Numerical Methods for Partial Differential Equations Seminar

Building 2, 449

Speaker: Ahmad Peyvan (Brown University)Title: Modeling High-Speed Flows: Approaches for Hypersonic Flows with Strong Shocks, Real Chemistry, and Interpretable Neural Operators for Riemann ProblemsAbstract:In this talk, we first explore the development of a high-order discontinuous Galerkin Spectral Element Method (DGSEM) for solving hypersonic flows. Various high-order methods such as Spectral Difference (SD), Flux Reconstruction (FR), and others are compared to select the most suitable method for simulating strong shock waves typical in hypersonic flows. Through simulations of the three-species Sod problem with simplified chemistry, we find that the entropy-stable DGSEM scheme exhibits superior stability, minimal numerical oscillations, and requires less computational effort for resolving reactive flow regimes with strong shock waves. Consequently, we extend this scheme to hypersonic Euler equations by deriving new entropy conservative fluxes for a five-species gas model.Secondly, we focus on interpretable neural operators capable of learning Riemann problems, particularly those involving strong shock waves. We utilize neural operators for compressible flows with extreme pressure jumps to address the challenge of accurately simulating high-speed flows with various discontinuities. We employ DeepONet, trained in a two-stage process, which significantly enhances accuracy, efficiency, and robustness compared to the vanilla version. This modified approach allows for a physical interpretation of results and accurately reflects flow features. We also compare results with another neural operator based on U-Net, which proves accurate for Riemann problems, especially with large pressure ratios, albeit computationally more intensive. Our study showcases the potential of simple neural network architectures, when properly trained, in achieving precise solutions for real-time forecasting of Riemann problems.