Feedback filtering in discontinuous Galerkin methods for Euler equations

A filtered discontinuous Galerkin method for the numerical solution of Euler equations is presented. The proposed approach makes it possible to capture discontinuities and to keep spurious oscillations under control while a high level of accuracy is maintained in smooth regions. The filter intensity is controlled in a dynamic way by the use of a sensor which measures the smoothness of the numerical solution. The signal obtained by the sensor is used to introduce a feedback of the numerical solution on the filter, adjusting its intensity with an iterative procedure. The described shock capturing approach makes it possible to perform solution reconstruction and filtering stabilisation using only information from inside the cell and so it is suitable for both structured and unstructured grids. Several 1D and 2D test cases were investigated in order to show the robustness of the scheme.