  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.
