A Roe-type approximate-flux Riemann solver without linear system resolution and application to lagrangian solid mechanics
Abstract
This paper introduces an approximate-flux version of Roe's Riemann solver for Lagrangian solid dynamics. The solver is based on a reformulation of the Riemann problem in flux variables and the use of orthonormal left and right eigenvectors of the Jacobian matrix which avoid the linear system resolution involved in the original Roe method. In the context of Godunov's type methods for solid dynamics, the flux computation enabled by the proposed solver provides computational effectiveness, a straightforward treatment of both Neumann and Dirichlet boundary conditions, and easy handling of material heterogeneities. Implemented within an in-house finite volume code as well as within the FEniCSx platform using a first-order Discontinuous Galerkin (DG) scheme in a total Lagrangian setting, the method demonstrates excellent performance for hyperelastic materials. Numerical benchmarks validate its effectiveness in 1D and 2D configurations including a coupled thermo-mechanical problem. The latter allows highlighting the suitability of a mixed Continuous/Discontinuous Galerkin framework using FEniCSx for complex solid dynamics problems coupled to other physics.
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BibTeX
@misc{renaud2026roetype,
title = {{A Roe-type approximate-flux Riemann solver without linear system resolution and application to lagrangian solid mechanics}},
author = {Adrien Renaud},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05734288v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Roe-type approximate-flux Riemann solver without linear system resolution and application to lagrangian solid mechanics
AU - Adrien Renaud
PY - 2026
DA - 2026/09/01
PB - HAL
UR - https://hal.science/hal-05734288v2
AB - This paper introduces an approximate-flux version of Roe's Riemann solver for Lagrangian solid dynamics. The solver is based on a reformulation of the Riemann problem in flux variables and the use of orthonormal left and right eigenvectors of the Jacobian matrix which avoid the linear system resolution involved in the original Roe method. In the context of Godunov's type methods for solid dynamics, the flux computation enabled by the proposed solver provides computational effectiveness, a straightforward treatment of both Neumann and Dirichlet boundary conditions, and easy handling of material heterogeneities. Implemented within an in-house finite volume code as well as within the FEniCSx platform using a first-order Discontinuous Galerkin (DG) scheme in a total Lagrangian setting, the method demonstrates excellent performance for hyperelastic materials. Numerical benchmarks validate its effectiveness in 1D and 2D configurations including a coupled thermo-mechanical problem. The latter allows highlighting the suitability of a mixed Continuous/Discontinuous Galerkin framework using FEniCSx for complex solid dynamics problems coupled to other physics.
ER -