@article{Droniou2024, 
author = {Jérôme Droniou and Jia Jia Qian},
title = {Two arbitrary-order constraint-preserving schemes for the Yang–Mills equations on polyhedral meshes},
year = {2024},
journal = {Mathematics in Engineering},
volume = {6},
number = {3},
pages = {468-493},
keywords = {discrete de Rham method, Yang–Mills equations, polytopal method, constraint-preserving scheme, energy estimate, 3D numerical tests},
url = {https://www.sciopen.com/article/10.3934/mine.2024019},
doi = {10.3934/mine.2024019},
abstract = {Two numerical schemes are proposed and investigated for the Yang–Mills equations, which can be seen as a nonlinear generalisation of the Maxwell equations set on Lie algebra-valued functions, with similarities to certain formulations of General Relativity. Both schemes are built on the Discrete de Rham (DDR) method, and inherit from its main features: an arbitrary order of accuracy, and applicability to generic polyhedral meshes. They make use of the complex property of the DDR, together with a Lagrange-multiplier approach, to preserve, at the discrete level, a nonlinear constraint associated with the Yang–Mills equations. We also show that the schemes satisfy a discrete energy dissipation (the dissipation coming solely from the implicit time stepping). Issues around the practical implementations of the schemes are discussed; in particular, the assembly of the local contributions in a way that minimises the price we pay in dealing with nonlinear terms, in conjunction with the tensorisation coming from the Lie algebra. Numerical tests are provided using a manufactured solution, and show that both schemes display a convergence in        L    2  -norm of the potential and electrical fields in        O    (      h          k      +      1        ) (provided that the time step is of that order), where    k is the polynomial degree chosen for the DDR complex. We also numerically demonstrate the preservation of the constraint.}
}