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(Generalized Wave Continuity Equation)
(Generalized Wave Continuity Equation)
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=== Generalized Wave Continuity Equation ===
 
=== Generalized Wave Continuity Equation ===
For the theory based on the WCE, stability is shown to be independent of the choice of [[TAU0]] (<math>\tau_0</math>). However, experience tell us that in the GWCE 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.  
+
For the theory based on the WCE, stability is shown to be independent of the choice of [[TAU0]] (<math>\tau_0</math>). However, experience tell us that in the GWCE a larger value of <math>\tau_0</math> tends 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.  
  
 
Kinnmark's 1986 monograph<ref name=Kinnmark3>Kinnmark, I., 1986. The Shallow Water Wave Equations: Formulation, Analysis and Application, Lecture Notes in Engineering. Springer Berlin Heidelberg, Berlin, Heidelberg. doi:10.1007/978-3-642-82646-7.</ref> does offer some analysis that backs up this empirical experience. For the consistent mass-matrix solver employing the centered <math>\theta</math> scheme it can be shown that unconditional stability is only possible if,
 
Kinnmark's 1986 monograph<ref name=Kinnmark3>Kinnmark, I., 1986. The Shallow Water Wave Equations: Formulation, Analysis and Application, Lecture Notes in Engineering. Springer Berlin Heidelberg, Berlin, Heidelberg. doi:10.1007/978-3-642-82646-7.</ref> does offer some analysis that backs up this empirical experience. For the consistent mass-matrix solver employing the centered <math>\theta</math> scheme it can be shown that unconditional stability is only possible if,

Revision as of 18:33, 9 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.

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

Wave Continuity Equation

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]. 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} [1][2]. Likely because of this fact, 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[3][4]:

  • 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 < \sqrt{6}/6} : consistent mass-matrix solved in 2-D

These conditions are for linear finite-elements (ADCIRC uses these) with even node spacings and constant bathymetry. Other conditions for quadratic finite-elements, uneven node spacings, and non-constant bathymetry are shown in Kinnmark and Gray (1985)[3].

In the explicit method case, 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).

Generalized Wave Continuity Equation

For the theory based on the WCE, 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, experience tell us that in the GWCE 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} tends 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.

Kinnmark's 1986 monograph[5] does offer some analysis that backs up this empirical experience. For the consistent mass-matrix solver employing the centered 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} scheme it can be shown that unconditional stability is only possible if,

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 \leq \tau}

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 \tau} is the linear friction coefficient. In the deep ocean when employing a quadratic drag law this could be a very restrictive requirement on 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} (e.g., 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 = C_d|u|/H \sim 0(10^{-7})} assuming 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 C_d \sim 0.001} , 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 |u| \sim 0.1} m/s, 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 H \sim 1000} m). In addition, it is in opposition to our desire to choose the recommended value, 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 \sim 1-10\tau_{max}} , for good mass conservation and dispersive properties[6]. An alternative way to view the criteria is to look for a Courant number which eliminates the criteria on 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 \tau_0} . It can be shown that this is the case in 1-D when,

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 < 4/3}

Assuming the same relationship between 1-D and 2-D as in the WCE, then the model should be stable for any 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} in 2-D when 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 < 2\sqrt{2}/3} . This means the stability requirement is slightly improved from the lumped explicit method of the WCE (unless one can choose a sufficiently small 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} to obtain greater stability).

Critique

One aspect missing from the theory above are different weighting schemes which are possibly non-centered, 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} \neq \mathrm{C00}} . Empirical evidence suggests stability of the GWCE could be improved if,

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{B00} = 0.5, \quad \mathrm{C00} = 0}

Moreover, the equations analyzed are always linearized (a requirement of the von Neumann stability analysis), thus stability may be more restricting in 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. 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. Comput. Phys. 52 (1983) 290–312.
  2. 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.
  3. 3.0 3.1 I.P.E. Kinnmark, W.G. Gray, Stability and accuracy of spatial approximations for wave equation tidal models, J. Comput. Phys. 60 (1985) 447–466. doi:10.1016/0021-9991(85)90030-0.
  4. Kinnmark, I.P.E., Gray, W.G., 1984. A Two-Dimensional Analysis of the Wave Equation Model for Finite Element Tidal Computations. Int. J. Numer. Methods Eng. 20, 369–383.
  5. Kinnmark, I., 1986. The Shallow Water Wave Equations: Formulation, Analysis and Application, Lecture Notes in Engineering. Springer Berlin Heidelberg, Berlin, Heidelberg. doi:10.1007/978-3-642-82646-7.
  6. Kolar, R.L., Westerink, J.J., Cantekin, M.E., Blain, C.A., 1994. Aspects of Nonlinear Simulations using Shallow-water Models based on the Wave Continuity Equation. Comput. Fluids 23, 523–538.