Development of high-order adaptive algorithms for simulating magnetic reconnection processes with uncertainty quantification

With the rapid advancement of technology and scientific knowledge, we witness every day tremendous progress in various fields of research and development leading to a better understanding of the world around us and opening up new avenues for exploration. Within the process, a growing reliance has emerged on cutting-edge numerical methods, as the most cost-effective tool, to conduct accurate simulations of complex problems. \\
The present work aims to contribute to this progress by developing novel and advanced numerical tools that can enhance the accuracy and reliability of such simulations. Practically, the research focuses on three main axes: modern high-order methods, automatic adaptive mesh refinement, and uncertainty quantification techniques. \\

The first research axis addresses the development and implementation of the first high-order Flux Reconstruction (FR) schemes specifically targeted towards high-speed flows on both straight and curved-edged 2D and 3D simplex elements within the open-source COOLFluiD (Computational Object-Oriented Libraries for Fluid Dynamics) platform. The proposed FR solver provides more accurate detection of complex flow features over relatively coarser meshes when compared to their low-order peers (particularly in presence of shock waves), is fully implicit and able to simulate compressible flow problems governed by the Euler or Navier-Stokes equations on triangular and tetrahedral meshes. In order to simulate cases with shocks, an improved and more case-independent shock-capturing scheme, previously developed for quadrilateral elements, was extended to tackle supersonic and hypersonic simulations. Extensive verification of the resulting FR solver (up to $7^{th}$ order, a.k.a P6, for the solution polynomial and $3^{rd}$ order a.k.a Q2 for the geometrical representation) has been performed on benchmark test cases with flow speeds ranging from subsonic to hypersonic, up to Mach 9.6~. The obtained results have been favorably compared to those available in the literature.\\

The second research axis focuses on the development and implementation of the first r-adaptive mesh refinement (r-AMR) algorithm for the high-order FR solver. The r-refinement consists of nodal re-positioning while keeping the number of mesh nodes and their connectivity frozen. The developed algorithm is based on physics-driven spring-analogies, where the mesh can be seen as a network of fictitious springs. While AMR increases the local mesh density, the high-order FR method potentially provides more accurate detection of complex flow features over a relatively coarser mesh when compared to low-order methods. This combination produced some very promising results of r-AMR applied to benchmark high-order steady-state high-speed flow simulations.\\

The final research axis addresses the coupling of an Uncertainty Quantification (UQ) toolkit, MultilevelEstimators, and COOLFLuiD. A Multi-Level Monte Carlo (MLMC) method, used for UQ, constructs levels using isotropic p-refinement instead of isotropic h-refinement, therefore requiring, ideally, just one initial ”coarse” mesh which will undergo polynomial upgrade for obtaining finer levels. Additionally, the range of applicability of the MLMC within the FR framework -- with and without r-AMR -- is extended to construct, in a general manner, low- to high-fidelity models that produce promising high-fidelity and high-accuracy results. \\