Séminaire de M. Abdelmalik (Eindhoven University of Technology Eindhoven, Netherlands)
Entropy-based Ansatz for Galerkin Approximations of the Boltzmann Equation
Abstract :
In this talk we develop an entropy-stable finite-element-moment method for the Boltzmann equation with binary collisions in bounded domains. The proposed method engenders a discontinuous-Galerkin method in position and temporal dependence, and a moment-method in velocity dependence. We base our moment-method on a converging sequence of approximations to the binary collision operator $\mathcal C$, denoted by $\mathcal C_N$. We associate with each member of the sequence a $\varphi$-divergence-based renormalization-map and entropy [1]. We show that each $\mathcal C_N$ inherits salient properties from $\mathcal C$, such as the preservation of the collision invariants and that the linearization of $\mathcal C_N$ coincides with the linearization of $\mathcal C$. We show that our proposed moment method engenders hierarchies of symmetric dissipative hyperbolic systems and the corresponding finite-element-moment method is entropy-stable. Finally, we apply our finite-element-moment method to the Boltzmann equation with collision operator $\mathcal C$ and demonstrate the corresponding approximation properties, using benchmark test cases, in comparison to Direct Simulation Monte Carlo.
[1] M. Abdelmalik, E.H. van Brummelen, Journal of Statistical Physics 164, 77–104 (2016).
