Arbeitsgruppe Kuiper - Recent changes [en]

MA Emilio Schmidt

Emilio.S — Tue, 28 Apr 2026 14:09:34 GMT

3D Hydrodynamic Reference Test: Cylindrical coordinates

File:MAES-CartesianMeanSpacingOverPeaksdifferentSeedsr0.png

Emilio.S — Mon, 27 Apr 2026 08:29:10 GMT

Emilio.S uploaded File:MAES-CartesianMeanSpacingOverPeaksdifferentSeedsr0.png

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Mean fragment spacing $\langle \Delta z \rangle$ as a function of $r_0$ for fixed $Z_{\max} = 3.2\,\text{pc}$ and the seeds $3$, $5$, $7$, $18$, $21$ and $77$. The blue points indicate the measured mean fragment spacing, while the error bars denote the standard deviation of the individual fragment spacings within the corresponding simulation. The gray dashed line marks the analytically expected dependence $\lambda_{\text{frag}}(r_0)$.

File:MAES-CartesianMeanSpacingOverPeaksdifferentSeedsZmax.png

Emilio.S — Mon, 27 Apr 2026 08:24:50 GMT

Emilio.S uploaded File:MAES-CartesianMeanSpacingOverPeaksdifferentSeedsZmax.png

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Mean fragment spacing $\langle \Delta z \rangle$ as a function of $Z_{\max}$ for $r_0 = 0.033\,\mathrm{pc}$ and the seeds $3$, $5$, $7$, $18$, $21$ and $77$. The blue points indicate the measured mean fragment spacing, while the error bars denote the standard deviation of the individual fragment spacings within the corresponding simulation. The gray dashed horizontal line marks the analytically expected value $\lambda_{\text{frag}}(r_0) = 0.257\,\text{pc}$.

File:MAES-CartesianDensityHeatmapzProfileZmax16.png

Emilio.S — Mon, 27 Apr 2026 08:19:46 GMT

Emilio.S uploaded File:MAES-CartesianDensityHeatmapzProfileZmax16.png

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Density distributions and corresponding axial density profiles for $Z_{\max} = 1.6\,\mathrm{pc}$ and $r_0 = 0.033\,\mathrm{pc}$ at a time close to $14.5\tau$, shown for the seeds $3$, $5$, $7$, $18$, $21$ and $77$. The upper row shows the density distribution in the $xz$-plane at $y \approx 0$, while the lower row shows the corresponding axial density profiles at $y \approx 0$. The gray horizontal dashed line marks the threshold $\rho_{\mathrm{thres}}$ used for peak detection, and the blue vertical dashed lines indicate the positions of the counted local maxima from which the mean fragment spacing is determined.

File:MAES-CartesianColormapDensityxzphi0t0t5e+6.png

Emilio.S — Mon, 27 Apr 2026 08:11:03 GMT

Emilio.S uploaded File:MAES-CartesianColormapDensityxzphi0t0t5e+6.png

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Two-dimensional density maps of $\rho(x,z)$ in the $xz$-plane at $y \approx 0\,\text{pc}$, shown for the initial state at $t=0\,\text{yr}$ (left) and at $t=5.2\times10^6\,\text{yr}$ (right).

User:Lothar.brendel

Lothar.brendel — Fri, 24 Apr 2026 15:12:59 GMT

Transformation of probability density: strictly decreasing

← Older revision		Revision as of 17:12, 24 April 2026
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	and		and
	\[		\[
	P_y(f(a+\text dx))-P_y(f(a)) = P_y(f(a)+f'(a)\text dx)-P_y(f(a)) = ~~\underbrace{~~p_x(f(a))~~}_{p_y(a)}~~\underbrace{f'(a)\text dx}_{\text dy}		P_y(f(a+\text dx))-P_y(f(a)) = P_y(f(a)+f'(a)\text dx)-P_y(f(a)) = p_x(f(a))\underbrace{f'(a)\text dx}_{\text dy}
	\]		\]

			Comparison $\Rightarrow p_y(f(a))=p_x(a)/f'(a)$

			For strictly decreasing $f$ the order is swapped, i.e. $P_x(b)-P_x(a) =P_y(f(a))-P_y(f(b))$, yielding the positive $-f'(a)$.

