Appunti di Numerical methods in engineering sciences

The course is divided in two parts, devoted essentially to the numerical approximation of boundary value problems for Partial Differential Equations (Pde's), and of initial value problems for Ordinary Differential Equations (Ode's). The basic common and necessary instruments to deal with both classes of problems are also developed. NUMERICAL SOLUTION OF BOUNDARY VALUE PROBLEMS FOR PARTIAL DIFFERENTIAL EQUATIONS (PDE): * Finite Difference method on a model problem in 1D. Consistency and Stability - Lax's Theorem for convergence of a numerical scheme. *Finite Element method on a model problem in 1D: Variational formulation, continuous piecewise linear finite element approximation, stability and convergence; construction of the final system and comparison with finite differences. *Finite Element method on a model problem in 2D: Variational Formulation, Continuous piecewise linear finite element discretization on triangular meshes; Explicit computation of the elementary stiffness matrix and right-hand side; Assembling and solution of the final system. *Various examples of boundary value problems in 2D. NUMERICAL SOLUTION OF INITIAL VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS (ODE): *One-step methods: Euler backward and forward, Crank-Nicolson, Heun; Stability and A-stability, consistency, convergence and order of convergence. *Multistep Methods: general structure, consistency and stability conditions; Explicit and Implicit Adams methods. *Runge-Kutta methods: consistency and stability conditions; example of construction of an explicit RK-method (Hints on predictor-corrector methods). *Systems of Ordinary Differential Equations: stiff problems. COMMON TOOLS: *Solution of linear systems of equations: direct and iterative methods. *Nonlinear equations: bisection and Newton's methods. Convergence, order of convergence, stopping criteria. Nonlinear systems of equations: Newton's method and variants. *Lagrange interpolation: interpolation error, piecewise Lagrange interpolation, order of approximation in various norms. *Least squares method for data fitting: linear regression and various examples. *Interpolatory quadrature formulas in 1D: midpoint, trapezoidal, Simpson and error analysis. Gaussian formulae.Extension to dimension 2 on rectangular domains. Quadrature formulas on triangular domains: barycenter, vertex, and midpoint of the edges.