In Space-Time Discontinuous Galerkin finite element methods for time-dependent partial differential equations time is considered as an additional dimension. By formulating the problem in space-time, the spatial and temporal variables can be simultaneously discretized using basis functions that are discontinuous both in space and in time. Space-time DG methods are well suited to solve a large class of partial differential equations on (time-dependent) domains and provide accurate and conservative numerical discretizations that are suitable for hp-mesh adaptation.

In this series of lectures, we will first consider space-time DG discretizations for simple hyperbolic and parabolic equations in order to explain the main concept of the space-time DG method. In the second part some examples from fluid mechanics will be discussed, such as the compressible Euler and Navier-Stokes equations, the incompressible Navier-Stokes equations, and if time permits, dispersed multiphase flows modeled by non-conservative hyperbolic partial differential equations.