Some hyperbolic systems are known to include implicit preservation of differential constraints: these are for example the time conservation of the curl or the divergence of a vector that appear as an implicit constraint. In this article, we show that this kind of constraint can be easily conserved at the discrete level with the classical discontinuous Galerkin method, provided the right approximation space is used for the vectorial space, and under some mild assumption on the numerical flux. For this, we develop a discrete differential geometry framework for some well chosen piece-wise polynomial vector approximation space. More precisely, we define the discrete Hodge star operator, the exterior derivative, and their adjoints. The discrete adjoint divergence and curl are proven to be exactly preserved by the discontinuous Galerkin method under a small assumption on the numerical flux. Numerical tests are performed on the wave system, the two dimensional Maxwell system and the induction equation, and confirm that the differential constraints are preserved at machine precision while keeping the high order of accuracy.

翻译：某些双曲系统已知包含隐式微分约束：例如作为隐式约束出现的向量旋度或散度的时间守恒性。本文证明，只要采用适当的向量空间逼近形式且数值通量满足温和假设，经典不连续伽辽金方法即可在离散层面轻松保持此类约束。为此，我们针对特定分段多项式向量逼近空间建立了离散微分几何框架。具体而言，定义了离散霍奇星算子、外微分算子及其伴随算子。在数值通量的微小假设下，证明了不连续伽辽金方法能够精确保持离散伴随散度和旋度。对波动系统、二维麦克斯韦方程组及感应方程进行的数值测试表明，该方法在保持高阶精度的同时，能够以机器精度维持微分约束。