Construction of high order finite volume and finite difference shock-capturing schemes for conservation laws. Topic 1 - illustrating how to construct shock capturing schemes for conservation laws. We focus on semi-discrete schemes based on the method of lines.

In this talk I report some results on the approximation of fluid-structure interaction problems using non matching grids. Our formulation originates from the Immersed Boundary Method and then moved toward the Fictitious Domain approach. The advantages of this formulation is that it avoids the difficulties related with mesh generation and it allows the treatment of fluid and solid in their natural Eulerian and Lagrangian framework. I present the well-posedness of our formulation at the continuous level in a simplified setting. Moreover, I shall discuss various time discretizations that provide unconditionally stable schemes and some computational details.

Eigenvalue problems associated with partial differential equations are key ingredients for the modeling and simulation of a variety of real world applications, ranging from fluid-dynamics, structural mechanics, acoustics, to electromagnetism and medical problems. We review some properties related to the approximation of eigenvalue problems. Starting from matrix algebraic problems, we present a series of examples and counterexamples showing that even extremely simple situations can produce unexpected results.