A novel optimization procedure for the generation of stability polynomials of stabilized explicit Runge-Kutta methods is devised. Intended for semidiscretizations of hyperbolic partial differential equations, the herein developed approach allows the optimization of stability polynomials with more than hundred stages. A potential application of these high degree stability polynomials are problems with locally varying characteristic speeds as found in non-uniformly refined meshes and different wave speeds. To demonstrate the applicability of the stability polynomials we construct 2N storage many-stage Runge-Kutta methods that match their designed second order of accuracy when applied to a range of linear and nonlinear hyperbolic PDEs with smooth solutions. The methods are constructed to reduce the amplification of round off errors which becomes a significant concern for these many-stage methods.

翻译：针对显式稳定Runge-Kutta方法稳定性多项式生成，提出了一种新型优化流程。本文发展的方法专为双曲型偏微分方程的半离散化设计，可对超过百级阶段的稳定性多项式进行优化。这些高阶稳定性多项式的潜在应用场景包括：非均匀加密网格与不同波速条件下出现的局部特征速度变化问题。为验证稳定性多项式的适用性，我们构造了2N存储型多阶段Runge-Kutta方法，该方法在应用于一系列具有光滑解的线性和非线性双曲型偏微分方程时，可保持其二阶精度。方法构造过程特别注重减小舍入误差的放大效应——这对多阶段方法而言是需重点关注的显著问题。