A00, B00, C00 - Revision history

Taylorgasher: Added category

2020-05-17T16:48:47Z

Added category

← Older revision		Revision as of 12:48, 17 May 2020
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	== References ==		== References ==
	<references />		<references />

			[[Category:Numerics]]

Taylorgasher at 16:48, 17 May 2020

2020-05-17T16:48:04Z

← Older revision		Revision as of 12:48, 17 May 2020
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	thus removing the restrictive condition in the centered-scheme (<math>\tau_0 \leq \Delta t</math>), instead requiring only that <math>\tau_0</math> is small enough for the given time step <math>\Delta t</math>.		thus removing the restrictive condition in the centered-scheme (<math>\tau_0 \leq \Delta t</math>), instead requiring only that <math>\tau_0</math> is small enough for the given time step <math>\Delta t</math>.

	Moreover, the equations analyzed are always linearized (a requirement of the ~~von Neumann stability analysis~~[https://en.wikipedia.org/wiki/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<ref>S. Tanaka, S. Bunya, J.J. Westerink, C. Dawson, R.A. Luettich, Scalability of an Unstructured Grid Continuous Galerkin Based Hurricane Storm Surge Model, J. Sci. Comput. 46 (2011) 329–358. doi:10.1007/s10915-010-9402-1</ref>.		Moreover, the equations analyzed are always linearized (a requirement of the [https://en.wikipedia.org/wiki/Von_Neumann_stability_analysis 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<ref>S. Tanaka, S. Bunya, J.J. Westerink, C. Dawson, R.A. Luettich, Scalability of an Unstructured Grid Continuous Galerkin Based Hurricane Storm Surge Model, J. Sci. Comput. 46 (2011) 329–358. doi:10.1007/s10915-010-9402-1</ref>.

	== References ==		== References ==
	<references />		<references />

Wpringle: /* Critique */

2020-01-22T20:17:18Z

Critique

← Older revision		Revision as of 16:17, 22 January 2020
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	<math>\mathrm{A00} = \mathrm{B00} = 0.5\theta, \quad \mathrm{C00} = 1-\theta</math>		<math>\mathrm{A00} = \mathrm{B00} = 0.5\theta, \quad \mathrm{C00} = 1-\theta</math>

	~~This~~ is unconditionally stable under the following conditions for the consistent mass-matrix solver,		which is unconditionally stable under the following conditions for the consistent mass-matrix solver,

	<math>2/3 \leq \theta \leq 1, \quad \tau_0\Delta t \leq (16/3)(3\theta/2-1)</math>		<math>2/3 \leq \theta \leq 1, \quad \tau_0\Delta t \leq (16/3)(3\theta/2-1)</math>

	~~which removes~~ the restrictive condition in the centered-scheme (<math>\tau_0 \leq \Delta t</math>), instead requiring only that <math>\tau_0</math> is small enough for the given time step <math>\Delta t</math>.		thus removing the restrictive condition in the centered-scheme (<math>\tau_0 \leq \Delta t</math>), instead requiring only that <math>\tau_0</math> is small enough for the given time step <math>\Delta t</math>.

	Moreover, the equations analyzed are always linearized (a requirement of the von Neumann stability analysis[https://en.wikipedia.org/wiki/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<ref>S. Tanaka, S. Bunya, J.J. Westerink, C. Dawson, R.A. Luettich, Scalability of an Unstructured Grid Continuous Galerkin Based Hurricane Storm Surge Model, J. Sci. Comput. 46 (2011) 329–358. doi:10.1007/s10915-010-9402-1</ref>.		Moreover, the equations analyzed are always linearized (a requirement of the von Neumann stability analysis[https://en.wikipedia.org/wiki/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<ref>S. Tanaka, S. Bunya, J.J. Westerink, C. Dawson, R.A. Luettich, Scalability of an Unstructured Grid Continuous Galerkin Based Hurricane Storm Surge Model, J. Sci. Comput. 46 (2011) 329–358. doi:10.1007/s10915-010-9402-1</ref>.

