Università Roma Tre

Fundamentals of the approximation of Ordinary Differential Equations systems. One-step methods of forward/backward Euler type and their convergence.
Convergence of numerical methods for time-dependent Partial Differential Equations, Lax-Richtmyer theorem.
Advection equation: analytical aspects. Representation formula via characteristics. Monotone methods for the advection equation: Upwind, Lax-Friedrichs.
Hyperbolic scalar conservation laws in one dimension: analytical aspects, basics on entropic solutions, Rankine-Hougoniot condition. Fundamentals on convergence theory for finite-volume approximations. Monotone finite-volume methods: Godunov, Lax-Friedrichs, Rusanov.
Linear and nonlinear hyperbolic systems: analytical aspects, characteristic decomposition. Central schemes for hyperbolic systems.
The Shallow Water Equations in one and two space dimensions. Central scheme approximation, basics on boundary conditions.
The heat equation: analytical aspects, domain of dependence, regularity. Explicit and implicit approximation in one and two space dimensions via centered second differences and Euler time discretization.
Modelling of incompressible fluids: the Navier-Stokes Equations. Approximate formulations (Euler, Stokes), derivation of the Shallow Water Equations. Finite difference numerical methods based on the Vorticity-Streamfunction formulation.

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press

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