Stability analysis of high-order finite-difference discretizations of the linearized forward-speed seakeeping problem

Mostafa Amini-Afshar*, Harry B. Bingham, William D. Henshaw

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review

Abstract

A high-order finite-difference method solution of the linearized, potential flow, seakeeping problem for a ship at steady forward speed was recently presented by Amini-Afshar et al. [1,2]. In this paper, we provide a detailed matrix-based eigenvalue stability analysis of this model, highlighting the sources of instability and the effects of possible remedies. In particular, we illustrate how both boundary treatment and grid stretching are important factors which are not typically captured by a von Neumann-type analysis. The new analysis shows that when grid stretching is used together with centered finite difference schemes, the method is generally unstable. The source of the instability can in some cases be traced to an effective downwinding of the convective terms. Stable solutions can be obtained either by introducing upwind-biased schemes for computing the convective derivatives on the free-surface, or by application of a mild filter at each time-step. A second source of instability is associated with the treatment of the convective derivatives of the free-surface elevation at points close to the domain boundaries. Here it is necessary to consider whether the surrounding fluid points lie in an upwind or a downwind direction. For upwinded points, ordinary one-sided differencing can be used, but for downwinded points we instead impose a Neumann-type boundary condition derived from the body and free-surface boundary conditions. As an example application to complement those already given in [1], [2], the method is applied to solve the steady wave resistance problem and comparison is made to reference solutions for a two-dimensional floating cylinder and a submerged sphere. Estimates of the wave resistance of the Wigley hull are also compared with experimental measurements.
Original languageEnglish
Article number101913
JournalApplied Ocean Research
Volume92
Number of pages16
ISSN0141-1187
DOIs
Publication statusPublished - 2019

Keywords

  • Stability analysis
  • Wave resistance
  • Neumann–Kelvin linearization
  • Finite difference
  • Overlapping grids
  • Forward speed

Cite this

@article{dd3cbf9b72a0481b9e2867103dc6a032,
title = "Stability analysis of high-order finite-difference discretizations of the linearized forward-speed seakeeping problem",
abstract = "A high-order finite-difference method solution of the linearized, potential flow, seakeeping problem for a ship at steady forward speed was recently presented by Amini-Afshar et al. [1,2]. In this paper, we provide a detailed matrix-based eigenvalue stability analysis of this model, highlighting the sources of instability and the effects of possible remedies. In particular, we illustrate how both boundary treatment and grid stretching are important factors which are not typically captured by a von Neumann-type analysis. The new analysis shows that when grid stretching is used together with centered finite difference schemes, the method is generally unstable. The source of the instability can in some cases be traced to an effective downwinding of the convective terms. Stable solutions can be obtained either by introducing upwind-biased schemes for computing the convective derivatives on the free-surface, or by application of a mild filter at each time-step. A second source of instability is associated with the treatment of the convective derivatives of the free-surface elevation at points close to the domain boundaries. Here it is necessary to consider whether the surrounding fluid points lie in an upwind or a downwind direction. For upwinded points, ordinary one-sided differencing can be used, but for downwinded points we instead impose a Neumann-type boundary condition derived from the body and free-surface boundary conditions. As an example application to complement those already given in [1], [2], the method is applied to solve the steady wave resistance problem and comparison is made to reference solutions for a two-dimensional floating cylinder and a submerged sphere. Estimates of the wave resistance of the Wigley hull are also compared with experimental measurements.",
keywords = "Stability analysis, Wave resistance, Neumann–Kelvin linearization, Finite difference, Overlapping grids, Forward speed",
author = "Mostafa Amini-Afshar and Bingham, {Harry B.} and Henshaw, {William D.}",
year = "2019",
doi = "10.1016/j.apor.2019.101913",
language = "English",
volume = "92",
journal = "Applied Ocean Research",
issn = "0141-1187",
publisher = "Pergamon Press",

}

Stability analysis of high-order finite-difference discretizations of the linearized forward-speed seakeeping problem. / Amini-Afshar, Mostafa; Bingham, Harry B.; Henshaw, William D.

In: Applied Ocean Research, Vol. 92, 101913, 2019.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - Stability analysis of high-order finite-difference discretizations of the linearized forward-speed seakeeping problem

AU - Amini-Afshar, Mostafa

AU - Bingham, Harry B.

AU - Henshaw, William D.

PY - 2019

Y1 - 2019

N2 - A high-order finite-difference method solution of the linearized, potential flow, seakeeping problem for a ship at steady forward speed was recently presented by Amini-Afshar et al. [1,2]. In this paper, we provide a detailed matrix-based eigenvalue stability analysis of this model, highlighting the sources of instability and the effects of possible remedies. In particular, we illustrate how both boundary treatment and grid stretching are important factors which are not typically captured by a von Neumann-type analysis. The new analysis shows that when grid stretching is used together with centered finite difference schemes, the method is generally unstable. The source of the instability can in some cases be traced to an effective downwinding of the convective terms. Stable solutions can be obtained either by introducing upwind-biased schemes for computing the convective derivatives on the free-surface, or by application of a mild filter at each time-step. A second source of instability is associated with the treatment of the convective derivatives of the free-surface elevation at points close to the domain boundaries. Here it is necessary to consider whether the surrounding fluid points lie in an upwind or a downwind direction. For upwinded points, ordinary one-sided differencing can be used, but for downwinded points we instead impose a Neumann-type boundary condition derived from the body and free-surface boundary conditions. As an example application to complement those already given in [1], [2], the method is applied to solve the steady wave resistance problem and comparison is made to reference solutions for a two-dimensional floating cylinder and a submerged sphere. Estimates of the wave resistance of the Wigley hull are also compared with experimental measurements.

AB - A high-order finite-difference method solution of the linearized, potential flow, seakeeping problem for a ship at steady forward speed was recently presented by Amini-Afshar et al. [1,2]. In this paper, we provide a detailed matrix-based eigenvalue stability analysis of this model, highlighting the sources of instability and the effects of possible remedies. In particular, we illustrate how both boundary treatment and grid stretching are important factors which are not typically captured by a von Neumann-type analysis. The new analysis shows that when grid stretching is used together with centered finite difference schemes, the method is generally unstable. The source of the instability can in some cases be traced to an effective downwinding of the convective terms. Stable solutions can be obtained either by introducing upwind-biased schemes for computing the convective derivatives on the free-surface, or by application of a mild filter at each time-step. A second source of instability is associated with the treatment of the convective derivatives of the free-surface elevation at points close to the domain boundaries. Here it is necessary to consider whether the surrounding fluid points lie in an upwind or a downwind direction. For upwinded points, ordinary one-sided differencing can be used, but for downwinded points we instead impose a Neumann-type boundary condition derived from the body and free-surface boundary conditions. As an example application to complement those already given in [1], [2], the method is applied to solve the steady wave resistance problem and comparison is made to reference solutions for a two-dimensional floating cylinder and a submerged sphere. Estimates of the wave resistance of the Wigley hull are also compared with experimental measurements.

KW - Stability analysis

KW - Wave resistance

KW - Neumann–Kelvin linearization

KW - Finite difference

KW - Overlapping grids

KW - Forward speed

U2 - 10.1016/j.apor.2019.101913

DO - 10.1016/j.apor.2019.101913

M3 - Journal article

VL - 92

JO - Applied Ocean Research

JF - Applied Ocean Research

SN - 0141-1187

M1 - 101913

ER -