A class of high-order Runge-Kutta-Chebyshev stability polynomials

TL;DR

This paper introduces a new class of high-order Runge-Kutta-Chebyshev stability polynomials with explicit schemes that are efficient for large, moderately stiff problems, verified through numerical experiments.

Contribution

The paper presents a novel analytic form of factorized Runge-Kutta-Chebyshev stability polynomials of arbitrary order, enabling efficient explicit schemes for stiff problems.

Findings

FRKC schemes meet all linear order conditions up to order 6.

Second-order unsplit FRKC2 and higher order split schemes are efficient for stiff problems.

Numerical experiments confirm stability and efficiency of the proposed methods.

Abstract

The analytic form of a new class of factorized Runge-Kutta-Chebyshev (FRKC) stability polynomials of arbitrary order $N$ is presented. Roots of FRKC stability polynomials of degree $L=MN$ are used to construct explicit schemes comprising $L$ forward Euler stages with internal stability ensured through a sequencing algorithm which limits the internal amplification factors to $\sim L^2$. The associated stability domain scales as $M^2$ along the real axis. Marginally stable real-valued points on the interior of the stability domain are removed via a prescribed damping procedure. By construction, FRKC schemes meet all linear order conditions; for nonlinear problems at orders above 2, complex splitting or Butcher series composition methods are required. Linear order conditions of the FRKC stability polynomials are verified at orders 2, 4, and 6 in numerical experiments. Comparative studies with existing methods show the second-order unsplit FRKC2 scheme and higher order (4 and 6) split FRKCs schemes are efficient for large moderately stiff problems.