On the accuracy and applicability of a new implicit Taylor method and the high-order spectral method on steady nonlinear waves

Mathias Klahn*, Per A. Madsen, David R. Fuhrman

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review

Abstract

This paper presents an investigation and discussion of the accuracy and applicability of an implicit Taylor (IT) method versus the classical higher-order spectral (HOS) method when used to simulate two-dimensional regular waves. This comparison is relevant, because the HOS method is in fact an explicit perturbation solution of the IT formulation. First, we consider the Dirichlet–Neumann problem of determining the vertical velocity at the free surface given the surface elevation and the surface potential. For this problem, we conclude that the IT method is significantly more accurate than the HOS method when using the same truncation order, M, and spatial resolution, N, and is capable of dealing with steeper waves than the HOS method. Second, we focus on the problem of integrating the two methods in time. In this connection, it turns out that the IT method is less robust than the HOS method for similar truncation orders. We conclude that the IT method should be restricted to M = 4, while the HOS method can be used with M ≤ 8. We systematically compare these two options and finally establish the best achievable accuracy of the two methods as a function of the wave steepness and the water depth.
Original languageEnglish
Article number20200436
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume476
Issue number2243
Number of pages27
ISSN1364-5021
DOIs
Publication statusPublished - 2020

Keywords

  • Nonlinear water waves
  • Spectral methods
  • Accuracy
  • Stability
  • Range of applicability

Fingerprint Dive into the research topics of 'On the accuracy and applicability of a new implicit Taylor method and the high-order spectral method on steady nonlinear waves'. Together they form a unique fingerprint.

Cite this