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Two numerical schemes are proposed and investigated for the Yang--Millsequations, which can be seen as a nonlinear generalisation of the Maxwellequations set on Lie algebra-valued functions, with similarities to certainformulations of General Relativity. Both schemes are built on the Discrete deRham (DDR) method, and inherit from its main features: an arbitrary order ofaccuracy, and applicability to generic polyhedral meshes. They make use of thecomplex property of the DDR, together with a Lagrange-multiplier approach, topreserve, at the discrete level, a nonlinear constraint associated with theYang--Mills equations. We also show that the schemes satisfy a discrete energydissipation (the dissipation coming solely from the implicit time stepping).Issues around the practical implementations of the schemes are discussed; inparticular, the assembly of the local contributions in a way that minimises theprice we pay in dealing with nonlinear terms, in conjunction with thetensorisation coming from the Lie algebra. Numerical tests are provided using amanufactured 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 polynomialdegree chosen for the DDR complex. We also numerically demonstrate thepreservation of the constraint.

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