Explicit Runge-Kutta-Chebyshev methods of second order with monotonic stability polynomial

TL;DR

The paper tackles solving mildly stiff ODEs with explicit stabilized integrators by enforcing a monotonic, positive stability function $R_s$ over the most negative real axis, maximizing the interval length $\rho_s$ and maintaining second-order accuracy via $R_s(0)=R'_s(0)=R''_s(0)=1$. The authors construct a Chebyshev-type stability polynomial using a shifted Chebyshev form for $R'_s$, derive $R_s$ through systematic order conditions, and implement the scheme with a three-term recurrence akin to RKC, ensuring internal stability via $0\le \tilde{R}_j(x)<1$ on $[-\rho_s,0)$. They quantify the monotonicity interval, derive a small error constant $C_s = (1 - R_s^{(3)}(0))/6$, and provide practical implementation details including stage-count estimation and embedded error estimation within a stabilized-RKC framework. Numerical experiments on standard stiff problems demonstrate that the monotonic Chebyshev methods are competitive with existing second-order stabilized methods, offering improved alignment with the exponential and smaller stability-function error constants, albeit with a larger number of internal stages in some cases. The work provides a robust, monotone-stable explicit method for mildly stiff systems, with practical guidance on error estimation and stage management, supported by open-source implementation.

Abstract

A new Chebyshev-type family of stabilized explicit methods for solving mildly stiff ODEs is presented. Besides conventional conditions of order and stability we impose an additional restriction on the methods: their stability function must be monotonically increasing and positive along the largest possible interval of negative real axis. Although stability intervals of the proposed methods are smaller than those of classic Chebyshev-type methods, their stability functions are more consistent with the exponent, they have more convex stability regions and smaller error constants. These properties allow the monotonic methods to be competitive with contemporary stabilized second-order methods, as the presented results of numerical experiments demonstrate.

Figures (3)

• Figure 1: Stability polynomials for ${s=9}$. Left: comparison of monotonic and RKC method. Right: graphs of the internal stability polynomials $\tilde{R}_j$ (colored) and $R_9$ (gray).
• Figure 2: Comparison of stability regions for second-order monotonic and RKC methods.
• Figure 3: Monotonic vs RKC methods. Left: lengths of stability intervals. Right: values of $R^{(3)}_s(0)$.