FC-based shock-dynamics solver with neural-network localized artificial-viscosity assignment

• Published in: arXiv.org 2021-11
• Main Authors: Bruno, Oscar P, Hesthaven, Jan S, Leibovici, Daniel V
• Format: Article
• Language: English
• Subjects: Algorithms; Boundary conditions; Conservation laws; Discontinuity; Domains; Euler-Lagrange equation; Neural networks; Oscillations; Periodic functions; Shock capturing; Spectra; Viscosity
• ISSN: 2331-8422

This paper presents a spectral scheme for the numerical solution of nonlinear conservation laws in non-periodic domains under arbitrary boundary conditions. The approach relies on the use of the Fourier Continuation (FC) method for spectral representation of non-periodic functions in conjunction with smooth localized artificial viscosity assignments produced by means of a Shock-Detecting Neural Network (SDNN). Like previous shock capturing schemes and artificial viscosity techniques, the combined FC-SDNN strategy effectively controls spurious oscillations in the proximity of discontinuities. Thanks to its use of a localized but smooth artificial viscosity term, whose support is restricted to a vicinity of flow-discontinuity points, the algorithm enjoys spectral accuracy and low dissipation away from flow discontinuities, and, in such regions, it produces smooth numerical solutions -- as evidenced by an essential absence of spurious oscillations in level set lines. The FC-SDNN viscosity assignment, which does not require use of problem-dependent algorithmic parameters, induces a significantly lower overall dissipation than other methods, including the Fourier-spectral versions of the previous entropy viscosity method. The character of the proposed algorithm is illustrated with a variety of numerical results for the linear advection, Burgers and Euler equations in one and two-dimensional non-periodic spatial domains.