A Collocation Continuation Method for the Onedimension Solution of the Boseeinstein Condensation

Continuation Method

More concretely the continuation method is used to compute a tight covering of the unstable manifolds of the two nontrivial steady state solutions.

From: Handbook of Dynamical Systems , 2002

Handbook of Dynamical Systems

Michael Dellnitz , Oliver Junge , in Handbook of Dynamical Systems, 2002

Error estimate

Observe that the convergence result in Proposition 3.1 does not require the existence of a hyperbolic structure along the unstable manifold. However, if we additionally assume its existence then we can establish results on the convergence behavior of the continuation method in a completely analogous way as in Dellnitz and Hohmann [ 8].

To this end assume that p is an element of an attractive hyperbolic set A. Then the unstable manifold of p is contained in A. Choose

$Q=\overline{\bigcup _{x\in A}{W}_{\eta }^s\left(x\right)}$

for some sufficiently small n > 0. Note that A = AQ. AS in (2.7) let $\rho ⩾1$ be a constant such that for every compact neighborhood $\tilde{Q}\subset Q$ of A Q we have

(3.2) $h\left(A_Q,\tilde{Q}\right)⩽\delta \Rightarrow \tilde{Q}\subset U_{\rho \delta }\left(A_Q\right).$

A proof of the following result can be found in Junge [26].

Proposition 3.2

Assume that in the initialization step of the continuation method we have

$h\left(W_0,C_0^{\left(k\right)}\right)⩽\zeta \text{diam}{C}_0^{\left(k\right)}$

for some constant $\zeta >0$ . If $C_j^{(k)}\subset W_{\eta }^s(W_j)$ for j =0,1,2,…, J, then

(3.3) $h\left(W_j,C_j^{\left(k\right)}\right)⩽\text{diam}{C}_j^{\left(k\right)}max\left(\zeta ,1+\beta +{\beta }^2+\cdot \cdot \cdot +{\beta }^j\zeta \right)$

for j = 1, 2,…, J. Here $\beta =C_{\lambda \rho }$ and C and λ are the characteristic constants of the hyperbolic set A (see Theorem 2.9).

The estimate (3.3) points up the fact that for a given initial level k and λ near 1 - corresponding to a weak contraction transversal to the unstable manifold - the approximation error may increase dramatically with an increasing number of continuations steps (increasing j).

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Special Volume: Computational Methods for the Atmosphere and the Oceans

Eric Simonnet , ... Michael Ghil , in Handbook of Numerical Analysis, 2009

3.5 Application potential

The early and best-known applications of continuation methods to atmospheric and ocean dynamics involved a very small number of degrees of freedom ( Lorenz [1963a,b], Veronis [1963, 1966]). For instance, the Lorenz model (Eq. (2.20) ) is probably one of the most studied dynamical systems of the past 40 years. As dynamical systems theory is being rapidly extended to infinite-dimensional systems governed by PDEs ( Temam [1997], pp. 643), the applications to atmospheric, oceanic, and climate dynamics are becoming more and more sophisticated ( Dijkstra [2005], pp. 532, Ghil and Childress [1987], pp. 485). Indeed, the computer power available to study bifurcation sequences is increasing rapidly so that more sophisticated numerical methods can be applied.

In the mid-1970s, it was possible to compute the first one or two bifurcations for spatially 1D energy balance models with ( Held and Suarez [1974], North [1975], North, Mengel and Short [1983]) or without ( Ghil [1976]) spectral truncation.

In the mid-1980s, truncations to tens of degrees of freedom of 2D geophysical flow problems could be approached in this manner ( Ghil and Childress [1987], pp. 485). In particular, continuation methods were first seen in the seminal paper of Legras and Ghil on atmospheric flow regimes ( Legras and Ghil [1985]). In the last few years, there has been a spectacular increase of bifurcation studies involving a large number of degrees of freedom. For instance, bifurcation sequences have been computed for 2D oceanic flows ( Cessi and Young [1992], Primeau [1998], Quon and Ghil [1992, 1995], Simonnet, Ghil, Ide, Temam and Wang [2003a,b], Simonnet, Temam, Wang, Ghil and Ide [1998], Speich, Dijkstra and Ghil [1995]) as well as 3D flows ( Chen and Ghil [1996], Dijkstra, Oksuzoglu, Wubs and Botta [2001], Ghil and Robertson [2000], Weijer, Dijkstra, Oksuzoglu, Wubs and Niet [2003]). All these studies involved tens of thousands of degrees of freedom with the noticeable exception of Weijer, Dijkstra, Oksuzoglu, Wubs and Niet [2003] which involved about 300,000 degrees of freedom. Simplified atmospheric, oceanic, or coupled GCMs have thus become amenable to a systematic study of their large-scale variability.

The applicability strongly depends on the availability of efficient solvers for the linear systems arising from the Newton-Raphson method. Recently, the development of targeted solvers for ocean models ( De Niet, Wubs, van Scheltinga and Dijkstra [2007], Wubs, De Niet and Dijkstra [2006]) has opened the way to tackle problems with up to 10⁶degrees of freedom. The latest solver is based on a block Gauss-Seidel preconditioner, which uses the special structure and properties of the hydrostatic and geostrophic balances.

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Leray–Schauder Theory and Mapping Degree

J. Mawhin , in Encyclopedia of Mathematical Physics, 2006

Introduction

The Leray–Schauder theory gives a powerful and versatile continuation method for proving the existence, multiplicity, and bifurcation of solutions of nonlinear operator, differential and integral equations. Let X and Y be topological spaces, A⊂X, f: X→Y, a continuous mapping, and y∈Y. The fundamental idea of a continuation method to solve the equation f(x) = y in A consists in embedding it into a one-parameter family of equations

[1] $F\left(x,\lambda \right)=z\left(\lambda \right)$

where the continuous functions F: X × [0, 1] → Y, z: [0, 1] → Y are chosen in such a way that F(·, 1) = f, z(1) = y and

1.

equation F(x, 0) = z(0) has a nonempty set of solutions in A;

2.

one of those solutions at least can be continued into a solution in A of [1] for each λ ∈ [0, 1].

Simple examples show that Assertion 2 can be violated when all solutions of [1] leave A after some λ* ∈ ]0, 1[. A way to avoid such a situation consists in "closing the boundary," through the "boundary condition":

$F\left(x,\lambda \right)\ne z\left(\lambda \right)\text{for each}\left(x,\lambda \right)\in \partial A\times \left[0,1\right]$

When this condition is satisfied, Assertion 2 can still fail when two existing solutions for λ small disappear after coalescing at some λ₀ < 1. Losing all solutions through this process can be eliminated by reinforcing Assumption 1 into

2′.

Equation F(x, 0) = z(0) has a "robust" nonempty set of solutions in A.

This statement can be made precise through the concept of topological degree of a mapping, an "algebraic" count of the number of its zeros. In a finite-dimensional setting, this concept was introduced by Kronecker for smooth mappings and by Brouwer for continuous mappings. Its extension by Leray and Schauder to some classes of mappings in Banach spaces made much wider applications to nonlinear differential and integral equations possible.

