Enhancing Low-Order Discontinuous Galerkin Methods with Neural Ordinary Differential Equations for Compressible Navier--Stokes Equations

However, it is still challenging to solve practical problems such as blood flows, atmospheric and ocean currents, wildfires, and wind turbines because of their multiscale nature. Resolving all scales is computationally infeasible. As a result, physical modeling is typically carried out on a coarse grid using the appropriate subgrid-scale (SGS) models. For example, large eddy simulation (LES) resolves large-scale turbulent motion on a grid, which carries most of the flow energy, while modeling the small scales that have relatively little influence on the mean flow [8]. The goal of SGS models is to capture the effect of the small-scale structures that cannot be resolved in the grid on the resolved scales and to guarantee numerical stability [9]. The static Smagorinsky model [10] and dynamic Smagorinsky model [11], which predict the dissipation of SGS energy, are the most widely used SGS models for turbulence. In high-order DG methods, both Collis [12] and Sengupta et al. [13] successfully used the static Smagorinksy model and dynamic Smagorinsky model, respectively. However, these Smagorinsky models perform poorly for certain flows [14, 15] because they are based on the assumption that eddy viscosity is always purely dissipative and thus are unable to account for energy flow from small scales to large scales (backscatter) [16, 11]. Alternatively, numerical dissipation can be used for modeling unresolved scales as an implicit SGS model.