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：本文提出了一种新型优化程序，用于生成稳定显式龙格-库塔方法的稳定性多项式。针对双曲型偏微分方程的半离散化问题，本文所发展的方法可实现对超过百个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及不同波速问题中存在的局部变化特征速度。为验证稳定性多项式的适用性，我们构建了2N存储多阶段龙格-库塔方法，该方法在应用于含光滑解的线性和非线性双曲型偏微分方程时，可达到设计的二阶精度。所构建的方法旨在减小舍入误差的放大效应——该问题在多阶段方法中尤为显著。