Hyperbolic moment equations using quadrature-based projection methods

Kinetic equations like the Boltzmann equation are the basis for various applications involving rarefied gases. An important problem of many approaches since the first developments by Grad is the desired global hyperbolicity of the emerging set of partial differential equations. Due to lack of hyperbolicity of Grad’s model equations, numerical computations can break down or yield nonphysical solutions. New hyperbolic PDE systems for the solution of the Boltzmann equation can be derived using quadrature-based projection methods. The method is based on a non-linear transformation of the velocity to obtain a Lagrangian velocity phase space description in order to allow for physical adaptivity, followed by a series expansion of the unknown distribution function in different basis functions and the application of quadrature-based projection methods. In this paper, we extend the proof for global hyperbolicity of the quadrature-based moment system system to arbitrary dimensions, utilizing quadrature-based projection methods for tensor product Hermite basis functions. The analytical computation of the eigenvalues shows the proposed correspondence to the Hermite quadrature points.