	== 2D flow in polar coordinates $(\rho,\phi)$ ==		== 2D flow in polar coordinates $(\rho,\phi)$ ==

Compile-belt

Lothar.brendel — Fri, 24 Apr 2026 13:02:35 GMT

Intel compiler + Intel MPI

← Older revision		Revision as of 15:02, 24 April 2026
(One intermediate revision by the same user not shown)
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	<syntaxhighlight lang="shell">		<syntaxhighlight lang="shell">
	~~python2~~ ./config/configure.py PETSC_ARCH=amplitude_intel_impi \		$python ./config/configure.py PETSC_ARCH=amplitude_intel_impi \
	--CFLAGS='' \		--CFLAGS='' \
	--COPTFLAGS=~~-O3~~ --FOPTFLAGS=~~-O3~~ \		--COPTFLAGS=$opt --FOPTFLAGS=$opt \
	--with-mpi-dir=$I_MPI_ROOT \		--with-mpi-dir=$I_MPI_ROOT \
	--with-blas-lib=[libmkl_intel_lp64.a,libmkl_sequential.a,libmkl_core.a] --with-lapack-lib=libmkl_core.a \		--with-blas-lib=[libmkl_intel_lp64.a,libmkl_sequential.a,libmkl_core.a] --with-lapack-lib=libmkl_core.a \

User:Lothar.brendel

Lothar.brendel — Fri, 24 Apr 2026 09:28:21 GMT

transf. of prob-density

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MA Emilio Schmidt

Emilio.S — Wed, 22 Apr 2026 12:26:03 GMT

3D Hydrodynamic Reference Test: Cylindrical coordinates: Minor mistake

← Older revision		Revision as of 14:26, 22 April 2026
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	\end{align}		\end{align}

	Thus, two candidate maxima are only counted as independent fragments if their separation is larger than $0.05\lambda_{\mathrm{frag}}(r_0)$. The motivation for relating $\Delta z_{\min}$ to $\lambda_{\mathrm{frag}}(r_0)$ is that the latter provides the natural physical length scale of the fragmentation process. Choosing the minimum separation as a fixed fraction of this scale ensures that the criterion adapts consistently when $r_0$ is varied. At the same time, the chosen fraction is small enough that genuinely distinct neighboring fragments are not merged, but large enough to suppress multiple counting within a single fragment.		Thus, two candidate maxima are only counted as independent fragments if their separation is larger than $0.05\lambda_{\mathrm{frag}}(r_0)$. The motivation for relating $\Delta z_{\min}$ to $\lambda_{\mathrm{frag}}(r_0)$ is that the latter provides the natural physical length scale of the fragmentation process. Choosing the minimum separation as a fixed fraction of this scale ensures that the criterion adapts consistently when $r_0$ is varied. At the same time, the chosen fraction is small enough that genuinely distinct neighboring fragments are not merged, but large enough to suppress multiple counting within a single fragment.

	Once the fragment positions $z_i$ have been identified in this way, the distances between neighboring fragments are computed as follows:		Once the fragment positions $z_i$ have been identified in this way, the distances between neighboring fragments are computed as follows:

User:Lothar.brendel

Lothar.brendel — Tue, 21 Apr 2026 16:02:29 GMT

+Polyfit

@@ Line 1: / Line 1: @@
-Lothar Brendel
+= Polynomial fitting and estimation of uncertainty =
-"Admin" of this Wiki
+== Background ==
 = Scratch Pad =
@@ Line 14: / Line 104: @@
 ** \(\sigma_{r\phi}=\sigma_{\phi r}=\mu\displaystyle\frac{\partial_\phi u_r-u_\phi}{r}+\mu\,\partial_r u_\phi\)
 ** \(\sigma_{\phi\phi}=-p+(\zeta-2\mu/3)\displaystyle\frac{\partial_r(ru_r)+\partial_\phi u_\phi}{r}+2\mu\displaystyle\frac{\partial_\phi u_\phi+u_r}{r}\)
 == Energy equation ==

