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.

翻译：双曲守恒律高阶数值近似中限制方法的使用通常需要为解定义容许区域/界。本文提出了一种通过玻尔兹曼方程提供的动力学表示来计算欧拉方程解界并进行限制的新方法，该方法能够将针对线性对流设计的限制器直接推广至欧拉方程。给定用于计算界的任意解值集合（如数值模板）及期望的线性对流限制器，所提方法可导出粒子分布函数值容许区域的解析表达式，该表达式可通过数值积分得到密度、动量和总能量的一组界。这些解界被证明能保持密度/压力/内能的正性，当与限制技术结合时，可在无需任何特殊修正（如放宽界限制）的情况下，鲁棒地解析强间断问题，同时在光滑区域恢复高阶精度。该方法在显式非结构高阶间断伽辽金/通量重构格式框架下，通过包括极端激波和激波-涡相互作用在内的多种气体动力学难题得到验证。此外，本研究为更复杂的宏观控制方程的限制技术奠定了基础——这些方程可从底层动力学表示推导而来，而其容许解界尚未被充分理解。