The solution of very large non-linear algebraic systems

The manuscript discusses the feasibility and the methods for solving systems of non-linear algebraic equations, the main numerical subjects being convergence tests, stop criteria, and expedients for large and sparse systems. After a detailed discussion on the features of large non-linear systems, the paper focuses on the numerical simulation of complex combustion devices and on the formation of macro- and micro-pollutants. This quantification is not possible by simply introducing a detailed kinetic scheme into a fluid dynamics (CFD) code, especially when considering turbulent flows. Actually, the resulting problem would reach a so huge dimension that is still in orders of magnitude larger than the feasible one (by means of modern computing devices). To overcome this obstacle it is possible to implement a separate and dedicated kinetic post-processor (KPP) that, starting from the CFD output data, allows simulating numerically the turbulent reactive systems by means of a detailed kinetic scheme. The resulting numerical problem consists of a very large, non-linear algebraic system comprising a few millions of unknowns and equations. The manuscript describes the KPP organization and structure as well as the numerical challenges and difficulties that one has to overcome to get the final numerical solution.