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：本文针对杨-米尔斯方程提出并研究了两种数值格式。该方程可视为麦克斯韦方程组的非线性推广，其定义在李代数值函数上，与广义相对论的某些表述具有相似性。两种格式均建立在离散德拉姆（DDR）方法基础上，继承了该方法的核心特征：可达到任意阶精度，且适用于通用多面体网格。通过结合DDR复形特性与拉格朗日乘子法，两种格式在离散层面保持了与杨-米尔斯方程相关的非线性约束条件。研究同时证明，格式满足离散能量耗散特性（耗散仅来源于隐式时间离散）。文中探讨了格式实际实施的相关问题，特别是局部贡献的集成方式——该方法在应对李代数张量化带来的非线性项时能最小化计算代价。通过构造解析解进行数值实验，结果表明：当时间步长精度匹配时，两种格式在势场和电场的$L^2$范数下均呈现$\mathcal O(h^{k+1})$阶收敛（其中$k$为DDR复形选取的多项式次数）。数值实验同时验证了约束条件的保持性。