We present a discretisation of the 3+1 formulation of the Yang-Mills equations in the temporal gauge, using a Lie algebra-valued extension of the discrete de Rham (DDR) sequence, that preserves the non-linear constraint exactly. In contrast to Maxwell's equations, where the preservation of the analogous constraint only depends on reproducing some complex properties of the continuous de Rham sequence, the preservation of the non-linear constraint relies for the Yang-Mills equations on a constrained formulation, previously proposed in [10]. The fully discrete nature of the DDR method requires to devise appropriate constructions of the non-linear terms, adapted to the discrete spaces and to the need for replicating the crucial Ad-invariance property of the $L^2$-product. We then prove some energy estimates, and provide results of 3D numerical simulations based on this scheme.

翻译：我们提出了一种在时间规范下对杨-米尔斯方程3+1形式化的离散化方法，该方法采用离散德拉姆（DDR）序列的李代数值扩展，能够精确保持非线性约束。与麦克斯韦方程中类比约束的保持仅依赖于再现连续德拉姆序列某些复杂性质不同，杨-米尔斯方程中非线性约束的保持需要采用先前文献[10]提出的约束形式化方法。DDR方法的完全离散特性要求对非线性项进行适当构造，使其适应离散空间并满足复制$L^2$积分的关键Ad不变性需求。最后，我们证明了若干能量估计，并给出了基于该格式的三维数值模拟结果。