Hyperbolic Conservation Laws and Compressible Fluids

Systems of hyperbolic conservation laws arise as models in various parts of continuum physics when dissipative effects are neglected, the most prominent example are the compressible Euler equations in fluid mechanics. There are many open mathematical questions associated with these equations. Indeed, strong solutions (in general) only exist for short times and weak (entropy) solutions are not unique (for systems in more than one space dimension). This non-uniqueness is believed to be connected to the onset of turbulence and has led to novel solution concepts such as statistical solutions. In this field we have contributed by providing a priori error estimates for scalar problems on manifolds and by providing a posteriori error estimates for discretisation errrors (in discontinuous Galerkin schemes) and for modelling errors.

Part of our research in this field is funded by a priority program of the DFG: Link to the priority program

  • Domschke, Kolb, Lang: Fast and Reliable Transient Simulation and Continuous Optimization of Large-Scale Gas Networks, Math. Meth. Oper. Res. (2022)
  • Egger, Giesselmann, Kunkel, Philippi: An asymptotic preserving discretization scheme for gas transport in pipe networks, IMA J. Numer. Anal. (2022)
  • Giesselmann, Krupa: Theory of shifts, shocks, and the intimate connections to L2-type a posteriori error analysis of numerical schemes for hyperbolic problems, Math. Comp. (2024)
  • Lang, Mindt: Entropy-Preserving Coupling Conditions for One-dimensional Euler Systems at Junctions, Netw. Heterog. Media (2018)
  • A posteriori error estimates for convection-diffusion equations (Giesselmann)