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1. Invariant-domain preserving and locally mass conservative approximation of the Lagrangian hydrodynamics equations

   https://doi.org/10.1016/j.cma.2025.117927  

   Guermond, Jean-Luc; Popov, Bojan; Saavedra, Laura; Sheridan, Madison (June 2025, Computer methods in applied mechanics and engineering)

   In this paper we construct an explicit approximation for the Lagrangian hydrodynamics equations equipped with an arbitrary equation of state. The approximation of the state variable is done with piecewise constant finite elements and the approximation of the mesh motion is done with higher-order continuous finite elements. The method is invariant-domain preserving and locally mass conservative. The purpose of this method is to be used in combination with higher-order methods to make them invariant domain preserving as well. 

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2. Second-Order Invariant Domain Preserving ALE Approximation of Euler Equations

   https://doi.org/10.1007/s42967-021-00165-y  

   Guermond, Jean-Luc; Popov, Bojan; Saavedra, Laura (June 2023, Communications on Applied Mathematics and Computation)

   Abstract An invariant domain preserving arbitrary Lagrangian-Eulerian method for solving non-linear hyperbolic systems is developed. The numerical scheme is explicit in time and the approximation in space is done with continuous finite elements. The method is made invariant domain preserving for the Euler equations using convex limiting and is tested on various benchmarks. 

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3. Efficient Parallel 3D Computation of the Compressible Euler Equations with an Invariant-domain Preserving Second-order Finite-element Scheme

   https://doi.org/10.1145/3470637  

   Maier, Matthias; Kronbichler, Martin (September 2021, ACM Transactions on Parallel Computing)

   We discuss the efficient implementation of a high-performance second-order collocation-type finite-element scheme for solving the compressible Euler equations of gas dynamics on unstructured meshes. The solver is based on the convex-limiting technique introduced by Guermond et al. (SIAM J. Sci. Comput. 40, A3211–A3239, 2018). As such, it is invariant-domain preserving ; i.e., the solver maintains important physical invariants and is guaranteed to be stable without the use of ad hoc tuning parameters. This stability comes at the expense of a significantly more involved algorithmic structure that renders conventional high-performance discretizations challenging. We develop an algorithmic design that allows SIMD vectorization of the compute kernel, identify the main ingredients for a good node-level performance, and report excellent weak and strong scaling of a hybrid thread/MPI parallelization. 

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4. thornado-transport: IMEX schemes for two-moment neutrino transport respecting Fermi-Dirac statistics

   Chu, R; Endeve, E; Hauck, C; Mezzacappa, A; Messer, B (July 2019, Journal of physics Conference series)

   We develop implicit-explicit (IMEX) schemes for neutrino transport in a background material in the context of a two-moment model that evolves the angular moments of a neutrino phase-space distribution function. Considering the upper and lower bounds that are introduced by Pauli’s exclusion principle on the moments, an algebraic moment closure based on Fermi-Dirac statistics and a convex-invariant time integrator both are demanded. A finite-volume/first-order discontinuous Galerkin(DG) method is used to illustrate how an algebraic moment closure based on Fermi-Dirac statistics is needed to satisfy the bounds. Several algebraic closures are compared with these bounds in mind, and the Cernohorsky and Bludman closure, which satisfies the bounds, is chosen for our IMEX schemes. For the convex-invariant time integrator, two IMEX schemes named PD-ARS have been proposed. PD-ARS denotes a convex-invariant IMEX Runge-Kutta scheme that is high-order accurate in the streaming limit, and works well in the diffusion limit. Our two PD-ARS schemes use second- and third-order, explicit, strong-stability-preserving Runge-Kutta methods as their explicit part, respectively, and therefore are second- and third-order accurate in the streaming limit, respectively. The accuracy and convex-invariance of our PD-ARS schemes are demonstrated in the numerical tests with a third-order DG method for spatial discretization and a simple Lax-Friedrichs flux. The method has been implemented in our high-order neutrino-radiation hydrodynamics (thornado) toolkit. We show preliminary results employing tabulated neutrino opacities. 

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5. A High-Accuracy, Low-Order Thermal Model of SiC MOSFET Power Modules Extracted from Finite Element Analysis via Model Order Reduction

   https://doi.org/10.1109/ECCE.2019.8912839  

   Entzminger, Cameron; Qiao, Wei; Qu, Liyan; Hudgins, Jerry L. (September 2019, 2019 IEEE Energy Conversion Congress and Exposition (ECCE))

   Silicon carbide (SiC) MOSFET power modules are being used for high power applications because of their superior thermal characteristics and high blocking voltage capabilities over traditional silicon power modules. This paper explores modeling the thermal process of a SiC MOSFET power module through a high-order finite element analysis (FEA) based thermal model and then reducing the order of the FEA thermal model using a Krylov subspace method. The low-order thermal model has a significantly reduced computation cost compared to the FEA model while preserving the accuracy of the model. The proposed method is applied to generate low-order thermal models for a SiC MOSFET, which are validated by computer simulations with respect to the FEA thermal model. 

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