Title: Runge-Kutta-Gegenbauer methods for advection-diffusion problems,
Abstract: In this paper, Runge-Kutta-Gegenbauer (RKG) stability polynomials of
arbitrarily high order of accuracy are introduced in closed form. The stability
domain of RKG polynomials extends in the the real direction with the square of
polynomial degree, and in the imaginary direction as an increasing function of
Gegenbauer parameter. Consequently, the polynomials are naturally suited to the
construction of high order stabilized Runge-Kutta (SRK) methods for systems of
PDEs of mixed hyperbolic-parabolic type.
We present SRK methods composed of $L$ ordered forward Euler stages, with
complex-valued stepsizes derived from the roots of RKG stability polynomials of
degree $L$. Internal stability is maintained at large stage number through an
ordering algorithm which limits internal amplification factors to $10 L^2$.
Test results for mildly stiff nonlinear advection-diffusion-reaction problems
with moderate ($\lesssim 1$) mesh Péclet numbers are provided at second,
fourth, and sixth orders, with nonlinear reaction terms treated by complex
splitting techniques above second order.