# Optimizing time-spectral solution of initial-value problems

## TL;DR

This paper demonstrates that the time spectral generalized weighted residual method (GWRM), combined with sparse matrix algorithms, significantly outperforms traditional time-stepping methods in solving initial-value PDEs with higher efficiency and lower resource requirements.

## Contribution

The work introduces enhanced algorithms for temporal and spatial subdomains that improve the performance of GWRM, making it more efficient than classical methods for high-accuracy solutions.

## Key findings

- GWRM outperforms explicit and implicit methods at high accuracy levels.
- GWRM reduces CPU time and memory usage by an order of magnitude for the forced wave equation.
- Scaling of computational resources with subdomains is favorable for complex MHD equations.

## 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) is compared to that of the explicit Lax-Wendroff method and the implicit Crank-Nicolson method. 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. For time-averaged solution of the two-time-scales, forced wave equation GWRM performance exceeds the finite difference methods by an order of magnitude both in terms of CPU time and memory requirement. Favourable scaling of CPU time and memory usage with the number of temporal and spatial subdomains is demonstrated for the MHD equations.

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Source: https://tomesphere.com/paper/1704.04065