We introduce a filtering technique for Discontinuous Galerkin approximations of hyperbolic problems. Following an approach already proposed for the Hamilton-Jacobi equations by other authors, we aim at reducing the spurious oscillations that arise in presence of discontinuities when high order spatial discretizations are employed. This goal is achieved using a filter function that keeps the high order scheme when the solution is regular and switches to a monotone low order approximation if it is not. The method has been implemented in the framework of the $deal.II$ numerical library, whose mesh adaptation capabilities are also used to reduce the region in which the low order approximation is used. A number of numerical experiments demonstrate the potential of the proposed filtering technique.

翻译：我们提出一种用于双曲问题间断伽辽金逼近的滤波技术。借鉴其他作者已提出的哈密顿-雅可比方程方法，我们旨在减少采用高阶空间离散时在间断处产生的伪振荡。该目标通过滤波器函数实现：当解光滑时保持高阶格式，否则切换为单调低阶近似。该方法已在$deal.II$数值库框架中实现，并利用其网格自适应能力缩小低阶近似的使用区域。大量数值实验展示了所提滤波技术的潜力。