficulty is to design accurate numerical methods for resolving the phase interface, as well as the simulation of the phenomena occurring at the interface, such as surface tension effects, heat transfer and friction. This is because of the interactions of the fluids at the phase interface and its complex geometry. So the accurate simulation of compressible multi-phase flows with surface tension effects is currently still one of the most challenging problems in computational fluid dynamics (CFD). In this work, we present a novel path-conservative finite volume discretization of the continuum surface force method (CSF) of Brackbill et al. to account for the surface tension effect due to curvature of the phase interface. This is achieved in the context of a diffuse interface approach, based on the seven equation Baer-Nunziato model of compressible multi-phase flows. Such diffuse interface methods for compressible multi-phase flows including capillary effects have first been proposed by Perigaud and Saurel. Regarding the high order accuracy of a diffuse interface approach, the interface is captured and allowed to travel across one single possibly refined cell, and is computed in the context of multi-dimensional high accurate space/time DG schemes with AMR and a posteriori sub-cell stabilization. The surface tension terms of the CSF approach are considered as a part of the non-conservative hyperbolic system. We propose to integrate the CSF source term as a non-conservative product and not simply as a source term, following the ideas on path conservative finite volume schemes put forward by Castro and Parés. For the validation of the current numerical methods, we compare our numerical results with those published previously in the literature.

In this work the numerical discretization of the partial differential governing equations for compressible and incompressible flows is dealt within the discontinuous Galerkin (DG) framework along space-time adaptive meshes. Two main research fields can be distinguished: (1) fully explicit DG methods on collocated grids and (2) semi-implicit DG methods on edge-based staggered grids. DG methods became increasingly popular in the last twenty years mainly because of three intriguing properties: i) non-linear L2 stability has been proven; ii) arbitrary high order of accuracy can be achieved by simply increasing the polynomial order of the chosen basis functions, used for approximating the state-variables; iii) high scalability properties make DG methods suitable for large-scale simulations on general unstructured meshes. It is a well known fact that a major weakness of high order DG methods lies in the difficulty of limiting discontinuous solutions, which generate spurious oscillations, namely the so-called ’Gibbs phenomenon’. Over the years, several attempts have been made to cope with this problem and different kinds of limiters have been proposed. Among them, a rather intriguing paradigm has been defined in the work of [71], in which the nonlinear stabilization of the scheme is sequentially and locally introduced only for troubled cells on the basis of a multidimensional optimal order detection (MOOD) criterion. In the present work the main benefits of the MOOD paradigm, i.e. the computational robustness even in the presence of strong shocks, are preserved and the numerical diffusion is considerably reduced also for the limited cells by resorting to a proper sub-grid. In practice the method first produces a so-called candidate solution by using a high order accurate unlimited DG scheme. Then, a set of numerical and physical detection criteria is applied to the candidate solution, namely: positivity of pressure and density, absence of floating point errors and satisfaction of a discrete maximum principle in the sense of polynomials. Then, in those cells where at least one of these criteria is violated the computed candidate solution is detected as troubled and is locally rejected. Next, the numerical solution of the previous time step is scattered onto cell averages on a suitable sub-grid in order to preserve the natural sub-cell resolution of the DG scheme. Then, a more reliable numerical solution is recomputed a posteriori by employing a more robust but still very accurate ADER-WENO finite volume scheme on the sub-grid averages within that troubled cell. Finally, a high order DG polynomial is reconstructed back from the evolved sub-cell averages. Moreover, handling typical multiscale problems, dynamic adaptive mesh refinement (AMR) and adaptive polynomial order methods are probably the two main ways of preserving accuracy and efficiency, and saving computational effort. The here adopted AMRapproach is the so called ’cell by cell’ refinement because of its formally very simple tree-type data structure. In the here-presented ’cell-by-cell’ AMR every single element is recursively refined, from a coarsest refinement level l0 = 0 to a prescribed finest (maximum) refinement level lmax, accordingly to a refinement-estimator function X that drives step by step the choice for recoarsening or refinement. The combination of the sub-cell resolution with the advantages of AMR allows for an unprecedented ability in resolving even the finest details in the dynamics of the fluid. First, the