Stephen O'Sullivan

Stephen O'Sullivan

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

Gavin Lee Goodship, Luis Miralles-Pechuan, Stephen O'Sullivan

Extended Stability Runge-Kutta (ESRK) methods are crucial for solving large-scale computational problems in science and engineering, including weather forecasting, aerodynamic analysis, and complex biological modelling. However, balancing accuracy, stability, and computational efficiency remains challenging, particularly for high-order, low-storage schemes. This study introduces a hybrid Genetic Algorithm (GA) and Reinforcement Learning (RL) approach for automated heuristic discovery, optimising low-storage ESRK methods. Unlike traditional approaches that rely on manually designed heuristics or exhaustive numerical searches, our method leverages GA-driven mutations for search-space exploration and an RL-inspired state transition mechanism to refine heuristic selection dynamically. This enables systematic parameter reduction, preserving fourth-order accuracy while significantly improving computational efficiency.The proposed GA-RL heuristic optimisation framework is validated through rigorous testing on benchmark problems, including the 1D and 2D Brusselator systems and the steady-state Navier-Stokes equations. The best-performing heuristic achieves a 25\% reduction in IPOPT runtime compared to traditional ESRK optimisation processes while maintaining numerical stability and accuracy. These findings demonstrate the potential of adaptive heuristic discovery to improve resource efficiency in high-fidelity simulations and broaden the applicability of low-storage Runge-Kutta methods in real-world computational fluid dynamics, physics simulations, and other demanding fields. This work establishes a new paradigm in heuristic optimisation for numerical methods, opening pathways for further exploration using Deep RL and AutoML-based heuristic search

Stephen O'Sullivan

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 Péclet numbers are provided at second, fourth, and sixth orders, with nonlinear reaction terms treated by complex splitting techniques above second order.