# Factorized Runge-Kutta-Chebyshev Methods

## TL;DR

This paper introduces the FRKC2 class of explicit, parallelizable Runge-Kutta-Chebyshev methods for large PDE systems, achieving high stability and accuracy with potential for extension to higher orders.

## Contribution

It presents a new factorized scheme that is easy to implement, highly stable, and suitable for large-scale parallel computations, with discussions on extending to fourth-order accuracy.

## Key findings

- Preserves 7 digits of accuracy at 16-digit precision.
- Achieves acceleration factors over 6000 compared to standard methods.
- Offers stability domains comparable or larger than existing schemes.

## Abstract

The second-order extended stability Factorized Runge-Kutta-Chebyshev (FRKC2) class of explicit schemes for the integration of large systems of PDEs with diffusive terms is presented. FRKC2 schemes are straightforward to implement through ordered sequences of forward Euler steps with complex stepsizes, and easily parallelised for large scale problems on distributed architectures. Preserving 7 digits for accuracy at 16 digit precision, the schemes are theoretically capable of maintaining internal stability at acceleration factors in excess of 6000 with respect to standard explicit Runge-Kutta methods. The stability domains have approximately the same extents as those of RKC schemes, and are a third longer than those of RKL2 schemes. Extension of FRKC methods to fourth-order, by both complex splitting and Butcher composition techniques, is discussed. A publicly available implementation of the FRKC2 class of schemes may be obtained from maths.dit.ie/frkc