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Modern Astrodynamics

MICHAEL DELLNILZ , OLIVER JUNGE , in Elsevier Astrodynamics Series, 2006

Proposition 5.4.2.

[23]. Assume that in the initialization step of the continuation method we have

$h\left(W_0,C_0^{\left(k\right)}\right)\le \varsigma \text{diam}{e}_0^{\left(k\right)}$

for some constant $\varsigma >0.If{C}_0^{\left(k\right)}\subset W_{\eta }^s\left(W_j\right)$ for j = 0, 1, 2,…,J, then

(5.9) $h\left(W_j,C_j^{\left(k\right)}\right)\le \text{diam}{e}_j^{\left(k\right)}\text{max}\left(\varsigma ,1+\beta +{\beta }^2+\cdots +{\beta }^j\varsigma \right)$

for j = 1, 2, …, J. Here β = Cλρ and C and λ are the characteristic constants of the hyperbolic set A.

The estimate (5.9) points up the fact that for a given initial level k and λ near 1—corresponding to a weak contraction transversal to the unstable manifold—the approximation error may increase dramatically with an increasing number of continuations steps (i.e., increasing j).

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Elliptic Boundary Value Problems of Second Order in Piecewise Smooth Domains

Mikhail Borsuk , Vladimir Kondratiev , in North-Holland Mathematical Library, 2006

9.4.3 About the solvability of (9.1.3) and (9.1.4) with a ₀ > 0

We set

(9.4.15) $\begin{array}{l}\mathrm{ℱ}\left(\mathrm{\lambda },a_0,{\omega }_0\right)=-{\theta }_0+\\ +{\displaystyle \underset{0}{\overset{+\infty }{\int }}\frac{\left[\left(m-1\right)y^2+{\mathrm{\lambda }}^2\right]{\left(y^2+{\mathrm{\lambda }}^2\right)}^{\frac{m-4}{2}}\mathit{dy}}{\left(m-1+q+\mu \right){\left(y^2+{\mathrm{\lambda }}^2\right)}^{\frac{m}{2}}+\mathrm{\lambda }\left(2-m+\mathsf{\tau }\right){\left(y^2+{\mathrm{\lambda }}^2\right)}^{\frac{m-2}{2}}-a_0}.}\end{array}$

By making the substitution: y = tλ, t ∈ (0, +∞) we obtain

$\mathrm{ℱ}\left(\mathrm{\lambda },a_0,{\omega }_0\right)=-{\theta }_0+{\displaystyle \underset{0}{\overset{+\infty }{\int }}\Lambda \left(\mathrm{\lambda },a_0,t\right)\mathit{dt}\text{,}}$

where

(9.4.16) $\begin{array}{l}\Lambda \left(\lambda ,a_0,t\right)\equiv \\ \equiv \frac{\left[\left(m-1\right)t^2+1\right]{\left(t^2+1\right)}^{\frac{m-4}{2}}}{\lambda \left(m-1+q+\mu \right){\left(t^2+1\right)}^{\frac{m}{2}}+\left(2-m+\tau \right){\left(t^2+1\right)}^{\frac{m-2}{2}}-a_0{\lambda }^{1-m}}.\end{array}$

Then the equation(9.1.3) takes the form

(9.4.17) $\mathrm{ℱ}\left(\lambda ,a_0,{\omega }_0\right)=0.$

According to the above, we have

(9.4.18) $\mathrm{ℱ}\left({\lambda }_0,0,{\omega }_0\right)=0.$

The direct calculations yield

(9.4.19)

(9.4.20) $\begin{array}{l}\frac{\partial \Lambda }{\partial a_0}=\frac{{\lambda }^{1-m}\left[\left(m-1\right)t^2+1\right]{\left(t^2+1\right)}^{\frac{m-4}{2}}}{{\left[\lambda \left(m-1+q+\mu \right){\left(t^2+1\right)}^{\frac{m}{2}}+\left(2-m+\tau \right){\left(t^2+1\right)}^{\frac{m-2}{2}}-a_0{\lambda }^{1-m}\right]}^2}\\ >0\forall t,\lambda ,a_0.\end{array}$

Therefore, we can apply the theorem about implicit functions. In a certain neighborhood of the point (λ₀, 0) the equation(9.4.17) (and so the equation(9.1.3) as well) determines λ = λ(a ₀,ω ₀) as a single-valued continuous function of a ₀, depending continuously on the parameter ω ₀ and having continuous partial derivatives $\frac{\partial \mathrm{\lambda }}{\partial a_0},\frac{\partial \mathrm{\lambda }}{\partial {\omega }_0}$ Now, we analyze the properties of λ as the function λ(a ₀,ω ₀). First from (9.4.17) we get:

$\frac{\partial \mathrm{ℱ}}{\partial \lambda }\frac{\partial \lambda }{\partial a_0}+\frac{\partial \mathrm{ℱ}}{\partial a_0}=0\mathrm{and}\frac{\partial \mathrm{ℱ}}{\partial \lambda }\frac{\partial \lambda }{\partial {\omega }_0}+\frac{\partial \mathrm{ℱ}}{\partial {\omega }_0}=0.$

Hence it follows that

(9.4.21) $\frac{\partial \lambda }{\partial a_0}=-\frac{\left(\frac{\partial \mathrm{ℱ}}{\partial a_0}\right)}{\left(\frac{\partial \mathrm{ℱ}}{\partial \lambda }\right)}\mathrm{and}\frac{\partial \lambda }{\partial {\omega }_0}=-\frac{\left(\frac{\partial \mathrm{ℱ}}{\partial {\omega }_0}\right)}{\left(\frac{\partial \mathrm{ℱ}}{\partial \lambda }\right)}.$

But, on the strength of (9.4.19), (9.4.20) we have

(9.4.22) $\begin{array}{l}\frac{\partial \mathrm{ℱ}}{\partial a_0}={\displaystyle \underset{0}{\overset{+\infty }{\int }}\frac{\partial \Lambda }{\partial a_0}\mathit{dt}>0,\frac{\partial \mathrm{ℱ}}{\partial \lambda }={\displaystyle \underset{0}{\overset{+\infty }{\int }}\frac{\partial \Lambda }{\partial \lambda }\mathit{dt}<0,}\frac{\partial \mathrm{ℱ}}{\partial {\theta }_0}=-1}\forall \left(\lambda ,a_0\right)\\ \mathrm{and}\frac{\partial \mathrm{ℱ}}{\partial {\omega }_0}=\frac{\partial \mathrm{ℱ}}{\partial {\theta }_0}\cdot \frac{d{\theta }_0}{d{\omega }_0}=\left\{\begin{array}{l}-\frac{1}{2},\mathrm{if}\mathrm{BVP}\mathrm{is}\mathrm{the}\mathrm{Dirichlet}\mathrm{problem},\hfill \\ -1,\mathrm{if}\mathrm{BVP}\mathrm{is}\mathrm{the}\mathrm{mixed}\mathrm{problem}.\hfill \end{array}\right.\end{array}$

From (9.4.21) and (9.4.22) we get:

(9.4.23) $\frac{\partial \lambda }{\partial a_0}>0\mathrm{and}\frac{\partial \lambda }{\partial {\omega }_0}<0\mathrm{for}\mathrm{any}{a}_0\ge 0.$

So, the function λ(a ₀, ω ₀) increases with respect to a ₀ and decreases with respect to ω ₀ . Applying the analytic continuation method, we obtain the solvability of the equation (9.1.3) ∀a ₀.

