Shock structure in numerical solutions of the Navier–Stokes and Bhatnagar–Gross–Krook equations

Abstract

The dynamics of compressible viscous flows can be modeled by the well-developed theory of thermodynamics and fluid mechanics leading to a continuum description of the process. At the same time, kinetic models aim to address the same phenomena from a more fundamental and completely different view point. In contrast to the continuum approach, kinetic theory allows a description of a system and its characteristics from the microscopic picture of the underlying processes based on collision dynamics. The task of comparing these two approaches and explaining how they are related is a major challenge for applied mathematics. It is known that kinetic models such as the Boltzmann and Bhatnagar–Gross–Krook equations can theoretically yield continuum models such as the equations of Euler or Navier–Stokes via the Hilbert and Chapman–Enskog expansions. Despite these theoretical results, mathematical modeling using the kinetic approach is not straightforward, in particular when it comes to numerical approximations. Various difficulties arise when attempting to adopt an appropriate kinetic numerical algorithm to obtain results comparable to solutions of continuum models. In this work we address a small aspect of this problem. The one-dimensional Bhatnagar–Gross–Krook equation and its shock forming initial and boundary conditions are considered. A numerical algorithm to simulate a shock formation is developed, discussed and the results are compared to a similar process based on a continuum model. Both approaches are evaluated and compared numerically.