Runge-Kutta-Gegenbauer explicit methods for advection-diffusion problems

TL;DR

This paper introduces high-order Runge-Kutta-Gegenbauer stability polynomials for explicit methods, enabling efficient and stable solutions for mixed hyperbolic-parabolic PDEs with improved stability domains.

Contribution

It presents a novel class of stability polynomials with arbitrarily high order, extending stability domains and constructing stabilized Runge-Kutta methods for complex PDE systems.

Findings

High-order RKG polynomials extend stability domains in real and imaginary directions.

Constructed SRK methods with complex stepsizes derived from RKG roots.

Test results show effectiveness for mildly stiff 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) explicit 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 P\'eclet numbers are provided at second, fourth, and sixth orders, with nonlinear reaction terms treated by complex splitting techniques above second order.