A novel optimization procedure for the generation of stability polynomials of stabilized explicit Runge-Kutta method 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：针对稳定显式龙格-库塔方法稳定性多项式的生成，本文提出了一种新型优化流程。该方法针对双曲型偏微分方程的半离散化设计，能够实现包含百级以上阶段的稳定性多项式优化。这些高次稳定性多项式的潜在应用场景包括：非均匀加密网格及不同波速条件下具有局部变化特征速度的问题。为证明稳定性多项式的适用性，我们构建了2N存储的多阶段龙格-库塔方法，当应用于具有光滑解的线性和非线性双曲型偏微分方程组时，该方法可匹配其二阶精度设计。我们特别优化了算法结构以降低舍入误差的放大效应——这一因素在多阶段方法中已成为重要考量。