Corollary 9.14

$\lambda =\lambda \left(a_0,{\omega }_0\right)\ge {\lambda }_0>0\mathit{for}\mathit{any}{a}_0\ge 0.$

Further, multiplying the equation of (StL) by Φ(ω) and integrating over the interval $\left(-\frac{{\omega }_0}{2},\frac{{\omega }_0}{2}\right)$ , we get

(9.4.24) $\begin{array}{l}\left(1-\mu \right){\displaystyle \underset{-\frac{{\omega }_0}{2}}{\overset{\frac{{\omega }_0}{2}}{\int }}|\Phi |{}^q{\left({\lambda }^2{\Phi }^2+{{\Phi }^{\prime }}^2\right)}^{\frac{m-2}{2}}{{\Phi }^{\prime }}^2\mathit{d\omega }=-a_0}{\displaystyle \underset{-\frac{{\omega }_0}{2}}{\overset{\frac{{\omega }_0}{2}}{\int }}|\Phi |{}^{q+m}\mathit{d\omega }+}\\ +\left[{\lambda }^2\left(m-1+q+\mu \right)+\lambda \left(2-m+\tau \right)\right]{\displaystyle \underset{-\frac{{\omega }_0}{2}}{\overset{\frac{{\omega }_0}{2}}{\int }}|\Phi |{}^{q+2}{\left({\lambda }^2{\Phi }^2+{{\Phi }^{\prime }}^2\right)}^{\frac{m-2}{2}}\mathit{d\omega }\ge }\\ \ge 〈{\lambda }^m\left(m-1+q+\mu \right)+{\lambda }^{m-1}\left(2-m+\tau \right)-a_0〉{\displaystyle \underset{-\frac{{\omega }_0}{2}}{\overset{\frac{{\omega }_0}{2}}{\int }}|\Phi |{}^{q+m}\mathit{d\omega }>0\text{,}}\end{array}$

by virtue of (9.1.4).

Lemma 9.15

We have the inequality

(9.4.25) ${\displaystyle \underset{-\frac{{\omega }_0}{2}}{\overset{\frac{{\omega }_0}{2}}{\int }}|\Phi |{}^q|{\Phi }^{\prime }|{}^m\mathit{d\omega }\le c\left(q,\mu ,m,\tau ,\lambda \right)}{\displaystyle \underset{-\frac{{\omega }_0}{2}}{\overset{\frac{{\omega }_0}{2}}{\int }}|\Phi |{}^{q+m}\mathit{d\omega }.}$

Proof. From (9.4.24), by Young's inequality with $p=\frac{m}{m-2},p^{\prime }=\frac{m}{2}$ , it follows that

$\begin{array}{l}\left(1-\mu \right){\displaystyle \underset{-\frac{{\omega }_0}{2}}{\overset{\frac{{\omega }_0}{2}}{\int }}|\Phi |{}^q|{\Phi }^{\prime }|{}^m\mathit{d\omega }\le }\\ \le \left[{\lambda }^2\left(m-1+q+\mu \right)+\lambda \left(2-m+\tau \right)\right]{\displaystyle \underset{-\frac{{\omega }_0}{2}}{\overset{\frac{{\omega }_0}{2}}{\int }}|\Phi |{}^q{\Phi }^2{\left({\lambda }^2{\Phi }^2+{{\Phi }^{\prime }}^2\right)}^{\frac{m-2}{2}}\mathit{d\omega }\le }\\ \le \mathrm{ϵ}{\displaystyle \underset{-\frac{{\omega }_0}{2}}{\overset{\frac{{\omega }_0}{2}}{\int }}|\Phi |{}^q{\left({\lambda }^2{\Phi }^2+{{\Phi }^{\prime }}^2\right)}^{\frac{m}{2}}\mathit{d\omega }+c_{\mathrm{ϵ}}}{\displaystyle \underset{-\frac{{\omega }_0}{2}}{\overset{\frac{{\omega }_0}{2}}{\int }}|\Phi {{|}^{q+}}^m\mathit{d\omega }\le }\\ \le \mathrm{ϵ}{\displaystyle \underset{-\frac{{\omega }_0}{2}}{\overset{\frac{{\omega }_0}{2}}{\int }}|\Phi |{}^q|{\Phi }^{\prime }|{}^m\mathit{d\omega }+c_{\mathrm{ϵ}}}{\displaystyle \underset{-\frac{{\omega }_0}{2}}{\overset{\frac{{\omega }_0}{2}}{\int }}|\Phi {{|}^{q+}}^m\mathit{d\omega }\forall \mathrm{ϵ}>0\text{,}}\end{array}$

since m ≥ 2. Choosing $\mathrm{ϵ}=\frac{1-\mu }{2}$ we obtain the required (9.4.25).

Lemma 9.16

Let the inequality (9.1.4) hold and, in addition,

(9.4.26) $q+\mu <1.$

Then

(9.4.27) ${\displaystyle \underset{-\frac{{\omega }_0}{2}}{\overset{\frac{{\omega }_0}{2}}{\int }}|{\Phi }^{\prime }|{}^m\mathit{d\omega }\le c\left(q,\mu ,m,\tau ,\lambda ,{\omega }_0\right).}$

Proof. For dividing the equation of (StL) by Φ|Φ| ^q–2, we get

$\begin{array}{l}\Phi \frac{d}{\mathit{d\omega }}\left[{\left({\lambda }^2{\Phi }^2+{{\Phi }^{\prime }}^2\right)}^{\frac{m-2}{2}}{\Phi }^{\prime }\right]+q{{\Phi }^{\prime }}^2{\left({\lambda }^2{\Phi }^2+{{\Phi }^{\prime }}^2\right)}^{\frac{m-2}{2}}+\\ +\lambda \left[\lambda \left(q+m-1\right)-m+2+\tau \right]{\Phi }^2{\left({\lambda }^2{\Phi }^2+{{\Phi }^{\prime }}^2\right)}^{\frac{m-2}{2}}=\\ =a_0|\Phi |{}^m-\mu {\left({\lambda }^2{\Phi }^2+{{\Phi }^{\prime }}^2\right)}^{\frac{m}{2}}.\end{array}$

On integrating the obtained equality we have

(9.4.28) $\begin{array}{l}\left(1-q-\mu \right){\displaystyle \underset{-\frac{{\omega }_0}{2}}{\overset{\frac{{\omega }_0}{2}}{\int }}{\left({\lambda }^2{\Phi }^2+{{\Phi }^{\prime }}^2\right)}^{\frac{m}{2}}\mathit{d\omega }+a_0}{\displaystyle \underset{-\frac{{\omega }_0}{2}}{\overset{\frac{{\omega }_0}{2}}{\int }}|\Phi |{}^m\mathit{d\omega }=}\\ =\lambda \left(\mathit{\lambda m}+2-m+\tau \right){\displaystyle \underset{-\frac{{\omega }_0}{2}}{\overset{\frac{{\omega }_0}{2}}{\int }}{\Phi }^2{\left({\lambda }^2{\Phi }^2+{{\Phi }^{\prime }}^2\right)}^{\frac{m-2}{2}}\mathit{d\omega }.}\end{array}$

