NA Seminar, Lea Versbach (Lund), "Introduction to entropy stable space-time Discontinuous Galerkin schemes for conservation laws"
Kontakt: philipp [dot] birken [at] na [dot] lu [dot] se
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In this lecture-style talk I will introduce the concept of space-time Discontinuous Galerkin (DG) schemes, where both space and time are discretized using DG. This leads to high order implicit schemes which are suitable numerical approximations of systems of nonlinear conservation laws.
We are interested in entropy stable space-time schemes which can be used to solve thermodynamic systems. In , an entropy stable space-time DG scheme is developed. This is achieved by noting that the discrete approximations to the derivative in space and time are summation-by-parts (SBP) operators, allowing that the discrete method mimics results from the continuous entropy analysis. The development of entropy stable DG schemes is typically done semi-discrete. The construction of entropy stable DG methods in time is similar to the spatial discrete entropy analysis, but has some differences which will be presented. The basic idea is to apply a nodal DG ansatz with the SBP property to the temporal approximation as well and ensure that the fully discrete space-time DG scheme remains entropy stable. This scheme bounds the mathematical entropy at any time.
 Lucas Friedrich, Gero Schnücke, Andrew R. Winters, David C. Del Rey Fernández, Gregor J. Gassner, and Mark H. Carpenter: Entropy Stable Space–Time Discontinuous Galerkin Schemes with Summation-by-Parts Property for Hyperbolic Conservation Laws, Journal of Scientific Computing 80 (2019), no. 1, 175–222.