High performance programming paradigms applied to computational fluid dynamic simulations

Most of the time spent to solve a numerical problem is taken by a linear system. There are cases in which the solution is straightforward: lower triangular, upper triangular or diagonal matrices. However, in every day life, solutions of linear systems are far for being trivial: usually partial differential equation (PDE) problems lead to complicated linear systems and it could be even worse when you have to deal with a system of PDEs. This is the situation where HPC is not enough and the starting point of my thesis. Literature and software libraries provide a huge amount of resources and it is very unlikely that you could do better than thousand of scientists and billions of lines of code improved for years. You might improve slightly your code, and not more but if you want to significantly improve your code, you will have to take into account something else: what you need is a new High Performance Point of View.