Euler equations of compressible gas dynamics and the magnetohydrodynamics (MHD) equations have been treated [281]. Then, the presented method has been readily extended to the special relativistic ideal MHD equations [280], but also the the case of diffusive fluids, i.e. fluid flows in the presence of viscosity, thermal conductivity and magnetic resistivity [116]. In particular, the adopted formalism is quite general, leading to a novel family of adaptive ADER-DG schemes suitable for hyperbolic systems of partial differential equations in which the numerical fluxes also depend on the gradient of the state vector because of the parabolic nature of diffusive terms. The presented results show clearly that the shock-capturing capability of the news schemes are significantly enhanced within the cell-by-cell Adaptive Mesh Refinement (AMR) implementation together with time accurate local time stepping (LTS). The resolution properties of the new scheme have been shown through a wide number of test cases performed in two and in three space dimensions, from low to high Mach numbers, from low to high Reynolds regimes. In particular, concerning MHD equations, the divergence-free character of the magnetic field is taken into account through the so-called hyperbolic ’divergence-cleaning’ approach which allows to artificially transport and spread the numerical spurious ’magnetic monopoles’ out of the computational domain. A special treatment has been followed for the incompressible Navier-Stokes equations. In fact, the elliptic character of the incompressible Navier-Stokes equations introduces an important difficulty in their numerical solution: whenever the smallest physical or numerical perturbation arises in the fluid flow then it will instantaneously affect the entire computational domain. Thus, a semi-implicit approach has been used. The main advantage of making use of a semi-implicit discretization is that the numerical stability can be obtained for large time-steps without leading to an excessive computational demand [117]. In this context, we derived two new families of spectral semi-implicit and spectral space-time DG methods for the solution of the two and three dimensional Navier-Stokes equations on edge-based staggered Cartesian grids [115], following the ideas outlined in [97] for the shallow water equations. The discrete solutions of pressure and velocity are expressed in the form of piecewise polynomials along different meshes. While the pressure is defined on the control volumes of the main grid, the velocity components are defined on edge-based dual control volumes, leading to a spatially staggered mesh. In the first family, high order of accuracy is achieved only in space, while a simple semi-implicit time discretization is derived by introducing an implicitness factor theta in [0.5, 1] for the pressure gradient in the momentum equation. The real advantages of the staggering arise after substituting the discrete momentum equation into the weak form of the continuity equation. In fact, the resulting linear system for the pressure is symmetric and positive definite and either block penta-diagonal (in 2D) or block hepta-diagonal (in 3D). As a consequence, the pressure system can be solved very efficiently by means of a classical matrix-free conjugate gradient method. Moreover, a rigorous theoretical analysis of the condition number of the resulting linear systems and the design of specific preconditioners, using the theory of matrix-valued symbols and Generalized Locally Toeplitz (GLT) algebra has been successfully carried out with promising results in terms of numerical efficiency [102]. The resulting algorithm is stable, computationally very efficient, and at the same time arbitrary high order accurate in both space and time. The new numerical method has been thoroughly validated for approximation polynomials of degree up to N = 11, using a large set of non-trivial test problems in two and three space dimensions, for which either analytical, numerical or experimental reference solutions exist. Moreover, the here mentioned semi-implicit DG method has been successfully extended to a novel edge-based staggered ’cell-by-cell’ adaptive meshes [114].

The main topic of the thesis is the synthesis of control laws for interacting agent-based dynamics and their mean-field limit. In particular, after a general introduction, in the second chapter a linearization-based approach is used for the computation of sub-optimal feedback laws obtained from the solution of differential matrix Riccati equations. Quantification of dynamic performance of such control laws leads to theoretical estimates on suitable linearization points of the nonlinear dynamics. Subsequently, the feedback laws are embedded into a nonlinear model predictive control framework where the control is updated adaptively in time according to dynamic information on moments of linear mean-field dynamics. The performance and robustness of the proposed methodology is assessed through different numerical experiments in collective dynamics. In the other chapters of the thesis I present some related projects, robustness of systems with uncertainties, a proximal gradient approach for sparse control and an application in crowd evacuation dynamics.

