A Data-Driven Reconstruction for Unstructured Finite Volume Method on Multi-Dimensional Compressible Flows

Abstract

This paper introduces a novel data-driven shock-capturing for unstructured finite volume method (FVM) on multi-dimensional compressible flows, aiming at enhancing accuracy in smooth region while ensuring robustness in discontinuous region, by machine-learning models. We achieve this objective via a two-step process based on a tree model and fully-connected neural network (FCNN) models. In the first step, a tree model is employed as a troubled-cell indicator to divide the computational domain into discontinuous and smooth regions. In the second step, we employ FCNN models to reconstruct solution variables at a cell interface, which are used to obtain numerical fluxes. Notably, two FCNN models are used for discontinuous and smooth regions, respectively, with the former designed for robust capturing of discontinuities, and the latter intended to enhance accuracy in smooth region. To train such models, we generate two types of datasets using various analytic functions to mimic discontinuous and smooth FVM solutions. In addition, we define specific input features to provide the information on solution distribution and coordinates, enabling the extension onto unstructured meshes. Finally, we systematically validate the proposed method through a comprehensive set of numerical experiments, ranging from scalar conservation laws, Euler equations to Navier-Stokes equations. Computational experiments demonstrate enhanced performance in terms of robustness and accuracy, highlighting the capability of data-driven reconstruction in complex flow simulations.