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：本文设计了一种生成稳定显式龙格-库塔方法稳定性多项式的新型优化流程。针对双曲型偏微分方程的半离散化，本文发展的方法可实现具有超过百个阶段的稳定性多项式的优化。这些高阶稳定性多项式的潜在应用场景包括非均匀细化网格及不同波速中出现的局部变特征速度问题。为验证稳定性多项式的适用性，我们构造了2N存储的多阶段龙格-库塔方法，在应用于具有光滑解的线性和非线性双曲型偏微分方程时，其能匹配设计中的二阶精度。这些方法被构造为可减少舍入误差的放大效应，而该问题对于此类多阶段方法尤为重要。