An Extended Discontinuous Galerkin Method for High-order shock treatment

We present an implicit high-order shock fitting approach based on a cut-cell method for the direct simulation of compressible flows. We formulate a suitable Constraint Optimization Problem and develop a Sequential Quadratic Program solver aiming to fit the shock front represented by the zero iso-contour of a Level Set function. In the Extended Discontinuous Galerkin method the approximation space is enriched by basis functions which are discontinuous alongside this interface, therefore a sharp representation of the shock-induced jumps can be obtained. As a consequence, no shock capturing has to be used and high-order accuracy is reached.

Further, two additional robustness measures inspired by numerical experiments and literature are discussed. Cell Agglomeration and solution reinitialization are used to prevent the optimization algorithm from stagnating at an undesired local minimum. Lastly, the methods capabilities are demonstrated on a sequence of two-dimensional problems of increasing difficulty (Space-Time Advection, Space-Time Burgers, Steady 2D Euler Equations) and discussed by means of a convergence study.