A discontinuous Galerkin method for shock capturing using a mixed high-order and sub-grid low-order approximation space

•(Non-linear) stability of high-order methods for conservation laws is an open issue.•This paper introduces a new discontinuous Galerkin method.•An approximation space is considered with high-order and sub-grid basis functions.•The high-order modes can additionally suppressed by penalty using a new sensor. This article considers a new discretization scheme for conservation laws. The discretization setting is based on a discontinuous Galerkin scheme in combination with an approximation space that contains high-order polynomial modes as well as piece-wise constant modes on a sub-grid. The high-order modes can continuously be suppressed with a penalty function that is based on a sensor which is intertwined with the approximation space. Numerical tests finally illustrate the performance of this scheme.