Time-Spectral Efficiency

TL;DR

This paper evaluates the efficiency of time-spectral methods, especially GWRM, for solving differential equations, highlighting their insensitivity to stiffness and chaoticity, but noting challenges with non-smooth solutions.

Contribution

It demonstrates the robustness of GWRM against stiffness and chaos and discusses limitations with steep or non-smooth solutions, suggesting areas for further research.

Findings

GWRM is insensitive to stiffness and chaoticity due to its implicit algorithm.

Explicit methods are less effective or slower for stiff and chaotic problems.

Smoothing algorithms are ineffective for non-smooth solutions.

Abstract

This study concerns the efficiency of time-spectral methods for numerical solution of differential equations. It is found that the time-spectral method GWRM demonstrates insensitivity to stiffness and chaoticity due to the implicit nature of the solution algorithm. Accuracy is thus determined primarily by numerical resolution of the solution shape. Examples of efficient solution of stiff and chaotic problems, where explicit methods fail or are significantly slower, are given. Non-smooth and partially steep solutions, however, remain challenging for convergence and accuracy. Some, earlier suggested, smoothing algorithms are shown to be ineffective in addressing this issue. Our findings underscore the need for further exploration of time-spectral approaches to enhance convergence and accuracy for steep or non-smooth solutions.