Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

High-order limiting methods using maximum principle bounds derived from the Boltzmann equation I: Euler equations

Tarik Dzanic, Luigi Martinelli

TL;DR

This work addresses the challenge of achieving robust, high-order accuracy for the Euler equations in the presence of strong shocks by leveraging a kinetic representation through the Boltzmann equation to define admissible solution bounds. By mapping local Boltzmann distribution bounds to macroscopic density, momentum, and energy bounds and applying a maximum principle–preserving limiter, the authors maintain positivity while preserving high-order accuracy in smooth regions. Their approach is demonstrated within an explicit unstructured discontinuous Galerkin/flux reconstruction framework across a suite of challenging gas-dynamics problems, including extreme shocks and shock–vortex interactions, showing stable convergence and minimal dissipation. The method offers a principled foundation for kinetic-based limiting that can extend to more complex macroscopic models derived from kinetic theory, with potential computational enhancements via adaptive velocity quadrature.

Abstract

The use of limiting methods for high-order numerical approximations of hyperbolic conservation laws generally requires defining an admissible region/bounds for the solution. In this work, we present a novel approach for computing solution bounds and limiting for the Euler equations through the kinetic representation provided by the Boltzmann equation, which allows for extending limiters designed for linear advection directly to the Euler equations. Given an arbitrary set of solution values to compute bounds over (e.g., numerical stencil) and a desired linear advection limiter, the proposed approach yields an analytic expression for the admissible region of particle distribution function values, which may be numerically integrated to yield a set of bounds for the density, momentum, and total energy. These solution bounds are shown to preserve positivity of density/pressure/internal energy and, when paired with a limiting technique, can robustly resolve strong discontinuities while recovering high-order accuracy in smooth regions without any ad hoc corrections (e.g., relaxing the bounds). This approach is demonstrated in the context of an explicit unstructured high-order discontinuous Galerkin/flux reconstruction scheme for a variety of difficult problems in gas dynamics, including cases with extreme shocks and shock-vortex interactions. Furthermore, this work presents a foundation for limiting techniques for more complex macroscopic governing equations that can be derived from an underlying kinetic representation for which admissible solution bounds are not well-understood.

High-order limiting methods using maximum principle bounds derived from the Boltzmann equation I: Euler equations

TL;DR

This work addresses the challenge of achieving robust, high-order accuracy for the Euler equations in the presence of strong shocks by leveraging a kinetic representation through the Boltzmann equation to define admissible solution bounds. By mapping local Boltzmann distribution bounds to macroscopic density, momentum, and energy bounds and applying a maximum principle–preserving limiter, the authors maintain positivity while preserving high-order accuracy in smooth regions. Their approach is demonstrated within an explicit unstructured discontinuous Galerkin/flux reconstruction framework across a suite of challenging gas-dynamics problems, including extreme shocks and shock–vortex interactions, showing stable convergence and minimal dissipation. The method offers a principled foundation for kinetic-based limiting that can extend to more complex macroscopic models derived from kinetic theory, with potential computational enhancements via adaptive velocity quadrature.

Abstract

The use of limiting methods for high-order numerical approximations of hyperbolic conservation laws generally requires defining an admissible region/bounds for the solution. In this work, we present a novel approach for computing solution bounds and limiting for the Euler equations through the kinetic representation provided by the Boltzmann equation, which allows for extending limiters designed for linear advection directly to the Euler equations. Given an arbitrary set of solution values to compute bounds over (e.g., numerical stencil) and a desired linear advection limiter, the proposed approach yields an analytic expression for the admissible region of particle distribution function values, which may be numerically integrated to yield a set of bounds for the density, momentum, and total energy. These solution bounds are shown to preserve positivity of density/pressure/internal energy and, when paired with a limiting technique, can robustly resolve strong discontinuities while recovering high-order accuracy in smooth regions without any ad hoc corrections (e.g., relaxing the bounds). This approach is demonstrated in the context of an explicit unstructured high-order discontinuous Galerkin/flux reconstruction scheme for a variety of difficult problems in gas dynamics, including cases with extreme shocks and shock-vortex interactions. Furthermore, this work presents a foundation for limiting techniques for more complex macroscopic governing equations that can be derived from an underlying kinetic representation for which admissible solution bounds are not well-understood.
Paper Structure (30 sections, 6 theorems, 80 equations, 18 figures, 6 tables)

This paper contains 30 sections, 6 theorems, 80 equations, 18 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Euler equations
  4. Boltzmann equation
  5. Discontinuous Galerkin methods
  6. Maximum principle preserving limiting
  7. Methodology
  8. Boltzmann representation of the Euler equations
  9. Maximum principle bounds for the Boltzmann equation
  10. From kinetic to macroscopic
  11. Limiting
  12. Relaxing the bounds
  13. Modeling internal energy
  14. Integrating the distribution function
  15. Bounding density and pressure
  16. ...and 15 more sections

Key Result

Theorem 4.1

The proposed bounds preserve constant states in the sense that if $\mathbf{w}(\mathbf{x}_i, t^n) = \mathbf{w}_0 \ \forall\ i \in S$ for some constant state $\mathbf{w}_0$, then $\mathbf{w}_{\min}^{n+1} = \mathbf{w}_{\max}^{n+1} = \mathbf{w}_0$.

Figures (18)

  • Figure 1: Schematic of an unstructured discontinuous Galerkin approximation with volume (blue) and surface (red) quadrature nodes.
  • Figure 1: Convergence of the $L^1$ norm of the density error for the Sod shock tube problem at $t = 0.2$ computed using varying approximation orders and mesh resolution with relaxed bounds. Average rate of convergence shown on bottom.
  • Figure 2: Various examples of choices for the stencil to compute bounds over ($S$, shown by solid red/blue nodes) as well as the nodes to enforce bounds over ($V$, shown by nodes with green outlines), including face/Voronoi neighbors (left), numerical stencil (middle), and nodal neighbors (right).
  • Figure 2: Convergence of the $L^1$ norm of the density error for the Sod shock tube problem at $t = 0.2$ computed using varying approximation orders and mesh resolution without relaxed bounds. Average rate of convergence shown on bottom.
  • Figure 3: Example schematic for a one-dimensional Euler solution within an element $\Omega_k$, including volume (solid blue circles) and surface (solid red circles) quadrature nodes for the local discrete stencil (denoted by $\{\mathbf{w}_1, \ldots, \mathbf{w}_7 \} \in S$). Corresponding equilibrium distribution function (in the velocity domain) for each solution node shown on the right.
  • ...and 13 more figures

Theorems & Definitions (7)

  • Theorem 4.1: Constant state preservation
  • Theorem 4.2: Positivity preservation
  • Theorem 4.3: Riemann-averaged states
  • Remark
  • Theorem 4.4: First-order scheme
  • Theorem 4.5: Mean bounds preservation
  • Corollary 4.5.1: Existence of solution