MA Emilio Schmidt

Emilio.S — Mon, 20 Apr 2026 11:09:15 GMT

3D Hydrodynamic Reference Test: Cylindrical coordinates: Fragment identification

@@ Line 2,132: / Line 2,132: @@
 For the first simulation, the domain in the \(z\)-direction is chosen to be approximately as large as in the radial direction, that is, \(Z_{\max} = R_{\max} = 0.2\,\text{pc}\). This gives \(N_z = 144\) cells in the \(z\)-direction. In the course of the simulation, however, a fragment forms. This is shown in the following figure. On the left, the density distribution in the \(rz\)-slice for \(\varphi \approx 0\) is shown at the initial time, while on the right the corresponding density distribution is shown at time \(t = 7.3 \cdot 10^{6}\,\mathrm{yr}\).
-\colorbox{yellow}{Colormap Density, Caption: Two-dimensional density maps of \(\rho(r,z)\) in the \(rz\)-plane at \(\varphi \approx 0\), shown for the initial state at \(t=0\,\text{yr}\) (left) and at \(t=7.3\times10^6\,\text{yr}\) (right).}
+[[File:MAES-ColormapDensityrzphi0t0t7e+6.png|600px|thumb|center|Two-dimensional density maps of \(\rho(r,z)\) in the \(rz\)-plane at \(\varphi \approx 0\), shown for the initial state at \(t=0\,\text{yr}\) (left) and at \(t=7.3\times10^6\,\text{yr}\) (right).]]
 This leads to the conclusion that numerical noise, caused for example by finite machine precision, round-off errors, or parallelization effects, is already sufficient as a perturbation to trigger an instability. In this context, the numerical noise can be understood as a superposition of many individual perturbations. In the simulation, however, the spectrum of modes available along the filament axis is restricted by the finite domain, since for a given \(Z_{\max}\) only certain axial wavelengths are compatible with the periodic boundary conditions. It is therefore useful to first perform a study in which the parameter \(Z_{\max}\) is varied for fixed \(r_0 = 0.033\,\text{pc}\). In this way, it can be investigated to what extent the measured fragment spacing in the simulations is affected by the finite domain and whether it converges to the theoretical expectation \(\lambda_{\text{frag}}(r_0)\) as \(Z_{\max}\) increases.
@@ Line 2,153: / Line 2,153: @@
 The prefactor must satisfy a compromise. On the one hand, it must not be too small, so that maxima caused merely by noise or similar effects are excluded. On the other hand, it must not be too large, so that already developed but still weakly pronounced fragments are not excluded. For the simulations considered here, the chosen value turns out to be a good compromise.
 This is illustrated exemplarily in the following figure for fixed \(Z_{\max} = 1.6\,\mathrm{pc}\) and \(r_0 = 0.033\,\mathrm{pc}\). Since the snapshots are written only at discrete times, the times at which the simulations are analyzed deviate slightly from \(12.5\tau\). However, this deviation is about two orders of magnitude smaller than \(12.5\tau\) and is therefore negligible. In the top row, the density distributions in the \(rz\)-slice at \(\varphi \approx 0\) are shown for the different seeds. In the bottom row, the corresponding axial density profiles at \(r \approx 0\) are shown for the different seeds, from which the mean fragment spacing is determined. The threshold above which local maxima are counted is indicated by the gray horizontal dashed line. The blue vertical dashed lines mark the positions of the counted local maxima from which the mean fragment spacing is determined.
-\colorbox{yellow}{Colormap Density, Axial Density Profile, Caption: Density distributions and corresponding axial density profiles for \(Z_{\max} = 1.6\,\mathrm{pc}\) and \(r_0 = 0.033\,\mathrm{pc}\) at a time close to \(12.5\tau\), shown for the seeds \(3\), \(5\), \(7\), \(18\), \(21\) and \(77\). The upper row shows the density distribution in the \(rz\)-plane at \(\varphi \approx 0\), while the lower row shows the corresponding axial density profiles at \(r \approx 0\). The gray horizontal dashed line marks the threshold \(\rho_{\mathrm{thres}}\) used for peak detection, and the blue vertical dashed lines indicate the positions of the counted local maxima from which the mean fragment spacing is determined.}
+[[File:MAES-ColormapDensityrzphi0AxialDensityProfile.png|1500px|thumb|center|Density distributions and corresponding axial density profiles for \(Z_{\max} = 1.6\,\mathrm{pc}\) and \(r_0 = 0.033\,\mathrm{pc}\) at a time close to \(12.5\tau\), shown for the seeds \(3\), \(5\), \(7\), \(18\), \(21\) and \(77\). The upper row shows the density distribution in the \(rz\)-plane at \(\varphi \approx 0\), while the lower row shows the corresponding axial density profiles at \(r \approx 0\). The gray horizontal dashed line marks the threshold \(\rho_{\mathrm{thres}}\) used for peak detection, and the blue vertical dashed lines indicate the positions of the counted local maxima from which the mean fragment spacing is determined.]]
 The figure clearly shows the presence of several fragments in each simulation. The positions and amplitudes of the overdensities depend on the respective seed. The mean fragment spacing, however, is of the same order of magnitude in all cases. The fact that the detailed fragment structure depends on the seed is physically plausible, since the seed determines the specific realization of the initial noise perturbation. The linear stability analysis predicts which axial modes are unstable and at what rate they grow, but not the exact positions at which fragments form. These positions are instead determined by the particular spatial distribution of the initial overdensities, that is, by the seed-dependent noise. The fact that the mean peak spacing nevertheless remains of the same order of magnitude for all seeds indicates that the scale of the fragment spacing is not set by the particular realization of the noise, but by the underlying physical instability.
@@ Line 2,168: / Line 2,186: @@
 The following figure shows the mean fragment spacing as a function of \(Z_{\max}\) for the different seeds. The points represent the measured mean values, while the error bars indicate the standard deviation of the individual fragment spacings within the respective simulation. The gray horizontal dashed line marks the analytically expected fragment spacing.
-\colorbox{yellow}{Mean Fragment Spacin Zmax Seeds, Caption: Mean fragment spacing \(\langle \Delta z \rangle\) as a function of \(Z_{\max}\) for \(r_0 = 0.033\,\mathrm{pc}\) and the seeds \(3\), \(5\), \(7\), \(18\), \(21\) and \(77\). The blue points indicate the measured mean fragment spacing, while the error bars denote the standard deviation of the individual fragment spacings within the corresponding simulation. The gray dashed horizontal line marks the analytically expected value \(\lambda_{\text{frag}}(r_0) = 0.257\,\mathrm{pc}\).}
+[[File:MAES-MeanFragmentSpacingZmaxSeeds.png|1500px|thumb|center|Mean fragment spacing \(\langle \Delta z \rangle\) as a function of \(Z_{\max}\) for \(r_0 = 0.033\,\mathrm{pc}\) and the seeds \(3\), \(5\), \(7\), \(18\), \(21\) and \(77\). The blue points indicate the measured mean fragment spacing, while the error bars denote the standard deviation of the individual fragment spacings within the corresponding simulation. The gray dashed horizontal line marks the analytically expected value \(\lambda_{\text{frag}}(r_0) = 0.257\,\mathrm{pc}\).]]

