PhD Graduate

PhD program:: XXXIV

supervisor: E. Carlini
advisor: E. Carlini

Thesis title: Semi-Lagrangian schemes for parabolic equations: second order accuracy and boundary conditions

This thesis deals with second order parabolic differential equations and some semi-Lagrangian methods to approximate their solutions. We start with a brief survey of the main theoretical results concerning linear and nonlinear parabolic equations, recalling some existence and uniqueness to the Cauchy problem on the entire space and to the Initial-Boundary value problem with Dirichlet and Neumann type boundary conditions. In the following three chapters, we present our approach to the numerical solution to three different problems. First, we introduce a semi-Lagrangian method for advection-diffusion-reaction systems of equations on bounded domains, with Dirichlet boundary conditions. Afterwards, we present a semi-Lagrangian technique for approximating the solution to Hamilton-Jacobi-Bellman equations on bounded domain, with Neumann-type boundary conditions. Finally, we present a Lagrange-Galerkin approximation of the Fokker-Planck equation, and we show how to apply such a method to obtain a second-order accurate solution to Mean Field Games. Every method is accompanied with numerical simulations.

Research products

11573/1667216 - 2023 - A semi-Lagrangian scheme for Hamilton–Jacobi–Bellman equations with oblique derivatives boundary conditions
Calzola, E.; Carlini, E.; Dupuis, X.; Silva, F. J. - 01a Articolo in rivista
paper: NUMERISCHE MATHEMATIK (Springer Verlag Germany:Tiergartenstrasse 17, D 69121 Heidelberg Germany:011 49 6221 3450, EMAIL: g.braun@springer.de, INTERNET: http://www.springer.de, Fax: 011 49 6221 345229) pp. 49-84 - issn: 0029-599X - wos: WOS:000903607300001 (2) - scopus: 2-s2.0-85144679285 (2)

11573/1552635 - 2021 - Second Order Fully Semi-Lagrangian Discretizations of Advection-Diffusion-Reaction Systems
Bonaventura, Luca; Calzola, Elisa; Carlini, Elisabetta; Ferretti, Roberto - 01a Articolo in rivista
paper: JOURNAL OF SCIENTIFIC COMPUTING (Dordrecht : Kluwer) pp. - - issn: 1573-7691 - wos: (0) - scopus: 2-s2.0-85107204506 (8)

11573/1392905 - 2020 - A Fully Semi-Lagrangian Method for the Navier–Stokes Equations in Primitive Variables
Bonaventura, L.; Calzola, E.; Carlini, E.; Ferretti, R. - 02a Capitolo o Articolo
book: Lecture Notes in Computational Science and Engineering - (978-3-030-30704-2; 978-3-030-30705-9)

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