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

TL;DR

This paper introduces a spectral shock-dynamics solver combining Fourier Continuation and neural-network-based localized artificial viscosity to accurately and efficiently handle discontinuities in nonlinear conservation laws.

Contribution

It presents a novel FC-SDNN method that achieves spectral accuracy with low dissipation and no need for problem-specific parameters, improving shock capturing in non-periodic domains.

Findings

Spectral accuracy achieved near discontinuities.

Lower dissipation compared to existing methods.

Effective shock detection with neural networks.

Abstract

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.