Involution-Preserving schemes through constrained mesh motion: a novel Arbitrary-Lagrangian-Eulerian framework
Abstract
We introduce a novel structure-preserving finite volume method that exploits mesh motion to enforce differential involutions arising in hyperbolic systems of conservation laws. The key idea is to leverage the geometric degrees of freedom of an Arbitrary-Lagrangian-Eulerian (ALE) framework to construct a constrained mesh motion that is intrinsically compatible with the involution constraints embedded in the governing equations. The method is developed within a cellcentered formulation on moving quadrilateral meshes, where discrete divergence and curl operators are constructed from vertex-based differential operators. Particular care is devoted to the consistent evolution of the mesh geometry and the satisfaction of the Geometric Conservation Law (GCL) The proposed framework is applied to two representative hyperbolic systems featuring distinct involution constraints: the ideal magnetohydrodynamics equations, endowed with the divergencefree condition on the magnetic field, and the hyperbolic Euler-Heat equations, subject to a curl-free constraint on the thermal impulse. Second-order accuracy in time is achieved through a Runge-Kutta discretization, with the geometric constraints consistently enforced at each stage. A comprehensive set of classical benchmark problems is presented to assess the accuracy and effectiveness of the method. The numerical results demonstrate that appropriately constrained mesh motion provides an effective mechanism for preserving differential involutions, establishing a novel connection between mesh geometry and the structure-preserving discretization of hyperbolic systems.
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BibTeX
@misc{cardonna2026involutionpreserving,
title = {{Involution-Preserving schemes through constrained mesh motion: a novel Arbitrary-Lagrangian-Eulerian framework}},
author = {Sacha Cardonna and Elena Bernardelli and Walter Boscheri},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05804768v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Involution-Preserving schemes through constrained mesh motion: a novel Arbitrary-Lagrangian-Eulerian framework
AU - Sacha Cardonna
AU - Elena Bernardelli
AU - Walter Boscheri
PY - 2026
DA - 2026/10/18
PB - HAL
UR - https://hal.science/hal-05804768v1
AB - We introduce a novel structure-preserving finite volume method that exploits mesh motion to enforce differential involutions arising in hyperbolic systems of conservation laws. The key idea is to leverage the geometric degrees of freedom of an Arbitrary-Lagrangian-Eulerian (ALE) framework to construct a constrained mesh motion that is intrinsically compatible with the involution constraints embedded in the governing equations. The method is developed within a cellcentered formulation on moving quadrilateral meshes, where discrete divergence and curl operators are constructed from vertex-based differential operators. Particular care is devoted to the consistent evolution of the mesh geometry and the satisfaction of the Geometric Conservation Law (GCL) The proposed framework is applied to two representative hyperbolic systems featuring distinct involution constraints: the ideal magnetohydrodynamics equations, endowed with the divergencefree condition on the magnetic field, and the hyperbolic Euler-Heat equations, subject to a curl-free constraint on the thermal impulse. Second-order accuracy in time is achieved through a Runge-Kutta discretization, with the geometric constraints consistently enforced at each stage. A comprehensive set of classical benchmark problems is presented to assess the accuracy and effectiveness of the method. The numerical results demonstrate that appropriately constrained mesh motion provides an effective mechanism for preserving differential involutions, establishing a novel connection between mesh geometry and the structure-preserving discretization of hyperbolic systems.
ER -