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 格式能够在空间离散方法保持不变的情况下实现多速率时间积分，使其易于在采用线法框架的现有代码中实施。研究表明，尽管动态重标标记变量会带来额外开销，且非线性稳定性特性受到限制，但相较于一系列先进的龙格-库塔方法仍可实现加速效果。该方法在粘性与无粘对流主导可压缩流动的多种模拟设置中均展现出有效性，我们为此提供了可复现性代码库。此外，我们针对配对显式龙格-库塔格式的非线性稳定性特性进行了深入研究，重点探讨了因底层空间离散格式单调性条件破坏所带来的局限性。同时，我们提出了一种新颖的方法，用于估计稳定性多项式优化所需的大型雅可比矩阵相关特征值。