Young Researchers Workshop: Ki-Net 2012-2019

Numerical approximation of statistical solutions for systems of hyperbolic conservation laws

Franziska Weber

Carnegie Mellon University


Fjordholm, Käppeli, Mishra and Tadmor (2016) demonstrated in numerical experiments that standard finite volume schemes may not converge to entropy weak solutions of systems of multi-dimensional hyperbolic conservation laws. Convergence to entropy measure-valued solutions could however be observed and proved. The issue is that entropy measure-valued solutions are not unique, not even in the scalar case. Motivated by these results, Fjordholm, Lanthaler and Mishra (2017) suggested statistical solutions as a notion of solutions for multi-dimensional systems of nonlinear conservation laws. In the scalar case, well-posedness of such solutions can be shown. We present numerical algorithms to approximate statistical solutions of systems of conservation laws and formulate conditions under which they converge. The conditions are inspired by Kolmogorov’s theory of turbulence (1941) and numerical experiments for the 2d compressible Euler equations show that the conditions might be satisfied at least in some cases. In addition, in the case of the incompressible Navier-Stokes equations, the statistical solutions introduced by Fjordholm, Lanthaler and Mishra can be shown to be equivalent to the statistical solutions as in the sense of Foias and Temam.