# METODI NUMERICI PER EQUAZIONI ALLE DERIVATE PARZIALI

Academic Year 2019/2020 - 1° Year
Teaching Staff: Sebastiano BOSCARINO
Credit Value: 6
Scientific field: MAT/08 - Numerical analysis
Taught classes: 35 hours
Exercise: 12 hours
Term / Semester:

## Learning Objectives

The object of the couse Numerical methods for partial differential equations" is to introduce the numericla methods for the numericla solution of partial differential equations, in particular for hyperbolic, elliptic and parabolic equations.

## Course Structure

Heat equation in one and multiple dimension. Elliptic equation, Poisson equation. Hyperbolic equations. Finite difference methods.

## Detailed Course Content

Poisson equation, heat equation and waves equation.

Stability Analysis: Von Neuman method.

Forward Euler method, Crank-Nicholson method.

Consistency, convergence and stability for finite difference methods for initial value problems. Lax Theorem.

Heat equation in multiple dimensions. Alternate Direction Implicit (ADI) method.

Elliptic Equation. Finite difference method for the Poisson equation on cartesian grid.

Level set methods and ghost point for arbitrary geometry. Multigrid method.

Hyperbolic equations. Characteristic method. Finite difference methods: upwind, Lax-Friedrichs e Lax-Wendroff. Consistency and stability. Modified equation. Dispersion and dissipation. Burgers equation.

## Textbook Information

--Randall Le Veque, Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM 2007.

--John Strickwerda, Finite Difference Schemes and Partial Differential Equations Paperback – September 30, 2007

--Robert D. Richtmyer, K. W. Morton, Difference methods for initial-value problems, Interscience Publishers, 1967 - 405 pages

--K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations, An Introduction, University of Oxford, UK, Second Edition