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# Difference between revisions of "A00, B00, C00"

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== Theory == | == Theory == | ||

− | Theory is dominated by analysis of the Wave Continuity Equation (WCE), a special case of the Generalized Wave Continuity Equation (GWCE) where the [[TAU0]] parameter is equal to the linear friction coefficient. In what has been determined to be a third-order method centered in time<ref name=Foreman>M.G.G. Foreman, An Analysis of the “Wave Equation” Model for Finite Element Tidal Computations, J. Computational Physics. 52 (1983) 290–312.</ref>, which was first introduced by Lynch and Gray (1979)<ref name=Lynch>D.R. Lynch, W.G. Gray, A Wave Equation Model for Finite Element Tidal Computations, Computers & Fluids. 7 (1979) 207–228.</ref>, the choice of A00, B00, C00 is reduced to depend on a single parameter, <math>\theta</math>: | + | Theory is dominated by analysis of the Wave Continuity Equation (WCE), a special case of the Generalized Wave Continuity Equation (GWCE) where the [[TAU0]] parameter is equal to the linear friction coefficient. In what has been determined to be a third-order accurate method centered in time<ref name=Foreman>M.G.G. Foreman, An Analysis of the “Wave Equation” Model for Finite Element Tidal Computations, J. Computational Physics. 52 (1983) 290–312.</ref>, which was first introduced by Lynch and Gray (1979)<ref name=Lynch>D.R. Lynch, W.G. Gray, A Wave Equation Model for Finite Element Tidal Computations, Computers & Fluids. 7 (1979) 207–228.</ref>, the choice of A00, B00, C00 is reduced to depend on a single parameter, <math>\theta</math>: |

<math>\mathrm{A00} = \mathrm{C00} = 0.5\theta, \quad \mathrm{B00} = 1-\theta</math> | <math>\mathrm{A00} = \mathrm{C00} = 0.5\theta, \quad \mathrm{B00} = 1-\theta</math> | ||

− | In other words, k+1 and k-1 weightings are | + | In other words, k+1 and k-1 weightings are chosen to be equal. It would however not appear that any restriction other than the requirement that A00, B00, C00 must sum to 1 is necessary to obtain second-order accuracy<ref name=Foreman></ref>. |

− | Essentially all studies, whether in 1-D <ref name=Foreman></ref><ref name=Lynch></ref> or ultimately 2-D with Coriolis and other terms (Kinnemark), found that unconditional stability is achieved with the prescription of <math>\theta \geq 0.5</math>. Likely in response to these findings, the typical choice for ADCIRC has become <math>\theta = 0.7</math>, i.e., <math>\mathrm{A00} = \mathrm{C00} = 0.35, \mathrm{B00} = 0.30</math> as noted above. | + | Essentially all studies, whether in 1-D<ref name=Foreman></ref><ref name=Lynch></ref> or ultimately 2-D with Coriolis and other terms (Kinnemark), found that unconditional stability is achieved with the prescription of <math>\theta \geq 0.5</math>. Likely in response to these findings, the typical choice for ADCIRC has become <math>\theta = 0.7</math>, i.e., <math>\mathrm{A00} = \mathrm{C00} = 0.35, \mathrm{B00} = 0.30</math> as noted above. Different values of <math>\theta</math> may be motivated by the following expression for optimal dispersive accuracy for the consistent mass-matrix solver<ref name=Foreman></ref>: |

+ | |||

+ | <math>\theta = \frac{1}{6}\left(1 + \frac{1}{Cr^2}\right)</math> | ||

+ | |||

+ | where <math>Cr = \sqrt{gh}\Delta t/\Delta x</math> is the Courant number based on the linear gravity wave speed. | ||

A purely explicit method (<math>\theta = 0</math>) for the WCE is found to be stable under the following conditions: | A purely explicit method (<math>\theta = 0</math>) for the WCE is found to be stable under the following conditions: | ||

− | * <math> | + | * <math>Cr < 1</math> : lumped mass-matrix solved in 1-D |

− | * <math> | + | * <math>Cr < \sqrt{3}/3</math>: consistent mass-matrix solved in 1-D |

− | * <math> | + | * <math>Cr < \sqrt{2}/2</math>: lumped mass-matrix solved in 2-D. |

− | * <math> | + | * <math>Cr < ??</math>: consistent mass-matrix solved in 2-D. |

− | As one can see, stability (and dispersive | + | As one can see, stability (and optimal dispersive accuracy<ref name=Foreman></ref>) is superior for the lumped mass-matrix solver versus the consistent mass-matrix solver, hence the lumped solver should always be chosen when employing an explicit method (see [[IM]] parameter for setting the solver type). |

== Critique == | == Critique == | ||

− | Since theory is based on the WCE instead of the GWCE, stability | + | Since theory is based on the WCE instead of the GWCE, stability is shown to be independent of the choice of [[TAU0]] (<math>\tau_0</math>). However, from experience a larger value of <math>\tau_0</math> always tend to be more unstable than a smaller value. This makes sense since the behavior of the equations will become more and more similar to the Primitive Continuity Equation with greater <math>\tau_0</math>, which is responsible for 2Δx instabilities - the motive for using the GWCE in the finite-element method. Further analysis of the GWCE is required to determine stability with respect to the choice of <math>\tau_0</math> and different weighting factors (possibly non-centered, i.e., <math>\mathrm{A00} \neq \mathrm{C00}</math>). |