IIn this work we present a new class of well-balanced, arbitrary high order accurate semi-implicit discontinuous Galerkin methods for the solution of the shallow water and incompressible Navier-Stokes equations on staggered unstructured curved meshes. Isoparametric finite elements are used to take into account curved domain boundaries. Regarding two-dimensional shallow water equations, the discrete free surface elevation is defined on a primal triangular grid, while the discrete total height and the discrete velocity field are defined on an edge-based staggered dual grid. Similarly, for the two-dimensional incompressible Navier-Stokes case, the discrete pressure is defined on the main triangular grid and the velocity field is defined on the edge-based staggered grid. While staggered meshes are state of the art in classical finite difference approximations of the incompressible Navier-Stokes equations, their use in the context of high order DG schemes is novel and still quite rare. High order (better than second order) in time can be achieved by using a space-time finite element framework, where the basis and test functions are piecewise polynomials in both space and time. Formal substitution of the discrete momentum equation on the dual grid into the discrete continuity equation on the primary grid yields a very sparse system for the scalar pressure involving only the direct neighbor elements, so that it becomes a block four-point system in 2D and a block five-point system for 3D tetrahedral meshes. The resulting linear system is conveniently solved with a matrix-free GMRES algorithm. Note that the same space-time DG scheme on a collocated grid would lead to ten non-zero blocks per element in 2D and seventeen non-zero blocks in 3D, since substituting the discrete velocity into the discrete continuity equation on a collocated mesh would involve also neighbors of neighbors. From numerical experiments we find that our linear system is well-behaved and that the GMRES method converges quickly even without the use of any preconditioner, which is a unique feature in the context of high order implicit DG schemes. A very simple and efficient Picard iteration is then used in order to derive a space-time pressure correction algorithm that achieves also high order of accuracy in time, which is in general a non-trivial task in the context of high order discretizations for the incompressible Navier-Stokes equations. The special case of high order in space low order in time allows us to recover further regularity about the main linear system for the pressure, such as the symmetry and the positive semi-definiteness in the general case. This allows us to use a very fast linear solver such as the conjugate gradient (CG) method. The flexibility and accuracy of high order space-time DG methods on curved unstructured meshes allows to discretize even complex physical domains with very coarse grids in both space and time. We will further extend the previous method to three-dimensional incompressible Navier-Stokes system using a tetrahedral main grid and a corresponding face-based hexaxedral dual grid. The resulting dual mesh consists in non-standard 5-vertex hexahedral elements that cannot be represented using tensor products of one dimensional basis functions. Indeed a modal polynomial basis will be used for the dual mesh. This new family of numerical schemes is verified by solving a series of typical numerical test problems and by comparing the obtained numerical results with available exact analytical solutions or other numerical reference data. Furthermore, the comparison with available experimental results will be presented for incompressible Navier-Stokes equations.

