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 recall a discrete de-Rham framework in which discontinuous approximation spaces for vectors fits. 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：已知某些双曲系统隐式保持微分约束：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，若在向量空间中使用恰当的近似空间，并在数值通量满足温和假设的条件下，此类约束可通过经典间断伽辽金方法在离散层面轻松保持。为此，我们回顾了适用于向量间断近似空间的离散德拉姆框架。在数值通量满足微小假设的前提下，间断伽辽金方法被证明能精确保持离散伴随散度与旋度。通过对波动系统、二维麦克斯韦系统及感应方程进行数值实验，证实微分约束可在保持机器精度的同时维持高阶精度。