Since q + μ < 1, λ(λm + 2 – m + τ) > 0 and m ≥ 2 we shall get the required (9.4.27), if we apply the Young inequality with $p=\frac{m}{m-2},p^{\prime }=\frac{m}{2},\forall \epsilon >0$ . Finally, if there were q + μ ≥ 1, then from (9.4.28) we would get

$\begin{array}{l}\left(q-1+\mu \right){\lambda }^m{\displaystyle \underset{-\frac{{\omega }_0}{2}}{\overset{\frac{{\omega }_0}{2}}{\int }}|\Phi |{}^m\mathit{d\omega }+{\lambda }^{m-1}\left(\mathit{\lambda m}+2-m+\tau \right)}{\displaystyle \underset{-\frac{{\omega }_0}{2}}{\overset{\frac{{\omega }_0}{2}}{\int }}|\Phi |{}^m\mathit{d\omega }\le }\\ \le a_0{\displaystyle \underset{-\frac{{\omega }_0}{2}}{\overset{\frac{{\omega }_0}{2}}{\int }}|\Phi |{}^m\mathit{d\omega }\text{,}}\end{array}$

which would contradict (9.1.4), by virtue of Φ ≢ 0. The lemma is proved.

(9.4.1), (9.1.3) give us the function w = r^λ Φ(ω), which will be a barrier for our boundary problem (BVP).

Lemma 9.17

Let ζ(r)   ∈ C ₀ ^∞[0, d]. Then

$\zeta \left(r\right)\omega \left(x\right)\in {\mathfrak{N}}_{m,q}^1\left(r^{\tau },r^{\tau -m},G_0^d,G_0^d\backslash {\Gamma }_2^d\right).$

If (9.1.4) and (9.4.26) hold, then

$\zeta \left(r\right)\omega \left(x\right)\in {\mathfrak{N}}_{m,0}^1\left(r^{\tau },r^{\tau -m},G_0^d,G_0^d\backslash {\Gamma }_2^d\right).$

Proof. At first, we observe that w  ∈ L _∞(G ₀ ^d ) since λ > 0. Now we shall prove that

(9.4.29) ${{\rm I}}_q\left[w\right]\equiv {\displaystyle \underset{G_0^d}{\int }\left(r^{\tau -m}|w|{}^{q+m}+r^{\tau }|w|{}^q|\nabla w|{}^m\right)\mathit{dx}<\infty }.$

The direct calculations give

(9.4.30) $|\nabla w|{}^m=r^{m\left(\lambda -1\right)}{\left({\lambda }^2{\Phi }^2+{{\Phi }^{\prime }}^2\right)}^{\frac{m}{2}}.$

Therefore

$\begin{array}{l}{{\rm I}}_q\left[w\right]={\displaystyle \underset{G_0^d}{\int }\left\{r^{\tau +m\left(\lambda -1\right)+\mathit{q\lambda }}{\Phi }^{m+q}\left(\omega \right)+r^{\tau +m\left(\lambda -1\right)+\mathit{q\lambda }}|\Phi |{}^q{\left({\lambda }^2{\Phi }^2+{{\Phi }^{\prime }}^2\right)}^{\frac{m}{2}}\right\}\mathit{dx}}\\ \le c\left(\lambda ,m\right){\displaystyle \underset{0}{\overset{d}{\int }}{r}^{\tau +\left(m+q\right)\lambda -m+1}\mathit{dr}}{\displaystyle \underset{-\frac{{\omega }_0}{2}}{\overset{\frac{{\omega }_0}{2}}{\int }}\left(|\Phi |{}^q|{\Phi }^{\prime }|{}^m+|\Phi |{}^{q+m}\right)\mathit{d\omega }.}\end{array}$

It is clear that, by virtue of Lemma 9.15 , I_q [w] is finite. To prove the second assertion of the lemma we have to demonstrate that

(9.4.31) ${\rm I}\left[w\right]\equiv {\displaystyle \underset{G_0^d}{\int }\left(r^{\tau -m}|w|{}^m+r^{\tau }|\nabla w|{}^m\right)\mathit{dx}<\infty }.$

We again have

$\begin{array}{l}{\rm I}\left[w\right]={\displaystyle \underset{G_0^d}{\int }\left\{r^{\tau +m\left(\lambda -1\right)}{\left({\lambda }^2{\Phi }^2+{{\Phi }^{\prime }}^2\right)}^{\frac{m}{2}}+r^{\tau +m\left(\lambda -1\right)}{\Phi }^m\left(\omega \right)\right\}\mathit{dx}}\le \\ \le c\left(\lambda ,m\right){\displaystyle \underset{0}{\overset{d}{\int }}{r}^{\tau +\mathit{m\lambda }+1-m}\mathit{dr}}{\displaystyle \underset{-\frac{{\omega }_0}{2}}{\overset{\frac{{\omega }_0}{2}}{\int }}\left\{{\left({\lambda }^2{\Phi }^2+{{\Phi }^{\prime }}^2\right)}^{\frac{m}{2}}+{\Phi }^m\right\}\mathit{d\omega }.}\end{array}$

I[w] is finite by Lemma 9.16 . Thus,

$I\left[w\right]\le c\left(m,\lambda ,N,q,\mu ,{\omega }_0,d\right).$

Lemma 9.17 is proved.

Example 9.18

Let m = 2 and we shall consider the boundary value problem (BVP)₀ for the equation

(9.4.32) $\begin{array}{l}\frac{d}{d{x}_i}\left(r^{\tau }|w|{}^q{w}_{x_i}\right)=a_0{r}^{\tau -2}w|w|{}^q-\mu r^{\tau }w|w|{}^{q-2}|\nabla w|{}^2,x\in G_0,\\ a_0\ge 0,0\le \mu <1,q\ge 0,\tau \ge 0.\end{array}$

From (9.4.8), (9.1.3) it follows that the solution of our problem is the function

$w\left(r,\omega \right)=r^{\lambda }\times \left\{\begin{array}{ll}{cos}^{\frac{1}{1+q+\mu }}\left(\frac{\mathit{\pi \omega }}{{\omega }_0}\right)\hfill & \mathrm{for}\mathrm{the}\mathrm{Dirichlet}\mathrm{problem},\hfill \\ {cos}^{\frac{1}{1+q+\mu }}\left(\frac{\mathit{\pi \omega }}{2{\omega }_0}-\frac{\pi }{4}\right)\hfill & \mathrm{for}\mathrm{the}\mathrm{mixed}\mathrm{problem}.\hfill \end{array}\right.$

where

$\lambda =\frac{\sqrt{{\tau }^2+{\left(\pi /{\theta }_0\right)}^2+4a_0\left(1+q+\mu \right)}-\tau }{2\left(1+q+\mu \right)}$

(see (9.4.10)). It is easy to check that for such a λ the inequality (9.1.4) is satisfied.

