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Abstract : 

The numerical simulation of fluid flows has become an essential part of the virtual prototyping in the aerospace industry. It is made possible through Computational Fluid Dynamics (CFD) tools. These tools rely on algebraic relations between the values stored at mesh points called numerical schemes, which must fulfill specific criteria to lead to reliable simulations. Namely, numerical schemes must be accurate and stable (among other conditions). Traditionally, accuracy and stability conditions have been derived for problems governed by linear equations on regular meshes. However, fluid motion equations are non-linear, and meshes used in industrial CFD can be irregular (especially when dealing with complex geometries). This thesis proposes a novel framework to analyze the accuracy and stability of numerical schemes for non-linear systems of equations and irregular meshes. This framework also establishes conditions for optimizing numerical schemes locally in time and space. It is named Local Transfer function Analysis (LTA). Under the light of LTA, each mesh element acts as an impedance block that resists to the solution propagation over time and space. The optimization of numerical schemes becomes an impedance-matching problem. It is solved by minimizing the value of an objective function. The LTA objective function measures the distance between the dynamics predicted by the numerical scheme and some reference dynamics. Using reference data in this optimization process turns LTA into a tool that generates accurate and stable data-driven numerical schemes. More recently, Machine Learning (ML) has emerged as a field dedicated to extracting knowledge from data. This thesis then proposes employing ML architectures to find optimal values for the parameters of a numerical scheme in the sense of the LTA objective function. The method is applied first on 1D problems modeled by the Convection/Burgers’ equations and finally extended to 2D applications governed by the Convection/Euler equations.