Optimal Runge–Kutta Methods for First OrderPseudospectral Operators

New Runge–Kutta methods for method of lines solution of systems of ordinary differential equations arising from discretizations of spatial derivatives in hyperbolic equations, by Chebyshev or modified Chebyshev methods, are introduced. These Runge–Kutta methods optimize the time step necessary for stable solutions, while holding dispersion and dissipation fixed. It is found that maximizing dispersion minimizes dissipation, and vice versa. Optimal methods with respect to large stability intervals on the imaginary axis and with respect to the eigenvalue spectra of the underlying pseudospectral discretizations are developed. In the latter case, stability regions are optimized to include the outliers of the spatial operators. Performance on a model problem in computational aeroacoustics is evaluated. The optimized schemes have two more function evaluations per timestep than the standard fourth order Runge–Kutta method, but allow timesteps up to 1.7 times larger. Moreover, dissipation and dispersion are reduced. c © 1999 Academic Press