By calculating Φ′(ω) one can readily see that all the properties of the function Φ(ω) hold. Moreover, we have:

(9.4.33) $\begin{array}{l}{\displaystyle \underset{-\frac{{\omega }_0}{2}}{\overset{\frac{{\omega }_0}{2}}{\int }}{{\Phi }^{\prime }}^2\left(\omega \right)\mathit{d\omega }=\frac{\pi }{{\left(1+q+\mu \right)}^2{\omega }_0}\cdot \frac{\Gamma \left(\frac{3}{2}\right)\Gamma \left(\frac{1-q-\mu }{2\left(1+q+\mu \right)}\right)}{\Gamma \left(\frac{2+q+\mu }{1+q+\mu }\right)}}\times \\ \times \left\{\begin{array}{l}1,\mathrm{if}\mathrm{BVP}\mathrm{is}\mathrm{the}\mathrm{Dirichlet}\mathrm{problem},\hfill \\ \frac{1}{4},\mathrm{if}\mathrm{BVP}\mathrm{is}\mathrm{the}\mathrm{Dirichlet}\mathrm{problem}\hfill \end{array}\right.\end{array}$

provided q +   μ < 1. This integral is nonconvergent, if q + μ ≥ 1. At the same time ∀q > 0 we have

(9.4.34) $\begin{array}{l}{\displaystyle \underset{-\frac{{\omega }_0}{2}}{\overset{\frac{{\omega }_0}{2}}{\int }}|\Phi \left(\omega \right)|{}^q{{\Phi }^{\prime }}^2\left(\omega \right)\mathit{d\omega }=\frac{\pi }{{\left(1+q+\mu \right)}^2{\omega }_0}\cdot \frac{\Gamma \left(\frac{3}{2}\right)\Gamma \left(\frac{1-\mu }{2\left(1+q+\mu \right)}\right)}{\Gamma \left(\frac{2+\frac{3}{2}q+\mu }{1+q+\mu }\right)}}\times \\ \times \left\{\begin{array}{l}1,\mathrm{if}\mathrm{BVP}\mathrm{is}\mathrm{the}\mathrm{Dirichlet}\mathrm{problem},\hfill \\ \frac{1}{4},\mathrm{if}\mathrm{BVP}\mathrm{is}\mathrm{the}\mathrm{Dirichlet}\mathrm{problem}\hfill \end{array}\right.\end{array}$

since μ < 1. This fact completely agrees with Lemmas 9.15-9.17 since

${\displaystyle \underset{-\frac{{\omega }_0}{2}}{\overset{\frac{{\omega }_0}{2}}{\int }}|\Phi \left(\omega \right)|{}^{q+2}\mathit{d\omega }=\frac{{\omega }_0}{\sqrt{\pi }}\cdot \frac{\Gamma \left(\frac{q+\frac{3+\mu }{2}}{\left(1+q+\mu \right)}\right)}{\Gamma \left(\frac{2+\frac{3}{2}q+\mu }{1+q+\mu }\right)}}.$

This fact demonstrates that $w\left(x\right)\in {\mathfrak{N}}_{2,0}^1\left(r^{\mathsf{\tau }},r^{\mathsf{\tau }‐m},G_0^d\right)$ , if q + μ < 1, and $w\left(x\right)\notin {\mathfrak{N}}_{2,0}^1\left(r^{\mathsf{\tau }},r^{\mathsf{\tau }‐m},G_0^d\right)$ , if q + μ ≥ 1. For the latter case we have $w\left(x\right)\in {\mathfrak{N}}_{2,q}^1\left(r^{\mathsf{\tau }},r^{\mathsf{\tau }‐m},G_0^d\right)$ .

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Context in Content Composition

Nicholas Asher , in Philosophy of Linguistics, 2012

1.1 From Discourse to the Lexicon

As work on formal semantics and, in particular, discourse semantics progressed, the need for a formal specification of the meanings of open class terms became more and more pressing to build up meanings of discourses compositionally. Formal semanticists working in discourse were no longer able simply to avoid thinking about the meanings of open class terms. The reasons why lexical meanings are so important to the composition of discourse meaning are a little involved. To explain them, I need to give a sketchy introduction to what the interpretation of a discourse involves.

An interpretation for a discourse depends not only on a compositional semantics for sentences but also on what one might call a "binder" rule. ¹ Let the structure of a text be the sequence of its constituent sentences. Given a category of sentences S and a "combinator", '.', we define the category of texts T as:

•

S→ T

•

T.S→ T

Thus, a sentence is a text and a text combined with a sentence is also a text. Where ||T|| is the meaning (or meaning representation) of T and ||S|| is the meaning of a sentence whose meaning is to be added to ||T||, a binder rule is an operation b that takes a text meaning, combines it with a sentence meaning and returns a new text meaning that can be integrated with further text meanings:

(1)

b: ||T|| × ||S|| → ||T||

All theories of discourse semantics have some form of binder rule. In a Stalnakerian semantics for discourse, where each sentence denotes a set of possible worlds, the operation b is set theoretic intersection. In Hans Kamp's dynamic semantics, Discourse Representation Theory (DRT), the operation b is an operation of merge over discourse representation structures (DRSs), DRT's meaning representations. ² In dynamic semantic systems like Predicate Dynamic Logic [Groenendijk and Stokhof, 1990], where the meaning of a sentence is a relation between an input assignment and an output assignment (relative to a fixed model), b is the operation of relational composition. For the continuation based discourse semantics of [de Groote, 2006] and [Barker and Shan, 2006], the binder rule is a bit more complicated but fits into the general pattern.

For dynamic semantic theories, such as Segmented Discourse Representation Theory (SDRT), that assign texts a meaning involving a rich discourse structure, the way ||S|| combines with ||T|| will sometimes depend on details of the lexical entries of the words in ||S||. The basic premise of a discourse semantics like SDRT is that the way ||S|| will combine with ||T|| will depend on the rhetorical or discourse function that ||S|| has in the context of ||T||. Discourse functions affect many aspects of discourse meaning, including the resolution of anaphoric expressions and ellipsis, temporal structure, presupposition, and the interpretation of adverbials [Hobbs, 1979; Asher, 1993; Lascarides and Asher, 1993; Hobbs et al., 1993; Asher and Lascarides, 2003; Vieu et al., 2005]. To compute these components of interpretation, we need to compute the discourse functions of discourse constituents (which for the moment we may continue to think of as sentences or clauses).

Sometimes, a relatively small class of adverbs or adverbial phrases, what [Knott, 1995] and others have called discourse connectors or discourse markers, suffices to determine the discourse functions and hence a method of combination in SDRT. ³ Syntactic constructions may also yield important clues as to discourse structure and how a new sentence, or rather discourse constituent, must combine with the text's meaning. But sometimes the method of combination will depend on open class words like verbs, their arguments and their modifiers. To illustrate consider:

(2)

a.

