TITLE:
Optimizing Time-Spectral Solution of Initial-Value Problems
AUTHORS:
J. Scheffel, K. Lindvall
KEYWORDS:
Time-Spectral, Spectral Method, GWRM, Chebyshev Polynomial, Initial-Value, Fluid Mechanics, MHD
JOURNAL NAME:
American Journal of Computational Mathematics,
Vol.8 No.1,
March
9,
2018
ABSTRACT: Time-spectral
solution of ordinary and partial differential equations is often regarded as an
inefficient approach. The associated extension of the time domain, as compared
to finite difference methods, is believed to result in uncomfortably many
numerical operations and high memory requirements. It is shown in this work
that performance is substantially enhanced by the introduction of algorithms
for temporal and spatial subdomains in combination with sparse matrix methods.
The accuracy and efficiency of the recently developed time spectral,
generalized weighted residual method (GWRM) are compared to that of the explicit Lax-Wendroff and implicit Crank-Nicolson
methods. Three initial-value PDEs are employed as model problems; the 1D Burger
equation, a forced 1D wave equation and a coupled system of 14 linearized ideal
magnetohydrodynamic (MHD) equations. It is found that the GWRM is more
efficient than the time-stepping methods at high accuracies. The advantageous
scalings Nt1.0Ns1.43 and Nt0.0Ns1.08 were obtained for CPU time and memory requirements,
respectively, with Nt and Ns denoting the number of temporal and spatial subdomains. For time-averaged
solution of the two-time-scales forced wave equation, GWRM performance exceeds
that of the finite difference methods by an
order of magnitude both in terms of CPU time and memory requirement. Favorable
subdomain scaling is demonstrated for the MHD equations, indicating a
potential for efficient solution of advanced initial-value problems in, for
example, fluid mechanics and MHD.