General-domain FC-based shock-dynamics solver I: Basic elements

TL;DR

This paper introduces a flexible, parallelizable spectral shock-dynamics solver for nonlinear conservation laws in complex domains, utilizing Fourier Continuation and neural network-based shock detection, demonstrated on high-speed flow simulations.

Contribution

It presents a novel multi-patch domain decomposition strategy for shock-capturing spectral methods applicable to arbitrary domains without problem-dependent parameters.

Findings

Effective shock detection and artificial viscosity localization

High-speed flow simulations up to Mach 25 successfully performed

Parallel implementation demonstrates computational efficiency

Abstract

This contribution, Part I in a two-part article series, presents a general-domain version of the FC-SDNN (Fourier Continuation Shock-detecting Neural Network) spectral scheme for the numerical solution of nonlinear conservation laws, which is applicable under arbitrary boundary conditions and in general domains. Like the previous simple-domain contribution (Journal of Computational Physics X 15, (2022)), the present approach relies on the use of the Fourier Continuation method for accurate spectral representation of non-periodic functions in conjunction with smooth artificial viscosity assignments localized in regions detected by means of a Shock-Detecting Neural Network (SDNN). Relying on such techniques, the present Part I paper introduces a novel multi-patch/subpatch artificial viscosity-capable domain decomposition strategy for complex domains with smooth boundaries, and it illustrates the methodology by means of a variety of computational results produced by an associated parallel implementation of the resulting shock-capturing algorithm in a present-day computing cluster. The subsequent Part II contribution then extends the algorithm to enable treatment of obstacles with non-smooth boundaries, it considers questions concerning parallelization and accuracy, and it presents comparisons with physical theory and prior experimental and computational results. The resulting multi-patch FC-SDNN algorithm does not require use of problem-dependent algorithmic parameters or positivity-preserving limiters, and, on account of its use of an overlapping-patch discretization, it is geometrically flexible and efficiently parallelized. A variety of numerical tests for the 2D Euler equations are presented, including the simulation of supersonic and hypersonic flows and shocks past physical obstacles at high speeds, such as Mach 25 re-entry flow speeds.