Implicit Lagrangian Hydrodynamics with High-Order Finite Elements
Abstract
We present an implicit time integration capability for high-order curvilinear finite element Lagrangian hydrodynamics. Starting from an existing explicit formulation, the implicit treatment builds directly on the original discretization and operator structure and does not alter the underlying spatial formulation or physics model. We demonstrate our implementation using the Laghos miniapp, which is built on the MFEM finite element library. To support gradient-based nonlinear solution methods, we compute Jacobian actions automatically using MFEM's ∂FEM interface together with Enzyme-based automatic differentiation, and apply the resulting Jacobian in a matrix-free or fully-assembled manner within a Newton-Krylov solver. To ensure robust and differentiable nonlinear solves in the presence of shocks, we introduce a smooth artificial viscosity treatment based on smooth approximations of non-differentiable pointwise operations. The differentiable artificial viscosity presently lacks a limiter to ensure high-order scaling away from shocks, but is sufficient for illustrating the benefits of implicit Lagrangian hydrodynamics. The behavior and performance of the implicit method are demonstrated on several standard benchmark problems. We verify high-order convergence on the smooth Taylor-Green vortex in the absence of artificial viscosity, show correct strong-shock behavior on the Sedov blast problem, and obtain significant improvements in accuracy-per-time-to-solution on the Triple Point problem where explicit stability constraints become increasingly severe for high-order discretizations.
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BibTeX
@misc{andrej2026implicit,
title = {{Implicit Lagrangian Hydrodynamics with High-Order Finite Elements}},
author = {Julian Andrej and John Camier and Veselin Dobrev and Tzanio Kolev and Boyan Lazarov and Ketan Mittal and Robert Rieben and Brandon Talamini and Vladimir Z. Tomov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16424},
eprint = {2609.16424},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16424},
}
RIS
TY - UNPB
TI - Implicit Lagrangian Hydrodynamics with High-Order Finite Elements
AU - Julian Andrej
AU - John Camier
AU - Veselin Dobrev
AU - Tzanio Kolev
AU - Boyan Lazarov
AU - Ketan Mittal
AU - Robert Rieben
AU - Brandon Talamini
AU - Vladimir Z. Tomov
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.16424
UR - https://arxiv.org/abs/2609.16424
AB - We present an implicit time integration capability for high-order curvilinear finite element Lagrangian hydrodynamics. Starting from an existing explicit formulation, the implicit treatment builds directly on the original discretization and operator structure and does not alter the underlying spatial formulation or physics model. We demonstrate our implementation using the Laghos miniapp, which is built on the MFEM finite element library. To support gradient-based nonlinear solution methods, we compute Jacobian actions automatically using MFEM's ∂FEM interface together with Enzyme-based automatic differentiation, and apply the resulting Jacobian in a matrix-free or fully-assembled manner within a Newton-Krylov solver. To ensure robust and differentiable nonlinear solves in the presence of shocks, we introduce a smooth artificial viscosity treatment based on smooth approximations of non-differentiable pointwise operations. The differentiable artificial viscosity presently lacks a limiter to ensure high-order scaling away from shocks, but is sufficient for illustrating the benefits of implicit Lagrangian hydrodynamics. The behavior and performance of the implicit method are demonstrated on several standard benchmark problems. We verify high-order convergence on the smooth Taylor-Green vortex in the absence of artificial viscosity, show correct strong-shock behavior on the Sedov blast problem, and obtain significant improvements in accuracy-per-time-to-solution on the Triple Point problem where explicit stability constraints become increasingly severe for high-order discretizations.
ER -