• 2 Year
  • Scientific Computing Pathway
  • University of L'Aquila Place
  • 60 ECTS Credits
  • Read here Qualification
  • Not available if you spent your Year 1 in L'Aquila Note

List of course units

Semester 1

  • Advanced Analysis 1 (6 credits)

    • ECTS credits 6
    • Semester 1
    • University University of L'Aquila
    • Prerequisites

      Basic notions of functional analysis, functions of complex values, standard properties of classical solutions of semilinear first order equations, heat equation, wave equation, Laplace and Poisson's equations.

    • Objectives

      Knowledge of mathematical methods that are widely used by researchers in the area of Applied Mathematics, as Sobolev Spaces, distributions. Application of this knowledge to a variety of topics, including the basic equations of mathematical physics and some current research topics about linear and nonlinear partial differential equations.

    • Topics

      Distributions. Locally integrable functions. The space of test function D(U). Distributions. Distributions associated to Locally integrable functions. Singular distributions. Examples. Operations on distributions: sum, products times functions, change of variables, restrictions, tensor product. Differentiation and his properties; comparison with classical derivatives. Differentiation of jump functions. Partition of unity. Support of a distribution; compactly supported distributions.

      Convolution. Convolution in Lp spaces. Regularity of the convolution. Regularizing sequences and smoothing by means of convolutions. Convolution between distributions and regularization of distributions. Denseness of D(U) in D'(U).

      Sobolev spaces. Definition of weak derivatives and his motivation. Sobolev spaces Wk,p(U) and their properties. Interior and global approximation by smooth functions. Extensions. Traces. Embeddings theorems: Gagliardo-Nirenberg-Sobolev inequality and Embedding theorem for p < n. Embedding theorem for p = n. Hölder spaces. Morrey inequality. Embedding theorem for p > n. Sobolev inequalities in the general case. Compact embeddings: Rellich-Kondrachov theorem, Poincaré inequalities. Characterization of the dual space H-1.

      Second order parabolic equations. Definition of parabolici operator. Weak solutions for linear parabolici equations. existence of weak solutions: Galerkin approximation, construction of approximating solutions, energy estimates, existence and uniqueness of solutions. Existence of solutions of viscous scalar conservation laws.

      First order nonlinear hyperbolic equations. Scalar conservation laws: derivation, examples. Weak solutions, Rankine-Hugoniot conditions, entropy conditions. L1 stability, uniqueness and comparison for weak entropy solutions. Convergence of the vanishing viscosity and existence of the weak, entropy solution. Riemann problem. Definition of hyperbolic system. Quasilinear hyperbolic systems, symmetric and symmetrizable systems. Existence of solutions: approximations, a priori estimate, local existence of classical solutions.

    • Books

      V.S. Vladimirov, Equations of Mathematical Physics. Marcel Dekker, Inc..

      C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics. Springer.

      L.C. Evans, Partial Differential Equations. AMS.

      M.E. Taylor, Partial Differential Equations, Nonlinear equations. Springer.

      H. Brezis, Sobolev Spaces and Partial Differential Equations. Springer.


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  • Mathematical fluid dynamics (6 credits)

    • ECTS credits 6
    • Semester 1
    • University University of L'Aquila
    • Prerequisites

      Basic notions of functional analysis, functions of complex values, standard properties of the  heat equation, wave equation, Laplace and Poisson's equations.

    • Objectives

      This course is designed to give an overview of fluid dynamics from a mathematical viewpoint and to introduce students to  the mathematical modeling of fluid dynamic type. At the end of the course students will be able to perform a  qualitative and quantitative analysis of  solutions for particular fluid dynamics problems and to use concepts and mathematical techniques learned from this course for analysis of other partial differential equations.

    • Topics

      Derivation of the governing equations: Euler and Navier-Stokes.

      Eulerian and Lagrangian description of fluid motion; examples of fluid flows.

      Vorticity equation in 2D and 3D.

      Dimensional analysis: Reynolds number, Mach Number, Frohde number.

      From compressible to incompressible models.

      Fluid dynamic modeling in various fields: biofluids, atmosphere and ocean, astrophysics.

      Existence of solutions for viscid and inviscid fluids.


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  • High performance computing laboratory and applications to differential equations (6 credits)

    • ECTS credits 6
    • Semester 1
    • University University of L'Aquila
    • Topics

      Linux/Unix OS and tools;

      Basic Fortran (or C);

      HPC architecture and libraries;

      Application (ex ODEs, PDEs, elastodynamics).


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  • Machine learning (6 credits)


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Semester 2

  • Kinetic and hydrodynamic models (6 credits)

    • ECTS credits 6
    • Semester 1
    • University University of L'Aquila
    • Prerequisites

      Mathematical Analysis, Fourier transform.

    • Objectives

      This course provides an introduction to the classical kinetic theory of gases and the principles of kinetic modeling.

      A special focus is given to the derivation of hydrodynamic equations from kinetic models by means of non-perturbative techniques and to the analysis of numerical schemes for the simulation of fluid flows.

      On successful completion of this module the student has the knowledge on the basic principles and the simulation strategies of kinetic models.

    • Topics

      Boltzmann equation and the principles of kinetic description.

      Kinetic models: BGK,Maxwell molecules, Vlasov equation and Fokker-Planck equation.

      The closure problem and methods of reduced description: Chapman-Enskog expansion, Grad's Moment method.

      Non-perturbative techniques in kinetic theory: the method of the slow invariant manifold.

      Overview on Lattice Boltzmann models.

      Monte Carlo simulations of lattice gas models.


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  • Master's thesis (UAQ) (30 credits)

    • ECTS credits 30
    • Semester 2
    • University University of L'Aquila
    • Objectives
      The topic of the thesis can be proposed to the student by the local InterMaths coordinator or by the student him/herself. In any case, the InterMaths executive committee is the responsible to approve the thesis project before its formal start. The taste and expectations of the students are respected whenever possible. The local InterMaths coordinator in the hosting institution is the responsible to provide an academic advisor to the student, although proposals from the students will always be heard in this respect.

      In some cases, after the agreement with the local InterMaths coordinator, the thesis topic can be related to a problem proposed by a private company. In this case, a tutor will be designated by the company as responsible person of the work of the student, especially if he/she is eventually working in the facilities of the company; however, the academic advisor is, in any case, the responsible to ensure the progress, adequacy and scientific quality of the thesis. The necessary agreements between the university and the company will be signed in due time, according to the local rules, in order that academic credits could be legally obtained during an internship, and the students be covered by the insurance against accidents outside the university.

      NOTE: Although the thesis is scheduled for the 4th semester, some preliminary work may be anticipated due to the local rules - such as preliminary local courses in the 3rd semester, ensuring that the student can follow the main courses of the 3rd semester without problems. In this point, the personalised attention to the students has to be intensified, and decisions taken case by case.

    • More information Students work on their Master's Thesis over the 4th semester following their agreement with their thesis advisor.

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