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.

翻译：针对杨-米尔斯方程，本文提出并研究了两种数值格式。该方程可视为定义在李代数取值函数上的麦克斯韦方程的非线性推广，其形式与广义相对论的某些表述具有相似性。两种格式均基于离散德拉姆（DDR）方法构建，并继承其主要特征：任意阶精度以及适用于一般多面体网格。通过利用DDR的复形性质，结合拉格朗日乘子法，两种格式在离散层面保留了与杨-米尔斯方程相关的非线性约束。本文同时证明，这些格式满足离散能量耗散性质（耗散仅源于隐式时间推进）。针对格式的实际实现问题进行了讨论，重点探讨如何通过局部贡献的组装最小化处理非线性项（结合李代数张量化引入的复杂性）的计算代价。采用解析解进行数值测试，结果表明两种格式在势场与电场$L^2$范数下均达到$\mathcal O(h^{k+1})$阶收敛（当时间步长保持同阶时），其中$k$为DDR复形选取的多项式阶数。数值实验进一步验证了约束的保持特性。