Prof. Raffaele D'Ambrosio 
DISIM - University of L'Aquila
Coppito 1, Room 1015
e-mail: raffaele.dambrosio@univaq.it

Contact the teacher

Tuesday, 16.30-18.30 (HPC Laboratory, Coppito 1)
Thursday, 9.30-13.30 (HPC Laboratory, Coppito 1)

(Numerical approximation of ODEs)
J. D. Lambert, Numerical methods for ordinary differential systems: the initial value problem, John Wiley (1991).

(Finite difference schemes) 
E. Isaacson, H.Keller, Analysis of numerical methods, Dover Publications (1994).

(Finite elements schemes) 
A. Quarteroni, A. Valli, Numerical Approximation of Partial Differential Equations, Springer (1994).

T. Rauber, G. Runger, Parallel Programming for Multicore and Cluster Systems, Springer (2013).

Mid-term: written test.

Final exam: for those who passed the mid-term test, the final exam is an oral test on the remaining part of the course, plus an individual project assigned during the course. For those who did not pass the mid-term test, the final exam is a written test on all the topics of the course, plus an individual project assigned during the course.  

Introduction to ODEs.
Hadamard well-posedness of a differential problem. Picard-Lindelof theorem (no proof). Picard iterations.

Discretization of ODEs.
Discretization of the domain. Discretization of the problem: difference equations. Existence and uniqueness of solutions. Linearly independent solutions. General solution of homogeneous linear difference equations: cases of distinct, multiple and complex conjugate roots of the characteristic polynomial. General solution of the inhomogeneous problem. 

One-step methods for ODEs.
Euler method and its analysis. Stability, consistency, convergence. Lax equivalence theorem for one-step methods (no proof). 

Finite difference methods for ODEs.
Linear multistep methods. Trapezoidal method. Adams-Bashforth and Adams-Moulton methods. Residual operators. Local truncation error. Consistency. Linear difference operator. Order conditions. Characteristic polynomials. Zero-stability. Root condition (no proof). Analysis of families of methods. Implicit methods. Convergence of fixed point iterations. Linear stability analysis. Dahlquist test equation. Absolute stability.

Polynomial interpolation.
The problem of data and function approximation. Vandermonde method. Existence and uniqueness of the interpolation polynomial on distinct nodes. Lagrange interpolation. Conditioning analysis. Chebychev points. Runge phenomenon. 

Numerical quadrature.
Interpolatory quadrature rules. Newton-Cotes quadrature rule. Trapezoidal rule and accuracy analysis. Cavalieri-Simpson rule and accuracy analysis.

Numerical differentiation. 
Finite differences and analysis of their accuracy. Applications to ODEs. 

Numerical integration of elliptic problems.
Laplace operator. Domain discretization. Discretized Poisson problem. 5-point stencil. Error estimates. Consistency and convergence. Liebmann method (not mandatory). 

Numerical integration of parabolic problems. 
Heat equation. Discretized heat equation. Stability and accuracy analysis. Crank-Nicolson method; accuracy and stability analysis. $\vartheta$-method; accuracy and stability analysis. 

Numerical integration of hyperbolic problems. 
Wave equation. Region of influence. Triangle and interval of dependence. Discretized domain. Discretized wave equation. Numerical domain of dependence. CFL condition. Accuracy analysis.