The current manuscript is a final report on the activities carried out under the Project LDRD-CIS #226834. In scientific terms, the work reported in this manuscript is a continuation of the efforts started with Project LDRD-express #223796 with final report of activities SAND2021-11481, see [83]. In this section we briefly explain what pre-existing developments motivated the current body of work and provide an overview of the activities developed with the funds provided. The overarching goal of the current project LDRD-CIS #226834 and the previous project LDRD-express #223796 is the development of numerical methods with mathematically guaranteed properties in order to solve the Euler-Maxwell system of plasma physics and generalizations thereof. Even though Project #223796 laid out general foundations of space and time discretization of Euler-Maxwell system, overall, it was focused on the development of numerical schemes for purely electrostatic fluid-plasma models. In particular, the project developed a family of schemes with mathematically guaranteed robustness in order to solve the Euler-Poisson model. This model is an asymptotic limit where only electrostatic response of the plasma is considered. Its primary feature is the presence of a non-local force, the electrostatic force, which introduces effects with infinite speed propagation into the problem. Even though instantaneous propagation of perturbations may be considered nonphysical, there are plenty of physical regimes of technical interest where such an approximation is perfectly valid.

Physics-informed neural network architectures have emerged as a powerful tool for developing flexible PDE solvers that easily assimilate data. When applied to problems in shock physics however, these approaches face challenges related to the collocation-based PDE discretization underpinning them. By instead adopting a least squares space-time control volume scheme, we obtain a scheme which more naturally handles: regularity requirements, imposition of boundary conditions, entropy compatibility, and conservation, substantially reducing requisite hyperparameters in the process. Additionally, connections to classical finite volume methods allows application of inductive biases toward entropy solutions and total variation diminishing properties. For inverse problems in shock hydrodynamics, we propose inductive biases for discovering thermodynamically consistent equations of state that guarantee hyperbolicity. This framework therefore provides a means of discovering continuum shock models from molecular simulations of rarefied gases and metals. The output of the learning process provides a data-driven equation of state which may be incorporated into traditional shock hydrodynamics codes.

This report summarizes the findings and outcomes of the LDRD-express project with title “Fluid models of charged species transport: numerical methods with mathematically guaranteed properties”. The primary motivation of this project was the computational/mathematical exploration of the ideas advanced aiming to improve the state-of-the-art on numerical methods for the one-fluid Euler-Poisson models and gain some understanding on the Euler-Maxwell model. Euler-Poisson and Euler-Maxwell, by themselves are not the most technically relevant PDE plasma-models. However, both of them are elementary building blocks of PDE-models used in actual technical applications and include most (if not all) of their mathematical difficulties. Outside the classical ideal MHD models, rigorous mathematical and numerical understanding of one-fluid models is still a quite undeveloped research area, and the treatment/understanding of boundary conditions is minimal (borderline non-existent) at this point in time. This report focuses primarily on bulk-behaviour of Euler-Poisson’s model, touching boundary conditions only tangentially.

We present a fully discrete approximation technique for the compressible Navier–Stokes equations that is second-order accurate in time and space, semi-implicit, and guaranteed to be invariant domain preserving. The restriction on the time step is the standard hyperbolic CFL condition, i.e. τ≲O(h)∕V where V is some reference velocity scale and h the typical meshsize.

Arndt, Daniel A.; Bangerth, Wolfgang B.; Blais, Bruno B.; Clevenger, Thomas C.; Fehling, Marc F.; Heister, Timo H.; Heltai, Luca H.; Maier, Matthias M.; Munch, Peter M.; Pelteret, Jean-Paul P.; Rastak, Reza R.; Tomas, Ignacio T.; Turcksin, Bruno T.; Wang, Zhuoran W.; Wells, David W.