On the development of accurate and efficient numerical methods for real-world applications of computational fluid dynamics

Numerical simulations of fluid flows are fundamental to advancing our understanding if a wide variety of physical systems. From aerodynamics and atmospheric studies to combustion engines, chemical reactors and medical applications. Both scientific research and industrial modeling demand practical, large-scale simulations that balance computational efficiency with high-accuracy results. In recent years, significant advances in this field are driven by supercomputers with new architectures such as GPUs, and also by the development of modern numerical methods.

The finite element method is proved to be a powerful and versatile computational technique for numerical simulations. Its main strengths lie in its flexibility to handle complex geometries using unstructured meshes, the support of higher-order approximations, and its solid mathematical foundation that ensures convergence. However, there are several difficulties for achieving a stable solution. Solving nonsymmetric and nonlinear advection terms causes instabilities. Also, strong gradients may produce oscillations. To address these problems, additional stabilization techniques are required.

The thesis develops a stabilization strategy well-suited for high-order elements, designed to effectively eliminate numerical instabilities while introducing minimal numerical dissipation to achieve a high accuracy solution. We focus on projection-based methods, which allows us to stabilize using the unresolved components of the solution. Based on recent literature, our stencil features a combination of a high- and low-order stabilization ruled by a smoothness sensor. High-order stabilization is applied in smooth regions to maintain physical profiles, while a low-order viscous stabilization is activated only near strong gradients to suppress non-physical oscillations.

The proposed numerical framework is applied to several problems. In combustion simulations, the approach handles tabulated chemistry and delivers a proper representation of the flame-front. For compressible flows, a smoothness sensor based on the Mach number has been developed, effectively solving complex flow features such as shock waves. For incompressible flows, the high-order stabilization term avoids instabilities, while at the same time allowing the accurate resolution of the most relevant scales of the fluid movement. The proposed framework achieves convergence and provides a robust and scalable tool for applied computational fluid dynamics.