In the present work we solve systems of partial differential equations (PDE) for hyperbolic conservation laws using semi-implicit numerical methods on staggered meshes applied both to the class of finite volume (FV) and both to the family of high order Discontinuous Galerkin (DG) finite elements schemes. In particular, we want to show that these new semi-implicit schemes can be applied in several fields of applied sciences, such as in geophysical flows and compressible fluids in compliant tubes. Inside this thesis we distinguish two big parts. First, we consider staggered semi-implicit schemes for compressible viscous fluids flowing in elastic pipes. This topic is very important in several practical applications of civil, environmental, industrial and biomedical engineering. Here, we analyse the accuracy and the computational efficiency of fully explicit and semi-implicit 1D and 2D finite volume schemes for the simulation of highly unsteady viscous compressible flows in laminar regime in axially symmetric rigid and elastic pipes. We consider two families of differential models that can be used to predict the pressure and velocity distribution along the tube. One is the so called 2Dxr PDE model which is derived from the full compressible Navier-Stokes equations under the assumptions of a hydrostatic pressure and an axially symmetric geometry. The second family is a simple 1D non-conservative PDE system based on the cross-sectionally averaged version of the Navier-Stokes equations in cylindrical coordinates. In this last case, the use of a simple steady friction model is not enough to simulate the wall friction phenomena in highly transient regime. As a consequence the wall friction model has to be frequency dependent and, following previous studies present in the literature, we consider the classes of convolution integral (CI) models and instantaneous acceleration (IA) models. We carry out a rather complete analysis of the previously-mentioned methods for the simulation of flows characterized by fast transient regime in rigid and compliant tubes. The numerical results show that the convolution integral models are clearly better than instantaneous acceleration models concerning accuracy. Moreover, for CI models, instead of computing the convolution integrals, which is very time- and memoryconsuming, we express these methods via a set of additional ODEs for appropriate auxiliary variables. This trick improves the computational efficiency of these methods substantially, since it avoids the direct computation of the convolution integral. In addition, semi-implicit finite volume methods are significantly superior to classical explicit finite volume schemes in terms of computational efficiency, however, providing the same level of accuracy. We then proceed by extending the finite volume discretization of the 1D and 2Dxr PDE models to arbitrary high-order of accuracy in space introducing a new SIDG scheme on staggered meshes. Both models include the effects of the viscosity and of the wall motion. The nonlinear convective terms are discretized explicitly by using a classical RKDG scheme of arbitrary high-order of accuracy in space and third order of accuracy in time. The continuity equation is integrated over the elements that belong to the main grid, while the momentum equation is integrated over the control volumes of the edgebased staggered dual grid. Inserting the discrete momentum equation into the discrete continuity leads to a mildly nonlinear algebraic system for the degrees of freedom of the pressure, which is solved by using the (nested) Newton method of Brugnano, Casulli and Zanolli. We use the -method in order to get second order of accuracy in time for the implicit part of the scheme. In addition, the schemes have to obey only a mild CFL condition based on the fluid velocity and not based on the sound speed; consequently these schemes work also in the low Mach number regime and even in the incompressible limit of the Navier-Stokes equations. This is a very important property, which is the so-called asymptotic preserving (AP) property of the scheme. We carry out several numerical tests in order to validate this novel family of numerical methods against available exact solutions and experimental data. We also report numerical convergence tables in order to show that the new schemes indeed achieve high order of accuracy in space. In the second part of the thesis, we present a new class of a posteriori sub-cell finite volume limiters for spatially high order accurate semi-implicit discontinuous Galerkin schemes on staggered Cartesian grids for the solution of the 1D and 2D shallow water equations (SWE) and of the Euler equations both expressed in conservative form. Here, the starting point is the unlimited arbitrary high order accurate staggered SIDG scheme proposed by Dumbser and Casulli (2013). For this metho d, the mass conservation equation and the momentum equations are integrated using a discontinuous finite element strategy on staggered control volumes, where the discrete free surface elevation is defined on the main grid and the discrete momentum is defined on edge-based staggered dual control volumes. According to the semi-implicit approach, pressure terms are discretized implicitly, while the nonlinear convective terms are discretized explicitly. Inserting the momentum equations into the discrete continuity equation leads to a well conditioned block diagonal linear system for the free surface elevation which can be efficiently solved with modern iterative methods. Furthermore, the staggered SIDG is also extended to the Euler equations of compressible gasdynamics. Here, the governing PDE are rewritten using a flux vector splitting technique. The convective terms are updated using an explicit Runge-Kutta DG integrator. Then, the discrete momentum equation, which is integrated again on the dual grid, is coupled with the discrete energy equation that is discretized on the control volumes of the main grid. The pressure is efficiently obtained solving a linear system combined with an iterative Picard iteration procedure.

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.

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.

