In this paper, we apply the Paired-Explicit Runge-Kutta (P-ERK) schemes by Vermeire et. al. (2019, 2022) to dynamically partitioned systems arising from adaptive mesh refinement. The P-ERK schemes enable multirate time-integration with no changes in the spatial discretization methodology, making them readily implementable in existing codes that employ a method-of-lines approach. We show that speedup compared to a range of state of the art Runge-Kutta methods can be realized, despite additional overhead due to the dynamic re-assignment of flagging variables and restricting nonlinear stability properties. The effectiveness of the approach is demonstrated for a range of simulation setups for viscous and inviscid convection-dominated compressible flows for which we provide a reproducibility repository. In addition, we perform a thorough investigation of the nonlinear stability properties of the Paired-Explicit Runge-Kutta schemes regarding limitations due to the violation of monotonicity properties of the underlying spatial discretization. Furthermore, we present a novel approach for estimating the relevant eigenvalues of large Jacobians required for the optimization of stability polynomials.

翻译：本文应用Vermeire等人(2019,2022)提出的配对显式龙格-库塔(P-ERK)格式，处理由自适应网格细化产生的动态分区系统。P-ERK格式在不改变空间离散方法的前提下实现多速率时间积分，使其易于集成到采用方法线方法的现有代码中。研究表明，尽管动态重新分配标记变量和限制非线性稳定性特性会带来额外开销，但相较于一系列先进的龙格-库塔方法仍可实现加速。该方法在粘性和无粘对流主导可压缩流动的多种模拟设置中展现了有效性，本文附有可复现性代码库。此外，我们对配对显式龙格-库塔格式的非线性稳定性特性展开了深入研究，重点关注因违背底层空间离散单调性条件而产生的局限性。进一步地，我们提出一种新方法，用于估算针对稳定性多项式优化所需的巨型雅可比矩阵的相关特征值。