Patrick Joly (INRIA Rocquencourt. France)

Variational methods for time-dependent wave propagation problems

Abstract

For the numerical simulation of a number of time-dependent wave propagation problems, one has to face difficulties linked to "coupling problems" in various senses: coupling of different physical models such as in fluid-solid interaction problems, coupling of different computational grids or methods such as for mesh refinement purpose, coupling between volume and surfaces for diffraction problems,...

In this article, we shall show how the systematic use of appropriate variational formulations of the evolution problem, especially of mixed type, lead to efficient and robust numerical methods in the sense that:

• the numerical schemes remain essentially explicit in time except maybe for unknowns associated to the "coupling surfaces"
• energy conservation (or decay) properties of the continuous model are respected at the discrete level
• the CFL stability condition of the global scheme is not affected by the coupling algorithm

This will be illustrated through the following applications:

• the fictitious domain method for wave diffraction problems
• conservative space-time mesh refinement methods for Maxwell's equations
• fluid solid interaction problems occuring in geophysics, medical applications, musical acoustics,...