In this work we develop a new class of high order accurate Arbitrary-Lagrangian-Eulerian (ALE) one-step finite volume schemes for the solution of nonlinear systems of conservative and non-conservative hyperbolic partial differential equations. The numerical algorithm is designed for two and three space dimensions, considering moving unstructured triangular and tetrahedral meshes, respectively. As usual for finite volume schemes, data are represented within each control volume by piecewise constant values that evolve in time, hence implying the use of some strategies to improve the order of accuracy of the algorithm. In our approach high order of accuracy in space is obtained by adopting a WENO reconstruction technique, which produces piecewise polynomials of higher degree starting from the known cell averages. Such spatial high order accurate reconstruction is then employed to achieve high order of accuracy also in time using an element-local space-time finite element predictor, which performs a one-step time discretization. Specifically, we adopt either the continuous Galerkin (CG) predictor, which does not allow discontinuities in time and is suitable for smooth time evolutions, or the discontinuous Galerkin (DG) predictor which can handle stiff source terms that might produce jumps in the local space-time solution. Since we are dealing with moving meshes the elements deform while the solution is evolving in time, hence making the use of a reference system very convenient. Therefore, within the space-time predictor, the physical element is mapped onto a reference element using a high order isoparametric approach, where the space-time basis and test functions are given by the Lagrange interpolation polynomials passing through a predefined set of space-time nodes. The computational mesh continuously changes its configuration in time, following as closely as possible the flow motion. The entire mesh motion procedure is composed by three main steps, namely the Lagrangian step, the rezoning step and the relaxation step. In order to obtain a continuous mesh configuration at any time level, the mesh motion is evaluated by assigning each node of the computational mesh with a unique velocity vector at each timestep. The node solver algorithm preforms the Lagrangian stage, while we rely on a rezoning algorithm to improve the mesh quality when the flow motion becomes very complex, hence producing highly deformed computational elements. A so-called relaxation algorithm is finally employed to partially recover the optimal Lagrangian accuracy where the computational elements are not distorted too much. We underline that our scheme is supposed to be an ALE algorithm, where the local mesh velocity can be chosen independently from the local fluid velocity. Once the vertex velocity and thus the new node location has been determined, the old element configuration is connected with the new one at the future time level with straight edges to represent the local mesh motion, in order to maintain algorithmic simplicity. The final ALE finite volume scheme is based directly on a space-time conservation formulation of the governing system of hyperbolic balance laws. The nonlinear system is reformulated more compactly using a space-time divergence operator and is then integrated on a moving space-time control volume. We adopt a linear parametrization of the space-time element boundaries and Gaussian quadrature rules of suitable order of accuracy to compute the integrals. In our algorithm either a simple and robust Rusanov-type numerical flux or a more sophisticated and less dissipative Osher-type numerical flux is employed. We apply the new high order direct ALE finite volume schemes to several hyperbolic systems, namely the multidimensional Euler equations of compressible gas dynamics, the ideal classical and relativistic magneto-hydrodynamics (MHD) equations and the non-conservative seven-equation Baer-Nunziato model of compressible multi-phase flows with stiff relaxation source terms. Numerical convergence studies as well as several classical test problems will be shown to assess the accuracy and the robustness of our schemes. Furthermore we focus on the following issues to improve the algorithm efficiency: the time evolution, the numerical flux computation across element boundaries and the high order WENO reconstruction procedure. First, a time-accurate local time stepping (LTS) algorithm for unstructured triangular meshes is derived and presented, where each element can run at its own optimal time step, given by a local CFL stability condition. Then, we propose a new and efficient quadrature-free formulation for the flux computation, in which the space-time boundaries of each element are split into simplex sub-elements. This leads to space-time normal vectors as well as Jacobian matrices that are constant within each sub-element, hence allowing the flux integrals to be evaluated on the space-time reference control volume once and for all analytically during a preprocessing step. Finally, we consider the very new a posteriori MOOD paradigm, recently proposed for the Eulerian framework, to overcome the expensive WENO approach on moving meshes. The MOOD technique requires the use of only one central reconstruction stencil because the limiting procedure is carried out a posteriori instead of a priori, as done in the WENO formulation.