File:MAES-MeanFragmentSpacingr0Seeds.png

Emilio.S — Sun, 19 Apr 2026 10:03:55 GMT

Emilio.S uploaded File:MAES-MeanFragmentSpacingr0Seeds.png

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Mean fragment spacing $\langle \Delta z \rangle$ as a function of $r_0$ for fixed $Z_{\max} = 3.2\,\mathrm{pc}$ and the seeds $3$, $5$, $7$, $18$, $21$ and $77$. The blue points indicate the measured mean fragment spacing, while the error bars denote the standard deviation of the individual fragment spacings within the corresponding simulation. The gray dashed line marks the analytically expected dependence $\lambda_{\text{frag}}(r_0)$.

File:MAES-MeanFragmentSpacingZmaxSeeds.png

Emilio.S — Sun, 19 Apr 2026 10:02:42 GMT

Emilio.S uploaded File:MAES-MeanFragmentSpacingZmaxSeeds.png

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File:MAES-ColormapDensityrzphi0AxialDensityProfile.png

Emilio.S — Sun, 19 Apr 2026 10:00:21 GMT

Emilio.S uploaded File:MAES-ColormapDensityrzphi0AxialDensityProfile.png

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Density distributions and corresponding axial density profiles for $Z_{\max} = 1.6\,\mathrm{pc}$ and $r_0 = 0.033\,\mathrm{pc}$ at a time close to $12.5\tau$, shown for the seeds $3$, $5$, $7$, $18$, $21$ and $77$. The upper row shows the density distribution in the $rz$-plane at $\varphi \approx 0$, while the lower row shows the corresponding axial density profiles at $r \approx 0$. The gray horizontal dashed line marks the threshold $\rho_{\mathrm{thres}}$ used for peak detection, and the blue vertical dashed lines indicate the positions of the counted local maxima from which the mean fragment spacing is determined.

