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

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

URL: http://arxiv.org/abs/2310.18897v2
Date: Tue, 30 Jan 2024 18:35:26 GMT
Title: Enhancing Low-Order Discontinuous Galerkin Methods with Neural Ordinary Differential Equations for Compressible Navier--Stokes Equations
Authors: Shinhoo Kang, Emil M. Constantinescu
Abstract summary: It is common to run a low-fidelity model with a subgrid-scale model to reduce the computational cost. We propose a novel method for learning the subgrid-scale model effects when simulating partial differential equations augmented by neural ordinary differential operators. Our approach learns the missing scales of the low-order DG solver at a continuous level and hence improves the accuracy of the low-order DG approximations.
Score: 0.18648070031379424
License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
Abstract: The growing computing power over the years has enabled simulations to become more complex and accurate. While immensely valuable for scientific discovery and problem-solving, however, high-fidelity simulations come with significant computational demands. As a result, it is common to run a low-fidelity model with a subgrid-scale model to reduce the computational cost, but selecting the appropriate subgrid-scale models and tuning them are challenging. We propose a novel method for learning the subgrid-scale model effects when simulating partial differential equations augmented by neural ordinary differential operators in the context of discontinuous Galerkin (DG) spatial discretization. Our approach learns the missing scales of the low-order DG solver at a continuous level and hence improves the accuracy of the low-order DG approximations as well as accelerates the filtered high-order DG simulations with a certain degree of precision. We demonstrate the performance of our approach through multidimensional Taylor-Green vortex examples at different Reynolds numbers and times, which cover laminar, transitional, and turbulent regimes. The proposed method not only reconstructs the subgrid-scale from the low-order (1st-order) approximation but also speeds up the filtered high-order DG (6th-order) simulation by two orders of magnitude.