Waves in fluid flows and related phenomena take several forms but usually involve the complex interaction of nonlinear low-order terms with high-order terms in the PDEs modelling the flow. An example of such interactions occurs with shock waves in viscous flows, where the first-order advective term tends to strengthen the waves while the second-order viscous dissipation term smoothen those one. In a somewhat related way, similar mechanisms take place in reaction-diffusion systems where zeroth-order reaction terms balance second-order terms to create travelling waves.
The numerical simulation of travelling and standing waves has been and is still a challenging task. Several issues have to be addressed, e.g. the stability, conservation, positivity, etc, of the numerical solutions. The talks will review some of those issues for our method of choice, namely the finite element method. More precisely, I will talk about:
- The issue of conservation in finite element methods, in particular for the computation of shock waves in viscous and inviscid flows.
- The connection between the computation of travelling waves in reaction-diffusion systems and viscous shock waves in flows, in terms of the computational requirements and stability.
- Numerical difficulties in computing multi-phase flows, such as air flows charged with dispersed particles. An Eulerian model of the particle phase is a prototype of a reaction-advection system, where the ``reaction'' term is in fact a nonlinear momentum transfer term coupling the particle flow with the air flow.