An entropy stable scheme for the non-linear Boltzmann equation

•A discretely (locally and globally) conservative high-order semi-discrete entropy-stable scheme has been presented.•The scheme conserves mass, momentum, energy, and satisfies the celebrated H-theorem at a discrete level.•To address low/high rarefaction flows, we couple the scheme with an iteration-...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:Journal of computational physics 2022-08, Vol.463, p.111289, Article 111289
1. Verfasser: Jaiswal, Shashank
Format: Artikel
Sprache:eng
Schlagworte:
Online-Zugang:Volltext
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
Beschreibung
Zusammenfassung:•A discretely (locally and globally) conservative high-order semi-discrete entropy-stable scheme has been presented.•The scheme conserves mass, momentum, energy, and satisfies the celebrated H-theorem at a discrete level.•To address low/high rarefaction flows, we couple the scheme with an iteration-free implicit-explicit time discretization.•The scheme yields stable solution for Riemann problems involving shocks even without using a shock-limiter. The kinetic equations govern the behavior of gaseous flows, compressibility, turbulence, reactions with internal energy exchange, that form the essence of numerous physical processes. Despite their wide applicability, their six-dimensional (seven including time) nature presents a huge computational challenge. With the advent of the modern high performance computing systems and few recent advances in numerical methods, it is now possible to numerically study the behavior of these systems. However, to understand the instability of rich molecular processes and find a structure in the apparent chaos, a scheme that is efficient in multi-dimensions, exhibits high parallel-efficiency, and foremost produces an entropy solution as per the theory of solutions of hyperbolic systems may prove useful. In this work, first, we construct an entropy stable flux for non-linear inhomogeneous (full) Boltzmann equation. Second, to ensure geometrical flexibility, we couple the scheme with a class of high order discontinuous Galerkin discretization (Jaiswal 2019 [41]) which satisfy summation-by-part (SBP) discretely. Third, utilizing SBP, we prove that the resulting semi-discrete scheme is locally and globally conservative in flat spaces; preserves the entropy-decay (H-theorem) property; efficient and simple; and therefore suitable for dealing with highly complex non-smooth flow problems. Fourth, we show that the fully-discrete kinetic scheme, utilizing an implicit-explicit time-discretization, in the limit of vanishing Knudsen number, becomes an entropy-stable explicit scheme applied to the Euler system. Fifth, we carry out a series of verification tests to illustrate the stability, accuracy, and conservation properties of the proposed method. These tests involve over 85 million degrees of freedom and over 50 billion operations per time-step.
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2022.111289