Master thesis and internship[BR]- Master's thesis : Implementation of a monolithic convex limiting approach for the stabilization of high speed turbulent simulations using a Discontinuous Galerkin method[BR]- Integration internship

Abstract

This work presents the implementation and evaluation of a positivity-preserving hybrid method that combines the entropy-stable Discontinuous Galerkin spectral element method (ESDGSEM) with first-order finite volume (FV) schemes for simulating compressible flows. Maintaining stability in the presence of strong discontinuities and under-resolved regions is a major challenge for high-order methods like ESDGSEM. The Monolithic Convex Limiting (MCL) method has been implemented in ForDGe, providing a subcell-wise limiting technique with two variations: local bounds, which prevent the creation of new extrema, and global bounds, which avoid density and pressure crossing zero. Several existing positivity-preserving approaches, such as APriori, APosteriori, and APosteriori Subcell, have already been implemented. Each method computes a blending coefficient either element-wise or subcell-wise to describe the proportion of ESDGSEM and FV. An analysis of the methods’ performance, in terms of stability, accuracy, and computa- tional cost, is presented across a series of benchmark test cases, including the Kelvin-Helmholtz instability, strong vortex-shock interaction, Daru-Tenaud shock tube, and three-dimensional Taylor-Green vortex. Results show that APriori and MCL provide robust and stable simu- lations, although with increased dissipation. In contrast, APosteriori and APosteriori Subcell better preserve fine-scale features, remaining close to the ESDGSEM solution, though occasional stability issues are observed with APosteriori. Notably, the computational cost is significantly impacted by whether the blending coefficient is computed element-wise or subcell-wise, with the subcell-wise approach resulting in the highest processing expense. Overall, the choice of method involves balancing computational efficiency, robustness, and accuracy, depending on the specific demands of the simulation.