John fell. He slipped.

b.

John fell. He got hurt.

c.

John fell. He went down hard, onto the pavement.

In each of (2a-c), the second sentence has a different rhetorical or discourse function, which is reflected in the way SDRT integrates its content with the discourse context. For example, SDRT would relate the discourse constituents denoted by the two sentences in (2a) by Explanation—John fell because he slipped. The contsituents in (2b) would be related by Result, while the constituents in (2c) would be related by Elaboration. In each case, the rhetorical function can be traced to the meanings of the verb phrases.

Each of these discourse functions is defeasibly inferred from the way interpreters understand the combinations of the open class words and how their meanings combine in the predications. To make generalizations about the binder rule for a theory of compositional discourse interpretation, like SDRT, we need to have a lexical theory about words and how these words interact within predication. In particular, we need to group words into general types that would furnish the appropriate generalizations. However, in order to account for the diverse array of discourse relations these general types must be much more specific than the ones assumed by most compositional semantics, in which all common nouns and intransitive verbs have the same type E → T, where E is the type of entities and T the type of truth values.

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From Zeolites to Porous MOF Materials - The 40th Anniversary of International Zeolite Conference

M.P. Vega , ... E.L. Lima , in Studies in Surface Science and Catalysis, 2007

3. Bifurcation Analysis and Dynamic Simulations

Bifurcation theory provides tools for a system stability analysis under its parametric changes. As the parameters undergo changes, the existence of multiple steady states, sustained oscillations and traveling waves might occur for highly nonlinear processes [8 ]. In this paper, the quality of the different models was evaluated by comparing their dynamic structure (attractors and respective stability characteristics) to the dynamic behavior of the "real" plant, the FCC unit. In order to do that, bifurcation and stability analyses were carried out to unveil attractors, employing well-known continuation methods. The computations were carried out with routines provided by AUTO [ 9]. Branches of steady state solutions and periodic solutions were calculated with the arc-length method developed by Keller [10]. Nonlinear system theory states that if all eigenvalues of the Jacobian matrix lie in the open left half of the complex plane, the system is stable. Conversely, the steady state is unstable if the Jacobian matrix has at least one eigenvalue in the open right half of the complex plane.

The empirical model, internal model based on neural networks (NNs), is described as a discrete model, so that the stability characteristics are determined by the eigenvalues of the Jacobian matrix of the nonlinear map, which relates present data with the future process output. The stability characteristics of the closed loop (discrete system) are also determined by the eigenvalues of the Jacobian matrix of the nonlinear map. If the NN is described as a discrete model in the form of Eq. (8), relating past and present data with the future process output, the stability characteristics of the model solutions are determined by the eigenvalues of the Jacobian matrix, Eq. (9), of the nonlinear map.

(8) $x(k+l)=g{[}_X\left(k{)}_X\right(k\text{-}l),\dots ,x(k\text{-}n\left)\right]$

As a result, the computation of the Jacobian matrix requires that the NN be derived in terms of the state variables (present and past NN input data) of the nonlinear map. Steady states are stable if all eigenvalues of the Jacobian matrix are inside the unity circle. If any of the eigenvalues (Floquet multipliers) is outside the unity circle, the solution is unstable. At a Limit Point, an eigenvalue becomes identically equal to +1. At this point, multiple steady state solutions usually appear and a change in stability occurs. At a Hopf (Thorus) Bifurcation Point, a pair of complex eigenvalues crosses the unit circle with non-zero imaginary component and a branch of oscillatory solutions may appear. At a Period Doubling Bifurcation Point an eigenvalue becomes equal to -1 and branches of periodic solutions usually develop.

AUTO automatically detects bifurcation points and provides routines for computation of the multiple steady state solutions, oscillatory and periodic solutions that arise at these special points. Unstable behavior usually occurs in the vicinities of these bifurcation points, as at least one of the eigenvalues crosses the unity circle.

The use of bifurcation theory for analyzing nonlinear controlled systems requires that the manipulated variable be removed from the continuation parameter set. For closed loop purposes, plant model mismatch, set points and controller parameters are candidate variables for being a continuation parameter. As a result, bonds of stable operation can be computed for safe closed loop operation. As the continuation parameter can be varied over finite ranges, Hahn et al. [3] pointed out that the analysis can be performed over an entire operation region of the process rather than for a particular value of the set point. However, as emphasized by Hahn and co-workers, the set point can not be changed with high frequency, as bifurcation analysis returns information of equilibrium points rather than stability under the influence of time varying forcing functions. Therefore, closed loop analysis implies that references, inputs, and disturbances vary quasistatically when confronted to the system dynamics.

The identification of the bifurcation diagram and the dynamic structure of open/closed loops may allow the understanding of how and why the empirical models fail at certain process operation conditions, even when allowing a satisfactory one step ahead prediction of process dynamics, required by traditional validation methods [11], producing spurious controller performances. In fact, the bifurcation tool offers the possibility to delimitate bonds of stability of a nonlinear process.

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Nonlinearity and Functional Analysis

In Pure and Applied Mathematics, 1977

5.5A The Dirichlet problem for quasilinear elliptic equations

Let Ω be a bounded domain in ℝ ^N with boundary δΩ, and consider the following system of equations defined on $\overline{\Omega }$ ,

(5.5.1) $\sum _{|\alpha |+|\beta |=2}{A}_{\alpha \beta }\left(x,u,Du\right)D^{\alpha }{D}^{\beta }u+A_0\left(x,u,Du\right)=0\text{in}\Omega ,$

(5.5.2) $u{|}_{\partial \Omega }=g.$

The classic Dirichlet problem for (5.5.2)–(5.5.3) consists in determining a function $u\in \left[C^2(\Omega )\cap C\left(\overline{\Omega }\right)\right]$ that satisfies (5.5.1)–(5.5.2) in the pointwise sense. The differential operator on the left-hand side of (5.5.2) is elliptic if there is a constant μ > 0 such that A _αβ(x, y, z)ξ_αξ_β ⩾ μ|ξ|² for |y|, |z| ≤ M, x ∈ $\overline{\Omega }$ . The examples given in Section 1.2 show that such quasilinear elliptic Dirchlet problems may not be solvable for a variety of reasons, including the shape and size of Ω or the rapidity of growth and sign of A ₀(x, y, z).

The question of the solvability of Dirichlet problems of this class was posed by Hilbert in 1900 in his famous address (Hilbert, 1900), and was studied extensively by S. Bernstein thereafter. In N = 2, Bernstein attempted to solve (5.5.1)–(5.5.2) by (a) introducing a parameter t explicitly into the system (5.5.1)–(5.5.2) to obtain a one-parameter family of systems P_t so that for t = 0, the system P ₀ is solvable, while for t = 1, the system P ₁ coincides with (5.5.1)–(5.5.2); and (b) showing that each P_t is solvable for t ∈ (0, 1], by continuation. The continuation method was greatly extended in 1934 by Leray and Schauder, who transformed it into a homotopy argument by means of the degree. This approach, however, requires difficult analytic a priori estimates in order to ensure that the degree of the mapping f can be defined. Once these estimates have been established, the basic idea is to apply the a priori bound principle discussed in (5.4.14).

