The PPE formulation - Revision history

Apwillis: /* PPE-formulation with correct boundary conditions */

2025-03-24T10:06:38Z

PPE-formulation with correct boundary conditions

← Older revision		Revision as of 03:06, 24 March 2025
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	== PPE-formulation with ~~correct~~ boundary conditions ==		== PPE-formulation with corrected boundary conditions ==

	Consider the case where the spatial discretisation splits the Navier--Stokes equations into a set one-dimensional problems, here into problems for radially-dependent Fourier modes.		Consider the case where the spatial discretisation splits the Navier--Stokes equations into a set one-dimensional problems, here into problems for radially-dependent Fourier modes.

Apwillis at 15:18, 20 March 2025

2025-03-20T15:18:36Z

← Older revision		Revision as of 08:18, 20 March 2025
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	where $\mat{A}=\mat{A}(\vec{g}(\vec{u}'))$ is a 4$\times$4 matrix and the 4-vector $\vec{a}$ is composed of the $a_j$. Thus, the appropriate coefficients required to satisfy the boundary conditions are recovered from solution of this small system for $\vec{a}$.		where $\mat{A}=\mat{A}(\vec{g}(\vec{u}'))$ is a 4$\times$4 matrix and the 4-vector $\vec{a}$ is composed of the $a_j$. Thus, the appropriate coefficients required to satisfy the boundary conditions are recovered from solution of this small system for $\vec{a}$.

	(Note that more complicated boundary conditions (e.g. partial slip) are easily accommodated, by simply redefining $\vec{g}(\vec{u})$. It may be desirable to change the boundary condition on $\bar{\vec{u}}$ to $\bar{\vec{u}}=\vec{u}^q$, so that it will be closer to the final value for $\vec{u}^{q+1}$.)		('''Note''' that more complicated boundary conditions (e.g. partial slip) are easily accommodated, by simply redefining $\vec{g}(\vec{u})$. It may be desirable to change the boundary condition on $\bar{\vec{u}}$ to $\bar{\vec{u}}=\vec{u}^q$, so that it will be closer to the final value for $\vec{u}^{q+1}$.)

	The error in the boundary conditions $g_j(\vec{u}^{q+1})$ using the influence-matrix technique is at the level of the machine epsilon, typically order 1e-16.		The error in the boundary conditions $g_j(\vec{u}^{q+1})$ using the influence-matrix technique is at the level of the machine epsilon, typically order 1e-16.

Apwillis at 15:15, 20 March 2025

2025-03-20T15:15:47Z

← Older revision		Revision as of 08:15, 20 March 2025
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	where $\mat{A}=\mat{A}(\vec{g}(\vec{u}'))$ is a 4$\times$4 matrix and the 4-vector $\vec{a}$ is composed of the $a_j$. Thus, the appropriate coefficients required to satisfy the boundary conditions are recovered from solution of this small system for $\vec{a}$.		where $\mat{A}=\mat{A}(\vec{g}(\vec{u}'))$ is a 4$\times$4 matrix and the 4-vector $\vec{a}$ is composed of the $a_j$. Thus, the appropriate coefficients required to satisfy the boundary conditions are recovered from solution of this small system for $\vec{a}$.

	(Note that~~, with this approach,~~ more complicated boundary conditions (e.g. partial slip) are easily accommodated, by simply redefining $\vec{g}(\vec{u})$. It may be desirable to change the boundary condition on $\bar{\vec{u}}$ to $\bar{\vec{u}}=\vec{u}^q$, so that it will be closer to the final value for $\vec{u}^{q+1}$.)		(Note that more complicated boundary conditions (e.g. partial slip) are easily accommodated, by simply redefining $\vec{g}(\vec{u})$. It may be desirable to change the boundary condition on $\bar{\vec{u}}$ to $\bar{\vec{u}}=\vec{u}^q$, so that it will be closer to the final value for $\vec{u}^{q+1}$.)

	The error in the boundary conditions $g_j(\vec{u}^{q+1})$ using the influence-matrix technique is at the level of the machine epsilon, typically order 1e-16.		The error in the boundary conditions $g_j(\vec{u}^{q+1})$ using the influence-matrix technique is at the level of the machine epsilon, typically order 1e-16.

