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General Domain FC-Based Shock Dynamics Solver


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


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.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Machine learning, Neural networks, Shock dynamics, Spectral methods, Artificial viscosity, Fourier continuation, Non-periodic domain, Multi-patch domain decomposition, High Performance Computing, MPI
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Applied And Computational Mathematics
Awards:The W.P. Carey and Co. Prize in Applied Mathematics, 2024.
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Bruno, Oscar P.
Thesis Committee:
  • Bruno, Oscar P.
  • Meiron, Daniel I. (chair)
  • Owhadi, Houman
  • Pullin, Dale Ian
Defense Date:22 January 2024
Non-Caltech Author Email:dleibovi (AT)
Funding AgencyGrant Number
Air Force Office of Scientific Research (AFOSR)FA9550-21-1-0373
Vannevar Bush Faculty Fellowship (VBFF)N00014-16-1-2808
Record Number:CaltechTHESIS:03152024-221312028
Persistent URL:
Related URLs:
URLURL TypeDescription adapted for Chapters 1, 2 and 3
Leibovici, Daniel Victor0009-0007-8267-4250
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:16329
Deposited By: Daniel Leibovici
Deposited On:27 Mar 2024 21:44
Last Modified:17 Jun 2024 18:41

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