As a simple example, we mention

(5.5.3) Theorem Suppose ∂Ω and g are of class C ³, while the functions A_αβ (x, y, z) and A ₀(x, y, z) are C ¹ in x, y, z. Then the Dirichlet problem for (5.5.1)–(5.5.2) is solvable provided any solution v_t , of the system obtained from (5.5.1) by replacing A ₀ by tA ₀ and g by tg for t ∈ [0, 1] satisfies the a priori estimates

(5.5.4) $\underset{\Omega }{sup}|v_t|\le M_1;\underset{\Omega }{sup}|▿v_1|\le M_2,$

where M ₁, and M ₂ are constants independent of t and v_t.

Proof Sketch: As mentioned above, we apply (5.4.14) to prove the result, but first it is necessary to determine an appropriate Banach space X for the operator. To this end, we follow the Schauder inversion method discussed in Section 2.2D. Indeed, the a priori estimates (5.5.4) show that any solution v_t of a member of the adjusted system has a Hölder continuous gradient with exponent α ∈ (0, 1) and independent of t and v_t. We let $X=C^{1,\alpha }\left(\overline{\Omega }\right)$ , and define a mapping $T:C^{1,\alpha }\left(\overline{\Omega }\right)$ into itself by fixing u ∈ X and considering the solution U of the linear elliptic Dirichlet problem

$\sum _{|\alpha |+|\beta |=2}{A}_{\alpha \beta }(x,u,Du)D^{\alpha }{D}^{\beta }U+A_0(x,u,Du)=0\text{on}\Omega ,{\text{u}|}_{\partial \Omega }=g.$

By the results of Section 2.2D, Tu = U maps X into itself, is bounded, and in fact, maps bounded sets of X into bounded sets of $C^{2,\alpha }\left(\overline{\Omega }\right)$ . Since $C^{2,\alpha }\left(\overline{\Omega }\right)$ is a compact subset of $C^{1,\alpha }\left(\overline{\Omega }\right)$ , the mapping T is compact.

Now we apply the a priori bound principle (5.4.14) with f(u, t) = tTu and $X=C^{1,\alpha }\left(\overline{\Omega }\right)$ . By hypothesis, if v satisfies u = tTu, then $v\in C^{2,\alpha }\left(\overline{\Omega }\right)$ , and also satisfies the adjusted system. Consequently, $|\upsilon |{|}_{C^1\left(\overline{\Omega }\right)}\le M_1+M_2$ . Furthermore, by the a priori estimate (5.5.4) mentioned at the beginning of the proof, and a Hölder continuity of Ladyhenskaya and Uralsteva (1968), there is a number α ∈ (0, 1),

$|▿v\left(x\right)-{▿}_v\left(y\right)|\le M_3|x-y{|}^{\alpha },$

where M ₃ is independent of t ∈ [0, 1] and v. Thus

$||v|{|}_{C^{1,\alpha }\left(\overline{\Omega }\right)}\le M_1+M_2+M_3$

implying that, by (5.4.14), (5.5.1)–(5.5.2) has a solution $u\in C^2\left(\overline{\Omega }\right)$ . See Ladyhenskaya and Uralsteva (1968) for more details.

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Geometric Function Theory

Rudolf Wegmann , in Handbook of Complex Analysis, 2005

9 Mapping to Riemann surfaces

The derivative Φ′ of a conformal mapping Φ of the disk to a simply connected region does not vanish in D. The differentiated boundary correspondence equation (242) shows that the winding number of Φ′(e^it ) is connected to the winding number of ${\displaystyle \dot{\eta }}$ by

(264) $\text{wind}\left({\Phi }^{\prime }\left({\text{e}}^{\text{i}t}\right)\right)=\text{wind}({\displaystyle \dot{\eta }})-1.$

Only in the case $\text{wind}({\displaystyle \dot{\eta }})=1$ , the winding number of Φ′ (e^it ) is zero, which ensures by the argument principle that Φ′ does not vanish in D. The same holds for exterior mappings: if $\text{wind}({\displaystyle \dot{\eta }})=1$ then Φ′ does not vanish in the exterior D ⁻ of the disk.

When the curve Γ has self intersections, it can still be the boundary of a simply connected region G on a suitable Riemann surface. If $\text{wind}({\displaystyle \dot{\eta }})=1$ , the branch points of the Riemann surface must be outside G. In this case the methods for mapping D (or D ⁻) to the region, as described in Sections 4 and 8.2, can be applied without modification.

Example 11. Figure 8 is parameterized by

(265) $\eta (s)=\cos s+\text{i}\left[0.5\sin s+0.8\sin (3s)\right].$

The region G outside of the curve of Example 11 covers a neighborhood of the origin twice. Figure 23 shows the conformal mapping from D ⁻ to this region calculated by the Wegmann method on a grid with N = 2048 points. The computation is done in combination with a continuation method. The factor d in front of the sin(3s) term is gradually enlarged in four steps from 0.2,0.4,0.6, to 0.8. For each value of d four iterations are performed. Then the value of d is enlarged and the method is started with the result of the last iteration as S ₀. This explains the sawtooth behavior of the changes δ on the right panel of Figure 23. The achievable accuracy deteriorates as the region becomes more and more difficult. Crowding occurs at the flanks of the Figure 8. The wasp tail is only poorly resolved.

Fig. 23. The mapping from the exterior of a disk to the Riemann surface with boundary curve (265).

Figure 24 shows the image of 16 concentric circles of radii 1 (0.01) 1.15. For greater clarity only the image of the sector re^it for |t + π/2| ⩾ π/128 is shown.

Fig. 24. Detail of Figure 23.

The situation changes when $\text{wind}({\displaystyle \dot{\eta }})=1+l,$ , with l > 0 for the interior and l < 0 for the exterior mapping problem. It follows from the argument principle that, for each mapping Φ, the derivative Φ′ must have l zeros in D or –l zeros in D ⁻. These zeros correspond to branch points of the Riemann surface where G resides. These branch points lie in the target region G.

If the curve Γ is parameterized by a 2π-periodic function η such that $m:=\text{wind}({\displaystyle \dot{\eta }})1+l>0,$ one can still try to satisfy the boundary correspondence equation (122) with a function Φ analytic in D. The derivative Φ′ of this function has zeros at l (not necessarily distinct) points ζ ₁,…,ζ _l ∈ D. The curve Γ is then the boundary of a region G which is part of the Riemann surface of the function Φ. The mapping z → Φ(z) from D to G is conformal everywhere with the exception of the zeros ζ _j which are mapped to branch points of G.

The mapping is unique when the values

(266) $\Phi (z_j)=w_j$

at m different points z _j ∈ D, j = 1,…, m, and at a boundary point z ₀ ∈ ∂D are prescribed. The values w _j are in G for j = 1,…, m and in Γ for j = 0. There are other possibilities. One can prescribe instead the position of the preimages of the branch points, i.e., the zeros ζ ₁,…, ζ _l of Φ′ in D.