Apwillis at 14:23, 20 March 2025

2025-03-20T14:23:30Z

← Older revision		Revision as of 07:23, 20 March 2025
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	where $\mat{A}=\mat{A}(\vec{g}(\vec{u}'))$ is a 4$\times$4 matrix and the 4-vector $\vec{a}$ is composed of the $a_j$. Thus, the appropriate coefficients required to satisfy the boundary conditions are recovered from solution of this small system for $\vec{a}$.		where $\mat{A}=\mat{A}(\vec{g}(\vec{u}'))$ is a 4$\times$4 matrix and the 4-vector $\vec{a}$ is composed of the $a_j$. Thus, the appropriate coefficients required to satisfy the boundary conditions are recovered from solution of this small system for $\vec{a}$.

	(Note that, with this approach, more complicated boundary conditions (e.g. partial slip) are easily accommodated, by simply redefining $\vec{g}(\vec{u})$. ~~Optionally, one~~ may change the boundary condition on $\bar{\vec{u}}$ to $\bar{\vec{u}}=\vec{u}^q$, so that it will be closer to the final value for $\vec{u}^{q+1}$.)		(Note that, with this approach, more complicated boundary conditions (e.g. partial slip) are easily accommodated, by simply redefining $\vec{g}(\vec{u})$. It may be desirable to change the boundary condition on $\bar{\vec{u}}$ to $\bar{\vec{u}}=\vec{u}^q$, so that it will be closer to the final value for $\vec{u}^{q+1}$.)

	The error in the boundary conditions $g_j(\vec{u}^{q+1})$ using the influence-matrix technique is at the level of the machine epsilon, typically order 1e-16.		The error in the boundary conditions $g_j(\vec{u}^{q+1})$ using the influence-matrix technique is at the level of the machine epsilon, typically order 1e-16.

Apwillis: /* PPE-formulation with correct boundary conditions */

2025-03-20T14:21:53Z

PPE-formulation with correct boundary conditions

← Older revision		Revision as of 07:21, 20 March 2025
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	\mat{A}\,\vec{a} = -\vec{g}(\bar{\vec{u}}) ,$		\mat{A}\,\vec{a} = -\vec{g}(\bar{\vec{u}}) ,$

	where $\mat{A}=\mat{A}(\vec{g}(\vec{u}'))$ is a 4$\times$4 matrix and the 4-vector $\vec{a}$ is composed of the $a_j$. Thus, the appropriate coefficients required to satisfy the boundary conditions are recovered from solution of this small system for $\vec{a}$.		where $\mat{A}=\mat{A}(\vec{g}(\vec{u}'))$ is a 4$\times$4 matrix and the 4-vector $\vec{a}$ is composed of the $a_j$. Thus, the appropriate coefficients required to satisfy the boundary conditions are recovered from solution of this small system for $\vec{a}$.

			(Note that, with this approach, more complicated boundary conditions (e.g. partial slip) are easily accommodated, by simply redefining $\vec{g}(\vec{u})$. Optionally, one may change the boundary condition on $\bar{\vec{u}}$ to $\bar{\vec{u}}=\vec{u}^q$, so that it will be closer to the final value for $\vec{u}^{q+1}$.)

	The error in the boundary conditions $g_j(\vec{u}^{q+1})$ using the influence-matrix technique is at the level of the machine epsilon, typically order 1e-16.		The error in the boundary conditions $g_j(\vec{u}^{q+1})$ using the influence-matrix technique is at the level of the machine epsilon, typically order 1e-16.

Apwillis at 11:31, 20 March 2025

2025-03-20T11:31:06Z

← Older revision		Revision as of 04:31, 20 March 2025
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	\end{array}\right.$		\end{array}\right.$