File:MAES-ColormapDensityrzphi0t0t7e+6.png

Emilio.S — Sun, 19 Apr 2026 09:58:55 GMT

Emilio.S uploaded File:MAES-ColormapDensityrzphi0t0t7e+6.png

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Two-dimensional density maps of $\rho(r,z)$ in the $rz$-plane at $\varphi \approx 0$, shown for the initial state at $t=0\,\text{yr}$ (left) and at $t=7.3\times10^6\,\text{yr}$ (right).

MA Emilio Schmidt

Emilio.S — Sat, 18 Apr 2026 19:14:23 GMT

Work Plan and Approach: 3D Hydrodynamic Reference Test: Cylindrical coordinates

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File:MAES-MaxEigenValuesKM0Xmax25N500.png

Emilio.S — Fri, 10 Apr 2026 15:54:20 GMT

Emilio.S uploaded File:MAES-MaxEigenValuesKM0Xmax25N500.png

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Maximal squared growth rate $\bar{\sigma}_{\max}^2(m=0,\bar{k})$ as a function of the dimensionless wavenumber $\bar{k}$ for the axisymmetric mode $m=0$. The red point marks the maximum of the curve, corresponding to the fastest-growing unstable mode, and the dashed vertical line indicates the associated dimensionless wavenumber.

File:MAES-MaxEigenValuesMKXmax25N500.png

Emilio.S — Fri, 10 Apr 2026 15:35:09 GMT

Emilio.S uploaded File:MAES-MaxEigenValuesMKXmax25N500.png

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Squared maximal growth rates $\bar{\sigma}_{\max}^2(m,\bar{k})$ for fixed pairs $(m,\bar{k})$, shown here for $m \in \left{0,1,2\right}$ and $\bar{k} \in [0,2]$. The black box highlights the axisymmetric mode $m=0$, for which positive values occur.

MA Emilio Schmidt

Emilio.S — Fri, 10 Apr 2026 14:12:07 GMT

Interpretation of the spectral structure: Pictures

← Older revision		Revision as of 16:12, 10 April 2026
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	In the following figure, $\Delta b(\bar\sigma^2)$ is shown as a function of $\bar\sigma^2$ for $(m,\bar k) = (0,1)$. The zeros of $\Delta b(\bar\sigma^2)$ correspond exactly to the eigenvalues.		In the following figure, $\Delta b(\bar\sigma^2)$ is shown as a function of $\bar\sigma^2$ for $(m,\bar k) = (0,1)$. The zeros of $\Delta b(\bar\sigma^2)$ correspond exactly to the eigenvalues.

	~~\colorbox{yellow}{Plot of $\Delta b(\bar\sigma^2)$, caption~~: Shooting analysis for $(m,\bar k)=(0,1)$: ~~on the left,~~ $\Delta b(\bar\sigma^2)$; on the ~~right,~~ $\|\Delta b(\bar\sigma^2)\|$ on a logarithmic scale. The red points mark the zeros of $\Delta b$, which correspond exactly to the eigenvalues. The dashed line indicates the threshold $-\bar k^2=-1$.}		[[File:Deltab Sigma2 M0 K1 Xmax12.png\|800px\|thumb\|center\|Shooting analysis for $(m,\bar k)=(0,1)$: $\Delta b(\bar\sigma^2)$ on the left and the absolute value of$\Delta b(\bar\sigma^2)$ on the right on a logarithmic scale. The red points mark the zeros of $\Delta b$, which correspond exactly to the eigenvalues. The dashed line indicates the threshold $-\bar k^2=-1$.]]

