Two arbitrary-order constraint-preserving schemes for the Yang–Mills equations on polyhedral meshes

Jérôme Droniou; Jia Jia Qian; Jérôme Droniou; Jia Jia Qian

doi:10.3934/mine.2024019

Mathematics in Engineering

2024, Volume 6, Issue 3: 468-493. doi: 10.3934/mine.2024019

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Two arbitrary-order constraint-preserving schemes for the Yang–Mills equations on polyhedral meshes

Jérôme Droniou ^{1,2
,
,},
Jia Jia Qian ^{2
,}

1.
IMAG, Univ. Montpellier, CNRS, Montpellier, France
2.
School of Mathematics, Monash University, Melbourne, Australia

Received: 15 June 2023 Revised: 10 May 2024 Accepted: 17 June 2024 Published: 24 June 2024

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 $ \mathcal 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.
- discrete de Rham method,
- Yang–Mills equations,
- polytopal method,
- constraint-preserving scheme,
- energy estimate,
- 3D numerical tests
Citation: Jérôme Droniou, Jia Jia Qian. Two arbitrary-order constraint-preserving schemes for the Yang–Mills equations on polyhedral meshes[J]. Mathematics in Engineering, 2024, 6(3): 468-493. doi: 10.3934/mine.2024019

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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 $ \mathcal 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.

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