Abstract
The second-order extended stability Factorized Runge–Kutta–Chebyshev (FRKC2) explicit schemes for the integration of large systems of PDEs with diffusive terms are presented. The schemes are simple 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 for acceleration factors in excess of 6000 with respect to standard explicit Runge-Kutta methods. The extent of the stability domain is approximately the same as that of RKC schemes, and a third longer than in the case of RKL2 schemes. Extension of FRKC methods to fourth-order, by both complex splitting and Butcher composition techniques, is also discussed.A publicly available implementation of FRKC2 schemes may be obtained from maths.dit.ie/frkc
Highlights
Factorized Runge–Kutta–Chebyshev (FRKC) methods are well suited to the numerical integration of problems where diffusion limits the efficiency of standard explicit techniques
The main use of extended stability Runge–Kutta (ESRK) methods is to fill the gap between unconditionally stable but operationally complex implicit methods, and implemented explicit schemes which suffer from stability constraints for stiff problems
ESRK methods are useful for problems involving diffusion, where the work required by standard explicit techniques goes as the inverse square of the mesh spacing, while for extended stability methods it goes as the inverse mesh spacing
Summary
Factorized Runge–Kutta–Chebyshev (FRKC) methods are well suited to the numerical integration of problems where diffusion limits the efficiency of standard explicit techniques. Such systems of PDEs may be presented as semi-discrete ordinary differential equations of the form w = f (t, w). The objective is to determine a closed form for the polynomial such that the extent of the stability domain along the negative real axis β is as great as possible It is shown in [16] that the FRKC stability polynomial of rank N , and degree L, is a sum of Chebyshev polynomials of the first kind given by.
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