− | Moreover, the equations | + | Moreover, the equations analyzed are always linearized (a requirement of the [[von Neumann stability analysis]]), thus unconditional stability may not be possible for the semi-implicit method when simulating real-world problems, especially those with fine grid sizes and where nonlinearities are non-trivial. In such cases where it is not possible to achieve time steps more than twice that possible with an explicit method it becomes preferable to employ the explicit lumped mass-matrix solver since it is computationally twice as fast per time step solve. |

== References == | == References == | ||

<references /> | <references /> |

## Revision as of 05:57, 7 July 2019

**A00, B00, C00** are the weighting factors (at time levels k+1, k, k-1, respectively) for the free surface and boundary fluxes in the GWCE, and must sum to 1. Most critically, the weighting factors are used in the discretization of the linear gravity wave (pressure gradient) term and are responsible for determining the inherent implicity (impacting solution stability), in addition to order of accuracy and dispersive characteristics of the numerical method.

## Contents

## Typical Values

If the consistent mass-matrix solver is chosen (see IM parameter) then a semi-implicit method is possible and encouraged. In this case the most common choice for the weighting factors are:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{A00} = \mathrm{C00} = 0.35, \quad \mathrm{B00} = 0.30}**

If the lumped mass-matrix solver is chosen then only an explicit method is possible (the weighting A00 must be zero as no matrix solve is conducted), and the weighting factors that are typically chosen become simply:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{A00} = \mathrm{C00} = 0, \quad \mathrm{B00} = 1}**

## Theory

Theory is dominated by analysis of the Wave Continuity Equation (WCE), a special case of the Generalized Wave Continuity Equation (GWCE) where the TAU0 parameter is equal to the linear friction coefficient. In what has been determined to be a third-order accurate method centered in time^{[1]}, which was first introduced by Lynch and Gray (1979)^{[2]}, the choice of A00, B00, C00 is reduced to depend on a single parameter, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta}**
:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{A00} = \mathrm{C00} = 0.5\theta, \quad \mathrm{B00} = 1-\theta}**

In other words, k+1 and k-1 weightings are chosen to be equal. It would however not appear that any restriction other than the requirement that A00, B00, C00 must sum to 1 is necessary to obtain second-order accuracy^{[1]}.
Essentially all studies, whether in 1-D^{[1]}^{[2]} or ultimately 2-D with Coriolis and other terms (Kinnemark), found that unconditional stability is achieved with the prescription of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta \geq 0.5}**
. Likely in response to these findings, the typical choice for ADCIRC has become **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta = 0.7}**
, i.e., **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{A00} = \mathrm{C00} = 0.35, \mathrm{B00} = 0.30}**
as noted above. Different values of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta}**
may be motivated by the following expression for optimal dispersive accuracy for the consistent mass-matrix solver^{[1]}:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta = \frac{1}{6}\left(1 + \frac{1}{Cr^2}\right)}**

where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Cr = \sqrt{gh}\Delta t/\Delta x}**
is the Courant number based on the linear gravity wave speed.

A purely explicit method (**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta = 0}**
) for the WCE is found to be stable under the following conditions:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Cr < 1}**: lumped mass-matrix solved in 1-D**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Cr < \sqrt{3}/3}**: consistent mass-matrix solved in 1-D**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Cr < \sqrt{2}/2}**: lumped mass-matrix solved in 2-D.**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Cr < ??}**: consistent mass-matrix solved in 2-D.

As one can see, stability (and optimal dispersive accuracy^{[1]}) is superior for the lumped mass-matrix solver versus the consistent mass-matrix solver, hence the lumped solver should always be chosen when employing an explicit method (see IM parameter for setting the solver type).

## Critique

Since theory is based on the WCE instead of the GWCE, stability is shown to be independent of the choice of TAU0 (**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau_0}**
). However, from experience a larger value of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau_0}**
always tend to be more unstable than a smaller value. This makes sense since the behavior of the equations will become more and more similar to the Primitive Continuity Equation with greater **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau_0}**
, which is responsible for 2Δx instabilities - the motive for using the GWCE in the finite-element method. Further analysis of the GWCE is required to determine stability with respect to the choice of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{A00} \neq \mathrm{C00}}**
).

Moreover, the equations analyzed are always linearized (a requirement of the von Neumann stability analysis), thus unconditional stability may not be possible for the semi-implicit method when simulating real-world problems, especially those with fine grid sizes and where nonlinearities are non-trivial. In such cases where it is not possible to achieve time steps more than twice that possible with an explicit method it becomes preferable to employ the explicit lumped mass-matrix solver since it is computationally twice as fast per time step solve.

## References

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}^{1.4}M.G.G. Foreman, An Analysis of the “Wave Equation” Model for Finite Element Tidal Computations, J. Computational Physics. 52 (1983) 290–312. - ↑
^{2.0}^{2.1}D.R. Lynch, W.G. Gray, A Wave Equation Model for Finite Element Tidal Computations, Computers & Fluids. 7 (1979) 207–228.