Order reduction of semilinear differential matrix and tensor equations

In this thesis, we are interested in approximating, by model order reduction, the solution to large-scale matrix- or tensor-valued semilinear Ordinary Differential Equations (ODEs). Under specific hypotheses on the linear operators and the considered domain, these ODEs often stem from the space discretization on a tensor basis of semilinear Partial Differential Equations (PDEs) with a dimension greater than or equal to two. The bulk of this thesis is devoted to the case where the discrete system is a matrix equation. We consider separately the cases of general Lipschitz continuous nonlinear functions and the Differential Riccati Equation (DRE) with a quadratic nonlinear term. In both settings, we construct a pair of left-right approximation spaces that leads to a reduced semilinear matrix differential equation with the same structure as the original problem, which can be more rapidly integrated with matrix-oriented integrators. For the DRE, under certain assumptions on the data, we show that a reduction process onto rational Krylov subspaces obtains significant computational and memory savings as opposed to current approaches. In the more general setting, a challenging difference lies in selecting and constructing the two approximation bases to handle the nonlinear term effectively. In addition, the nonlinear term also needs to be approximated for efficiency. To this end, in the framework of the Proper Orthogonal Decomposition (POD) methodology and the Discrete Empirical Interpolation Method (DEIM), we derive a novel matrix-oriented reduction process leading to a practical, structure-aware low order approximation of the original problem. In the final part of the thesis, we consider the multidimensional setting. Here we extend the matrix-oriented POD-DEIM algorithm to the tensor setting and illustrate how we can apply it to systems of such equations. Moreover, we discuss how to integrate the reduced-order model and, in particular, how to solve the resulting tensor-valued linear systems

Finite volume central schemes for a two-phase compressible model with application to gas-liquid Riemann problems and magma ascent dynamics.

In this thesis a multidimensional compressible two-phase flow model with two pressures, two velocities and a single temperature is studied. The system of partial differential equations, derived using the theory of thermodynamically compatible systems, is written in conservative form and the hyperbolicity of the system is shown. Furthermore, for a more complete analysis, the characteristic polynomial is computed. The single temperature model is also compared with the classical Baer-Nunziato model, showing that, although they can be written in similar way, they present important differences. In order to solve the governing equations, several numerical schemes for the discretization in space, based on the finite volume methods, are considered. Several numerical fluxes, the Lax-Friedrichs, the Generalized FORCE, the Kurganov and Tadmor, the Kurganov, Noelle and Petrova and finally a modification of the Lax-Friedrichs numerical fluxes are introduced. For the discretization in time, two approaches for solving the system of partial differential equations, the splitting and the Runge-Kutta approach, are presented. All the numerical schemes are implemented using the open source CFD (Computational Fluid Dynamics) toolbox "OpenFOAM" (Open source Field Operation And Manipulation, developed by OpenCFD Ltd). After the definition of the physical model and of the numerical methods, the robustness, the correctness and the accuracy of the proposed schemes are investigated performing several monodimensional and multidimensional well-known numerical tests in presence of rarefaction and shock waves. Two shock-bubble interaction experiments have been reproduced numerically, comparing the results with laboratory observations. The results show a really good agreement between the simulation and the laboratory experiments. The physical model appears to be accurate and the numerical methods robust, being able to properly resolve detailed flow features as shock-wave refractions, reflections and diffractions. Finally, an application of the single temperature model to the volcanological field is presented, proposing a new model for the initial stages of magma ascent in a conduit during explosive eruptions. The magma ascent model is derived from the single temperature model adding a new transport equation for the gas dissolved in the liquid phase and showing through the calculation of the characteristic polynomial that the new equation does not alter the hyperbolicity of the system. Using one of the numerical schemes proposed, the initial phases of an explosive eruption at Soufriére Hills Volcano are simulated, focusing the attention on the effect of disequilibrium processes. To conclude, in the appendix, the derivation of the single temperature model using the thermodynamically compatible systems is presented and some details regarding the implementation of the numerical schemes proposed in this thesis using the OpenFOAM framework are given

Rapporto 2013 sullo stato della ricerca scientifica all’Università di Trieste

Università degli Studi di Trieste Commissione per la Valutazione della ricerca "Rapporto 2013 sullo stato della ricerca scientifica all’Università di Trieste. Primo esercizio di valutazione delle pubblicazioni scientifiche 2008-2011", Trieste, EUT Edizioni Università di Trieste, 201