The acquired skills allow to continue studying numerical analysis at advanced level, and providemathematical tools useful in various applications fields. Infact differential models real phenomena which arise in physics, engineering, medicine, biology, economics.
The RK methods: consistency and convergence. Stiff Problems. Stability analysis: A-stability, AN-stability, algebraic stability and BN-stability. Implementation aspects of RK-methods.
Linear Multistep (LM) methods: consistency, zero-stability and convergence. Predictor-Corrector methods. Stability analysis. Implementation aspects of LM-methods.
In the second part the course treats Boundary Value Problems for Ordinary Differential Equations. The shooting method. Elementary finite difference methods. Collocation methods. Weak formulation. Lax- Milgram Theorem. Sobolev space. Galerkin finite-element method. Strang’s lemma.
In the last part an overview of the numerical methods for Partial Differential Equations (PDEs). The course includes laboratory activities with MATLAB and the numerical solution of a case study, for instance the Black-Scholes model, which has many applications in mathematical finance, structured models whcih arise in ecology and epidemiology.
P. Deufhard, F. Bornemann Scientific Computing with Ordinary Differential Equations. Springer , New York 2002.
E. Hairer, S.P. Norsett, G. Wanner: Solving Ordinary Differential Equations I Sringer, Berlin 1987.
E. Hairer, G. Wanner: Solving Ordinary Differential Equations II Springer, Berlin 1996
E. Hairer, C. Lubich, G. Wanner: Geometric Numerical Integration Springer, Berlin 2001.
J.D. Lambert: Numerical methods for ordinary differential equations. Wiley 1991.
A. Quarteroni Modellistica Numerica per problemi differenziali. Springer, New York 2000.