Essentially optimal explicit Runge-Kutta methods with application to hyperbolic-parabolic equations

Optimal explicit Runge-Kutta methods consider more stages in order to include a particular spectrum in their stability domain and thus reduce time-step restrictions. This idea, so far used mostly for real-line spectra, is generalized to more general spectra in the form of a thin region. In thin regions the eigenvalues may extend away from the real axis into the imaginary plane. We give a direct characterization of optimal stability polynomials containing a maximal thin region and calculate these polynomials for various cases. Semi-discretizations of hyperbolic-parabolic equations are a relevant application which exhibit a thin region spectrum. As a model, linear, scalar advection-diffusion is investigated. The second-order-stabilized explicit Runge-Kutta methods derived from the stability polynomials are applied to advection-diffusion and compressible, viscous fluid dynamics in numerical experiments. Due to the stabilization the time step can be controlled solely from the hyperbolic CFL condition even in the presence of viscous fluxes.