SIAM Review, Volume 65, Issue 4, Page 1031-1073, November 2023.
Solutions to many partial differential equations satisfy certain bounds or constraints. For example, the density and pressure are positive for equations of fluid dynamics, and in the relativistic case the fluid velocity is upper bounded by the speed of light, etc. As widely realized, it is crucial to develop bound-preserving numerical methods that preserve such intrinsic constraints. Exploring provably bound-preserving schemes has attracted much attention and has been actively studied in recent years. This is, however, still a challenging task for many systems, especially those involving nonlinear constraints. Based on some key insights from geometry, we systematically propose an innovative and general framework, referred to as geometric quasilinearization (GQL), which paves a new effective way for studying bound-preserving problems with nonlinear constraints. The essential idea of GQL is to equivalently transform all nonlinear constraints to linear ones, by properly introducing some free auxiliary variables. We establish the fundamental principle and general theory of GQL via the geometric properties of convex regions and propose three simple effective methods for constructing GQL. We apply the GQL approach to a variety of partial differential equations and demonstrate its effectiveness and remarkable advantages for studying bound-preserving schemes, using diverse challenging examples and applications which cannot be easily handled by direct or traditional approaches.

中文翻译：

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用于分析和设计保界方案的几何拟线性化框架

SIAM Review，第 65 卷，第 4 期，第 1031-1073 页，2023 年 11 月。
许多偏微分方程的解满足某些边界或约束。例如，流体动力学方程的密度和压力为正，在相对论情况下，流体速度的上限受光速等限制。正如人们广泛认识到的那样，开发保界数值方法至关重要，该方法可以保持这种内在的限制。近年来，探索可证明的保界方案引起了广泛关注并得到了积极研究。然而，对于许多系统，尤其是涉及非线性约束的系统来说，这仍然是一项具有挑战性的任务。基于几何的一些关键见解，我们系统地提出了一种创新的通用框架，称为几何拟线性化（GQL），为研究非线性约束的保界问题铺平了新的有效途径。GQL的基本思想是通过适当引入一些自由辅助变量，将所有非线性约束等价地转化为线性约束。我们通过凸区域的几何性质建立了GQL的基本原理和一般理论，并提出了三种简单有效的构造GQL的方法。我们将 GQL 方法应用于各种偏微分方程，并使用直接或传统方法无法轻松处理的各种具有挑战性的示例和应用来证明其在研究保界方案方面的有效性和显着优势。