Wpringle: /* Critique */

2020-01-22T20:16:34Z

Critique

← Older revision		Revision as of 16:16, 22 January 2020
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	=== Critique ===		=== Critique ===
	One aspect missing from the theory above are different weighting schemes which are possibly non-centered, i.e., <math>\mathrm{A00} \neq \mathrm{C00}</math>. Pringle et al.<ref>Pringle t al., Global Ocean-to-Coastal Storm Tide Modeling in ADCIRC v55: Unstructured Mesh Design, in preparation (2020)</ref> shows that stability of the GWCE is improved if,		One aspect missing from the theory above are different weighting schemes which are possibly non-centered, i.e., <math>\mathrm{A00} \neq \mathrm{C00}</math>. Pringle et al.<ref>Pringle et al., Global Ocean-to-Coastal Storm Tide Modeling in ADCIRC v55: Unstructured Mesh Design, in preparation (2020)</ref> shows that stability of the GWCE is improved if,

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

Wpringle: /* Critique */

2020-01-22T20:16:20Z

Critique

← Older revision		Revision as of 16:16, 22 January 2020
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	=== Critique ===		=== Critique ===
	One aspect missing from the theory above are different weighting schemes which are possibly non-centered, i.e., <math>\mathrm{A00} \neq \mathrm{C00}</math>. Pringle et al.<ref>Pringle t al., Global Ocean-to-Coastal Storm Tide Modeling in ADCIRC v55: Unstructured Mesh Design, in preparation (~~2019~~)</ref> shows that stability of the GWCE is improved if,		One aspect missing from the theory above are different weighting schemes which are possibly non-centered, i.e., <math>\mathrm{A00} \neq \mathrm{C00}</math>. Pringle et al.<ref>Pringle t al., Global Ocean-to-Coastal Storm Tide Modeling in ADCIRC v55: Unstructured Mesh Design, in preparation (2020)</ref> shows that stability of the GWCE is improved if,

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

Wpringle: /* Critique */

2020-01-22T20:14:19Z

Critique

← Older revision		Revision as of 16:14, 22 January 2020
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	<math>2/3 \leq \theta \leq 1, \quad \tau_0\Delta t \leq (16/3)(3\theta/2-1)</math>		<math>2/3 \leq \theta \leq 1, \quad \tau_0\Delta t \leq (16/3)(3\theta/2-1)</math>

	which ~~is much less~~ restrictive ~~than~~ the ~~condition~~ <math>\tau_0 \leq \Delta t</math>.		which removes the restrictive condition in the centered-scheme (<math>\tau_0 \leq \Delta t</math>), instead requiring only that <math>\tau_0</math> is small enough for the given time step <math>\Delta t</math>.

	Moreover, the equations analyzed are always linearized (a requirement of the von Neumann stability analysis[https://en.wikipedia.org/wiki/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<ref>S. Tanaka, S. Bunya, J.J. Westerink, C. Dawson, R.A. Luettich, Scalability of an Unstructured Grid Continuous Galerkin Based Hurricane Storm Surge Model, J. Sci. Comput. 46 (2011) 329–358. doi:10.1007/s10915-010-9402-1</ref>.		Moreover, the equations analyzed are always linearized (a requirement of the von Neumann stability analysis[https://en.wikipedia.org/wiki/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<ref>S. Tanaka, S. Bunya, J.J. Westerink, C. Dawson, R.A. Luettich, Scalability of an Unstructured Grid Continuous Galerkin Based Hurricane Storm Surge Model, J. Sci. Comput. 46 (2011) 329–358. doi:10.1007/s10915-010-9402-1</ref>.

Wpringle: /* Critique */

2020-01-22T20:12:05Z

Critique

← Older revision		Revision as of 16:12, 22 January 2020
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	which is much less restrictive than the condition <math>\tau_0 \leq \Delta t</math>.		which is much less restrictive than the condition <math>\tau_0 \leq \Delta t</math>.

	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<ref>S. Tanaka, S. Bunya, J.J. Westerink, C. Dawson, R.A. Luettich, Scalability of an Unstructured Grid Continuous Galerkin Based Hurricane Storm Surge Model, J. Sci. Comput. 46 (2011) 329–358. doi:10.1007/s10915-010-9402-1</ref>.		Moreover, the equations analyzed are always linearized (a requirement of the von Neumann stability analysis[https://en.wikipedia.org/wiki/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<ref>S. Tanaka, S. Bunya, J.J. Westerink, C. Dawson, R.A. Luettich, Scalability of an Unstructured Grid Continuous Galerkin Based Hurricane Storm Surge Model, J. Sci. Comput. 46 (2011) 329–358. doi:10.1007/s10915-010-9402-1</ref>.