Wegmann [283] showed that his method (see equations (175) to (180)) can be adapted in the following way.

Let $m:=\text{wind}({\displaystyle \dot{\eta }})$ be a positive number.

Start with S ₀(t) such that S ₀(t) – t is a 2π-periodic function in W. When S _k is determined for some k ⩾ 0, the change DS is calculated from the condition (171), which is equivalent to the RH problem (172). The RH problem is solved with Theorem 9. One has to calculate the functions

(267) $\upsilon (t):=\theta \left(S_k(t)\right)-mt=\arg \left[{\text{e}}^{-\text{i}mt}{\displaystyle \dot{\eta }}\left(S_k(t)\right)\right],$

(268) $w:=K\upsilon ,$

(269) $g:=r\left(S_k(t)\right)\exp \left(w(t)\right)\mathrm{Im}\frac{\eta (S_k(t))}{{\displaystyle \dot{\eta }}(S_k(t))},$

(270) $h:=Kg.$

There are analytic functions Y and Ξ in D with boundary values

(271) $\begin{array}{cc}Y\left({\text{e}}^{\text{i}t}\right)=-w+\text{i}\upsilon ,& \Xi \end{array}\left({\text{e}}^{\text{i}t}\right)=-h+\text{i}g.$

According to Theorem 9, the general solution of the RH problem (172) is

(272) $\Psi (z)=z^m\exp \left(Y(z)\right)\left(\Xi (z)+P_m(z)\right)$

with a Laurent polynomial P _m of form

(273) $P_m(z)={\displaystyle {\displaystyle \sum _{j=-m}^m}{p}_j{z}^j}$

with complex coefficients p _j satisfying

(274) $\begin{array}{cc}{p}_{-j}={\displaystyle \overline{p_j}}& \begin{array}{cc}\text{for}& j=0,\dots ,m.\end{array}\end{array}$

When the calculation is done on a grid of N = 2n points, then the functions Y and Ξ are obtained as polynomials of degree n. The coefficients of these polynomials are obtained as a by-product of the process of conjugation. Therefore, the values of Y and Ξ at the points z _j , j = 0,…, m, can be easily calculated. Using the condition that Ψ must satisfy the interpolation conditions (266), the values of P _m at these points are evaluated as

(275) $P_m(z_j)=\frac{w_j}{z_j^m\exp (Y(z_j))}-\Xi (z_j).$

The condition (274) is equivalent to

(276) $P_m(1/{\displaystyle \overline{z_j}})={\displaystyle \overline{P_m(z_j).}}$

Equations (275) and (276) yield interpolation conditions at the 2m + 1 points $z_0,z_1,\dots ,z_m,1/{\displaystyle \overline{z_1}},\dots ,1/{\displaystyle \overline{z_m}}$ for the polynomial z ^m P _m (z) of degree ⩽ 2m. Therefore, P _m is uniquely defined by (275) and (276). Inserting (272) into (171) gives the change

(277) $DS(t)=\frac{-h(t)+P_m({\text{e}}^{\text{i}t})}{r(S_k(t))\exp (w(t))}-\mathrm{Re}\frac{\eta (S_k(t))}{{\displaystyle \dot{\eta }}(S_k(t))}.$

Note that in view of (274) the function P _m (e^it ) is real.

Example 12. Trefoil with boundary parameterization

(278) $\eta (s)=\cos (2s)+0.2\sin s+\text{i}\left[\sin (2s)+0.4\cos s\right].$

Figure 25 shows the result of a calculation with this method for the mapping Φ of the unit disk to the trefoil of Example 12 satisfying the interpolation conditions (266) with

Fig. 25. Mapping from the unit disk to the trefoil of Example 12. The upper (lower) panel shows the image of the upper (lower) semidisc. The interpolation points (279) are indicated by diamonds. The graph on the right shows δ, α, ε during the iteration.

(279) $\begin{array}{l}\begin{array}{ccc}{z}_0=1,& z_1=0.8\text{i,}& z_2=-0.8\text{i}\end{array},\hfill \\ \begin{array}{cc}{w}_0=1+0.4\text{i,}& w_1=w_2=-\mathrm{0.64.}\end{array}\hfill \end{array}$

The calculation is done with 256 grid points.

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A flexible symmetry-preserving Galerkin/POD reduced order model applied to a convective instability problem

Francisco Pla , ... José M. Vega , in Computers & Fluids, 2015

6 Concluding remarks

A Galerkin method based on POD has been presented to construct the bifurcation diagram giving the steady states, as the Rayleigh number is varied, in a two-dimensional Rayleigh–Bénard convection problem for large Prandtl number in a rectangular box. Because of an appropriate combination of slip/noslip boundary conditions, this problem exhibits a horizontal reflection symmetry. The POD modes have been obtained from snapshots computed by the Legendre solver at just one value of the Rayleigh number. The snapshots have been calculated as unconverged states of iterates converging to a steady state (either a conductive or convective steady state), either using a time-dependent solver or a Newton method. Maximizing the information contained in these unconverged snapshots has required using an improved version of the standard method to calculate the POD modes. The modified method provides good approximations of the POD modes associated with very small singular values. In addition, the POD modes have been obtained in two ways, namely ignoring the horizontal reflection symmetry or taking this symmetry into account.

Using POD modes calculated in the various above mentioned ways, several Galerkin systems have been obtained that were combined with a basic continuation method to compute the bifurcation diagram in the Rayleigh number range $1000⩽R\le 3000$ . As general conclusions from this analysis, it has been shown that:

•

Calculating the snapshots from outcomes of the Legendre solver at just one value of the bifurcation parameter means that the computational cost of the preprocess is very small. In fact, the whole computational cost (proprocess   +   online operation) of the method is at least four times smaller than that the Legendre solver.

•

The specific value of the Rayleigh number where snapshots are calculated is not critical, but it can be chosen with great flexibility.

•

The number of snapshots calculated by the time-dependent solver is larger than when using the Newton method.

•

The reflection symmetry allows for extracting a larger number of useful POD modes from a set of un-symmetric snapshots.

•

The approximated bifurcation diagram calculated with the various Galerkin systems is reasonably good. Departure from its counterpart calculated by the full Legendre solver is less than $O({10}^{-7})$ near the values of the Rayleigh number where the snapshots were calculated and it smoothly worsens as the Rayleigh number is increased.

The performance of the ROM could be improved in various ways. On the one hand, a more efficient projection of the nonlinear terms onto the POD modes, which has been made here upon reconstruction of the nonlinear terms and projection using the standard $L^2$ -inner product, which involves a fairly large computational cost to project the nonlinear terms. Instead, the Galerkin projection can be performed using a scalar product based on a few amount of points [39,51] or the so-called discrete empirical interpolation [12] can be used. Also, an adaptive method can be used to combine on demand the low-dimensional ROM system and the Legendre numerical solver. None on these improvements have been used here, where the main focus has been to illustrate how the usual way to calculate snapshots in the preprocess can be substituted by a more flexible and computational inexpensive method. Any improvement in the online operation of the ROM would further increase its overall performance.

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