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.