General Domain FC-Based Shock Dynamics Solver

Citation

Leibovici, Daniel Victor (2024) General Domain FC-Based Shock Dynamics Solver. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/bd5r-4q30. https://resolver.caltech.edu/CaltechTHESIS:03152024-221312028

Abstract

This thesis presents a novel FC-SDNN (Fourier Continuation Shock-detecting Neural Network) spectral scheme for the numerical solution of nonlinear conservation laws in general domains and under arbitrary boundary conditions, without the limiting CFL constraints inherent in other spectral schemes for general domains. The approach relies on the use of the Fourier Continuation (FC) method for 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). 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 contour levels. 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, especially in the vicinity of contact discontinuities. The approach, which does not require the use of otherwise ubiquitous positivity-preserving limiters, enjoys a great geometrical flexibility on the basis of an overlapping-patch discretization. This allows its application for the simulation of supersonic and hypersonic flows and shocks, including Euler simulations at significantly higher speeds than previously achieved, such as e.g. Mach 25 re-entry flow speeds, impinging upon complex physical obstacles. This multi-domain approach is suitable for efficient parallelization on large computer clusters, and the MPI implementation proposed in this thesis enjoys high parallel scalability and in particular perfect weak scaling, as demonstrated by simulations on general complex domains. The character of the proposed algorithm is demonstrated through a variety of numerical tests for the linear advection, Burgers and Euler equations in one and two-dimensional non-periodic spatial domains, with results in accordance with physical theory and prior experimental and computational results up to and including both supersonic and hypersonic regimes.

Thesis Files