We introduce effective splitting methods for implementing optimization-based limiters to enforce the invariant domain in gas dynamics in high order accurate numerical schemes. The key ingredients include an easy and efficient explicit formulation of the projection onto the invariant domain set, and also proper applications of the classical Douglas-Rachford splitting and its more recent extension Davis-Yin splitting. Such an optimization-based approach can be applied to many numerical schemes to construct high order accurate, globally conservative, and invariant-domain-preserving schemes for compressible flow equations. As a demonstration, we apply it to high order discontinuous Galerkin schemes and test it on demanding benchmarks to validate the robustness and performance of both $\ell^1$-norm minimization limiter and $\ell^2$-norm minimization limiter.

翻译：本文介绍了在高阶精确数值格式中实施基于优化的限制器以保持气体动力学不变域的有效分裂方法。关键要素包括：不变域集合上投影的简便高效显式表述，以及经典Douglas-Rachford分裂及其最新扩展Davis-Yin分裂的恰当应用。这种基于优化的方法可应用于多种数值格式，为可压缩流动方程构建高阶精确、全局守恒且保持不变域的数值格式。作为示例，我们将其应用于高阶间断Galerkin格式，并通过具有挑战性的基准测试验证了$\ell^1$-范数最小化限制器和$\ell^2$-范数最小化限制器的鲁棒性与性能。