	== References ==		== References ==
	<references />		<references />

Wpringle: /* Critique */

2020-01-22T20:08:53Z

Critique

← Older revision		Revision as of 16:08, 22 January 2020
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	<math>\mathrm{A00} = \mathrm{B00} = 0.5\theta, \quad \mathrm{C00} = 1-\theta</math>		<math>\mathrm{A00} = \mathrm{B00} = 0.5\theta, \quad \mathrm{C00} = 1-\theta</math>

	~~Which~~ is ~~now~~ unconditionally stable under the following conditions for the consistent mass matrix,		This is unconditionally stable under the following conditions for the consistent mass-matrix solver,

	<math>2/3 \leq \theta \leq 1, \quad \tau_0\Delta t \leq (16/3)(3\theta/2-1)</math>		<math>2/3 \leq \theta \leq 1, \quad \tau_0\Delta t \leq (16/3)(3\theta/2-1)</math>

			which is much less restrictive than the condition <math>\tau_0 \leq \Delta t</math>.

	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<ref>S. Tanaka, S. Bunya, J.J. Westerink, C. Dawson, R.A. Luettich, Scalability of an Unstructured Grid Continuous Galerkin Based Hurricane Storm Surge Model, J. Sci. Comput. 46 (2011) 329–358. doi:10.1007/s10915-010-9402-1</ref>.		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<ref>S. Tanaka, S. Bunya, J.J. Westerink, C. Dawson, R.A. Luettich, Scalability of an Unstructured Grid Continuous Galerkin Based Hurricane Storm Surge Model, J. Sci. Comput. 46 (2011) 329–358. doi:10.1007/s10915-010-9402-1</ref>.

Wpringle: /* Critique */

2020-01-22T20:07:42Z

Critique

← Older revision		Revision as of 16:07, 22 January 2020
Line 53:		Line 53:
	<math>\mathrm{A00} = \mathrm{B00} = 0.5\theta, \quad \mathrm{C00} = 1-\theta</math>		<math>\mathrm{A00} = \mathrm{B00} = 0.5\theta, \quad \mathrm{C00} = 1-\theta</math>

	Which is now unconditionally stable under the following conditions,		Which is now unconditionally stable under the following conditions for the consistent mass matrix,
	<math>2/3 \leq \theta \leq 1, \quad \~~tau0~~\Delta t \leq 4(~~2-m~~)(2/3\theta-1)</math>		<math>2/3 \leq \theta \leq 1, \quad \tau_0\Delta t \leq (16/3)(3\theta/2-1)</math>

	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<ref>S. Tanaka, S. Bunya, J.J. Westerink, C. Dawson, R.A. Luettich, Scalability of an Unstructured Grid Continuous Galerkin Based Hurricane Storm Surge Model, J. Sci. Comput. 46 (2011) 329–358. doi:10.1007/s10915-010-9402-1</ref>.		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<ref>S. Tanaka, S. Bunya, J.J. Westerink, C. Dawson, R.A. Luettich, Scalability of an Unstructured Grid Continuous Galerkin Based Hurricane Storm Surge Model, J. Sci. Comput. 46 (2011) 329–358. doi:10.1007/s10915-010-9402-1</ref>.

Wpringle: /* Critique */

2020-01-22T20:05:50Z

Critique

← Older revision		Revision as of 16:05, 22 January 2020
Line 49:		Line 49:

	=== Critique ===		=== Critique ===
	One aspect missing from the theory above are different weighting schemes which are possibly non-centered, i.e., <math>\mathrm{A00} \neq \mathrm{C00}</math>. ~~Empirical evidence and inspection of Kinnmark's stability equations~~<ref ~~name=Kinnmark3~~></ref> ~~suggests~~ stability of the GWCE is improved if,		One aspect missing from the theory above are different weighting schemes which are possibly non-centered, i.e., <math>\mathrm{A00} \neq \mathrm{C00}</math>. Pringle et al.<ref>Pringle t al., Global Ocean-to-Coastal Storm Tide Modeling in ADCIRC v55: Unstructured Mesh Design, in preparation (2019)</ref> shows that stability of the GWCE is improved if,

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

			Which is now unconditionally stable under the following conditions,
			<math>2/3 \leq \theta \leq 1, \quad \tau0\Delta t \leq 4(2-m)(2/3\theta-1)</math>

	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<ref>S. Tanaka, S. Bunya, J.J. Westerink, C. Dawson, R.A. Luettich, Scalability of an Unstructured Grid Continuous Galerkin Based Hurricane Storm Surge Model, J. Sci. Comput. 46 (2011) 329–358. doi:10.1007/s10915-010-9402-1</ref>.		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<ref>S. Tanaka, S. Bunya, J.J. Westerink, C. Dawson, R.A. Luettich, Scalability of an Unstructured Grid Continuous Galerkin Based Hurricane Storm Surge Model, J. Sci. Comput. 46 (2011) 329–358. doi:10.1007/s10915-010-9402-1</ref>.