	with boundary conditions $u'_+=1$, $u'_-=1$, $u'_z=\mathrm{i}$ on $r=R$ (see [[Equations_and_parameters#Decoupling_the_equations\|Decoupling_the_equations]]). The system $(4)$ provides a linearly independent function $\vec{u}'_1$ that may be added to $\bar{\vec{u}}$ without affecting the right-hand side in $(3)$, but altering (to correct) the boundary condition applied.		with boundary conditions $u'_+=1$, $u'_-=1$, $u'_z=\mathrm{i}$ on $r=R$ (see [[Equations_and_parameters#Decoupling_the_equations\|Decoupling_the_equations]]). The system $(4)$ provides a linearly independent function $\vec{u}'_4$ that may be added to $\bar{\vec{u}}$ without affecting the right-hand side in $(3)$, but altering (to correct) the boundary condition applied.
	<!--		<!--
	($\mat{X}$ has been dropped on the left-hand side of $(4)$ since $\mat{X}$ represents a combination of the identity and the Laplace operator, but $\bnabla^2(\bnabla p')=\bnabla(\nabla^2 p')-\bnabla \wedge\bnabla \wedge\bnabla p'=0$, i.e., only the identity is left behind.)		($\mat{X}$ has been dropped on the left-hand side of $(4)$ since $\mat{X}$ represents a combination of the identity and the Laplace operator, but $\bnabla^2(\bnabla p')=\bnabla(\nabla^2 p')-\bnabla \wedge\bnabla \wedge\bnabla p'=0$, i.e., only the identity is left behind.)
	-->		-->
	Similarly the system $(5)$ provides a further three functions $\vec{u}'_j$ ~~(where~~ $j=2,3,4$~~; the~~ +,- ~~and~~ $z$ ~~components may be considered separately)~~. The superposition		Similarly the system $(5)$ provides a further three functions $\vec{u}'_j$ for $j=1,2,3$, where each has only one non-zero component, $u_+$, $u_-$ or $u_z$. The superposition

	$\label{eq:usuperpos}		$\label{eq:usuperpos}

Apwillis at 16:12, 10 March 2020

2020-03-10T16:12:13Z

← Older revision		Revision as of 09:12, 10 March 2020
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	Consider the case where the spatial discretisation splits the Navier--Stokes equations into a set one-dimensional problems, here into problems for radially-dependent Fourier modes.		Consider the case where the spatial discretisation splits the Navier--Stokes equations into a set one-dimensional problems, here into problems for radially-dependent Fourier modes.

	Let $\vec{u}$ denote the velocity, $\vec{N}$ denote nonlinear terms, and $\mat{X}$ and $\mat{Y}$ be matrices associated with implicit timestepping of the viscous terms for a particular mode.		Let $\vec{u}$ denote the velocity, $\vec{N}$ denote nonlinear terms, and $\mat{X}$ and $\mat{Y}$ be matrices associated with implicit timestepping of the viscous terms for a particular Fourier mode.
	The time-discretised Navier–Stokes equations for this mode may be written in the form		The time-discretised Navier–Stokes equations for this mode may be written in the form

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	$\label{eq:pressure}		$\label{eq:pressure}
	(8) \qquad		(8a) \qquad
	p = \bar{p} + a_1\, p' - \~~frac{1}{2}\vec{u}\cdot\vec{u}~~\, ,$		\tilde{p} = \bar{p} + a_1\, p' \, \qquad$ (for each Fourier mode, then)

	~~where the last term~~ arises when the Navier-Stokes equations are in rotational form (see [[Equations_and_parameters]]).		$
			(8b) \qquad p = \tilde{p} - \frac{1}{2}\vec{u}\cdot\vec{u}\, \qquad$ (in physical space).

			The adjustment in $(8b)$ arises when the Navier-Stokes equations are in rotational form (see [[Equations_and_parameters]]).

Apwillis at 14:21, 10 March 2020

2020-03-10T14:21:32Z

← Older revision		Revision as of 07:21, 10 March 2020
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	$\label{eq:pressure}		$\label{eq:pressure}
	(8) \qquad		(8) \qquad
	p~~^{q+\frac{1}{2}}~~ = \bar{p} + a_1\, p' - \frac{1}{2}\vec{u}\cdot\vec{u}\, ,$		p = \bar{p} + a_1\, p' - \frac{1}{2}\vec{u}\cdot\vec{u}\, ,$

	where the last term arises when the Navier-Stokes equations are in rotational form (see [[Equations_and_parameters]]).		where the last term arises when the Navier-Stokes equations are in rotational form (see [[Equations_and_parameters]]).

Apwillis at 13:45, 10 March 2020

2020-03-10T13:45:35Z

← Older revision		Revision as of 06:45, 10 March 2020
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	$\label{eq:pressure}		$\label{eq:pressure}
	(8) \qquad		(8) \qquad
	p^{q+\frac{1}{2}} = \bar{p} + a_1\, p' \, .$		p^{q+\frac{1}{2}} = \bar{p} + a_1\, p' - \frac{1}{2}\vec{u}\cdot\vec{u}\, ,$

			where the last term arises when the Navier-Stokes equations are in rotational form (see [[Equations_and_parameters]]).

Apwillis at 12:34, 10 March 2020

2020-03-10T12:34:32Z

← Older revision		Revision as of 05:34, 10 March 2020
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	\end{array}\right.$		\end{array}\right.$

	with boundary conditions $u'_+=1$, $u'_-=1$, $u'_z=\mathrm{i}$ on $r=R$ (see [[Equations_and_parameters#Decoupling_the_equations\|Decoupling_the_equations]]). The system $(1)$ provides a linearly independent function $\vec{u}'_1$ that may be added to $\bar{\vec{u}}$ without affecting the right-hand side in $(3)$, but altering (to correct) the boundary condition applied.		with boundary conditions $u'_+=1$, $u'_-=1$, $u'_z=\mathrm{i}$ on $r=R$ (see [[Equations_and_parameters#Decoupling_the_equations\|Decoupling_the_equations]]). The system $(4)$ provides a linearly independent function $\vec{u}'_1$ that may be added to $\bar{\vec{u}}$ without affecting the right-hand side in $(3)$, but altering (to correct) the boundary condition applied.
	<!--		<!--
	($\mat{X}$ has been dropped on the left-hand side of $(4)$ since $\mat{X}$ represents a combination of the identity and the Laplace operator, but $\bnabla^2(\bnabla p')=\bnabla(\nabla^2 p')-\bnabla \wedge\bnabla \wedge\bnabla p'=0$, i.e., only the identity is left behind.)		($\mat{X}$ has been dropped on the left-hand side of $(4)$ since $\mat{X}$ represents a combination of the identity and the Laplace operator, but $\bnabla^2(\bnabla p')=\bnabla(\nabla^2 p')-\bnabla \wedge\bnabla \wedge\bnabla p'=0$, i.e., only the identity is left behind.)
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	\vec{u}^{q+1} = \bar{\vec{u}} + \sum_{j=1}^4 a_j\, \vec{u}'_j \,$		\vec{u}^{q+1} = \bar{\vec{u}} + \sum_{j=1}^4 a_j\, \vec{u}'_j \,$

	may be formed in order to satisfy the four boundary conditions, $\vec{u}^{q+1}=\vec{0}$ and $\bnabla\cdot\vec{u}^{q+1}=0$ on $r=R$. Let $\vec{g}(\vec{u})$ be a 4-vector composed of these boundary conditions, such that they are satisfied when $\vec{g}(\vec{u})=\vec{0}$. Substituting the superposition (6) into the boundary conditions, they may be written		may be formed in order to satisfy the four boundary conditions, $\vec{u}^{q+1}=\vec{0}$ and $\bnabla\cdot\vec{u}^{q+1}=0$ on $r=R$. Let $\vec{g}(\vec{u})$ be a 4-vector composed of these boundary conditions, such that they are satisfied when $\vec{g}(\vec{u})=\vec{0}$. Substituting the superposition $(6)$ into the boundary conditions, they may be written

	$\label{eq:velBCs}		$\label{eq:velBCs}
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	The functions $u'_j(r)$, the matrix $\mat{A}$ and its inverse may all be precomputed. The boundary conditions for $\vec{u}'$ have been chosen so that that $u'_\pm$ are pure real, $u'_z$ is pure imaginary, and $\mat{A}$ is real. For each timestep, this application of the influence matrix technique requires only evaluation of the deviation from the boundary condition, multiplication by a 4$\times$4 real matrix, and the addition of only two functions to each component of $\vec{u}$, each either pure real or pure imaginary. Compared to the evaluation of nonlinear terms, the computational overhead is negligible.		The functions $u'_j(r)$, the matrix $\mat{A}$ and its inverse may all be precomputed. The boundary conditions for $\vec{u}'$ have been chosen so that that $u'_\pm$ are pure real, $u'_z$ is pure imaginary, and $\mat{A}$ is real. For each timestep, this application of the influence matrix technique requires only evaluation of the deviation from the boundary condition, multiplication by a 4$\times$4 real matrix, and the addition of only two functions to each component of $\vec{u}$, each either pure real or pure imaginary. Compared to the evaluation of nonlinear terms, the computational overhead is negligible.

	~~<!--~~
	If needed, the pressure may be calculated		If needed, the pressure may be calculated
	using the same $~~a_j~~$:		using the same $a_1$ as in $(6)$ using the $p'$ from $(4)$:

	$\label{eq:pressure}		$\label{eq:pressure}
	(8) \qquad		(8) \qquad
	p^{q+\frac{1}{2}} = \bar{p} + ~~\sum_{j=1}^4 a_j~~\, p'_j \, .$		p^{q+\frac{1}{2}} = \bar{p} + a_1\, p' \, .$
	~~-->~~