	The numerical evaluation of the shooting method shows that $\Delta b(\bar\sigma^2)$ has several zeros below the threshold $-\bar k^2$, whereas only a single zero occurs above this threshold. This behavior is consistent with the spectrum discussed above, since several eigenvalues occur below the threshold, whereas only discrete eigenvalues are possible above the threshold. For the case considered here, $(m,\bar k) = (0,1)$, exactly one eigenvalue above $-\bar k^2$ is obtained on the finite computational domain.		The numerical evaluation of the shooting method shows that $\Delta b(\bar\sigma^2)$ has several zeros below the threshold $-\bar k^2$, whereas only a single zero occurs above this threshold. This behavior is consistent with the spectrum discussed above, since several eigenvalues occur below the threshold, whereas only discrete eigenvalues are possible above the threshold. For the case considered here, $(m,\bar k) = (0,1)$, exactly one eigenvalue above $-\bar k^2$ is obtained on the finite computational domain.
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	In the following figure, $\|\psi(x_{\max})\|$ is shown as a function of $x_{\max}$ for several values of $\bar\sigma^2$ in the vicinity of the zero found above the threshold. For each value of $\bar\sigma^2$, the value $b_\psi$ was chosen such that $u(x_{\max})=0$ is satisfied. According to the argument above, only for the actual eigenvalue should $\psi(x_{\max})=0$ additionally hold, whereas for neighboring values of $\bar\sigma^2$, $u(x_{\max})$ generally vanishes, but $\psi$ still contains a small, nonvanishing contribution of the asymptotically growing branch.		In the following figure, $\|\psi(x_{\max})\|$ is shown as a function of $x_{\max}$ for several values of $\bar\sigma^2$ in the vicinity of the zero found above the threshold. For each value of $\bar\sigma^2$, the value $b_\psi$ was chosen such that $u(x_{\max})=0$ is satisfied. According to the argument above, only for the actual eigenvalue should $\psi(x_{\max})=0$ additionally hold, whereas for neighboring values of $\bar\sigma^2$, $u(x_{\max})$ generally vanishes, but $\psi$ still contains a small, nonvanishing contribution of the asymptotically growing branch.

			[[File:MAES-PsiXmaxXmaxM0K1Xmax12.png\|400px\|thumb\|center\|Behavior of the absolute value of $\psi(x_{\max})$ as a function of $x_{\max}$ for different values of $\bar\sigma^2$ in the vicinity of the eigenvalue found above the threshold $-\bar k^2$, here for $(m,\bar k)=(0,1)$. For each $\bar\sigma^2$, $b_\psi$ was chosen such that $u(x_{\max})=0$ holds.]]

	It turns out that $\|\psi(x_{\max})\|$ remains very small as $x_{\max}$ increases only for one distinguished value of $\bar\sigma^2$, which indicates that the prefactor of the growing asymptotic branch vanishes there. By contrast, even small deviations from the actual eigenvalue lead to a strong growth of $\|\psi(x_{\max})\|$.		It turns out that $\|\psi(x_{\max})\|$ remains very small as $x_{\max}$ increases only for one distinguished value of $\bar\sigma^2$, which indicates that the prefactor of the growing asymptotic branch vanishes there. By contrast, even small deviations from the actual eigenvalue lead to a strong growth of $\|\psi(x_{\max})\|$.

File:MAES-PsiXmaxXmaxM0K1Xmax12.png

Emilio.S — Fri, 10 Apr 2026 14:11:32 GMT

Emilio.S uploaded File:MAES-PsiXmaxXmaxM0K1Xmax12.png

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Behavior of the absolute value of $\psi(x_{\max})$ as a function of $x_{\max}$ for different values of $\bar\sigma^2$ in the vicinity of the eigenvalue found above the threshold $-\bar k^2$, here for $(m,\bar k)=(0,1)$. For each $\bar\sigma^2$, $b_\psi$ was chosen such that $u(x_{\max})=0$ holds.

File:Deltab Sigma2 M0 K1 Xmax12.png

Emilio.S — Fri, 10 Apr 2026 14:06:05 GMT

Emilio.S uploaded File:Deltab Sigma2 M0 K1 Xmax12.png

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Shooting analysis for $(m,\bar k)=(0,1)$: $\Delta b(\bar\sigma^2)$ on the left and $|\Delta b(\bar\sigma^2)|$ on the right on a logarithmic scale. The red points mark the zeros of $\Delta b$, which correspond exactly to the eigenvalues. The dashed line indicates the threshold $-\bar k^2=-1$.

File:MAES-SpectrumM0K1Xmax25N500.png

Emilio.S — Fri, 10 Apr 2026 14:00:09 GMT

Emilio.S reverted File:MAES-SpectrumM0K1Xmax25N500.png to an old version Reverted to version as of 15:57, 10 April 2026 (CEST)

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Spectrum of the numerically determined eigenvalues $\bar{\sigma}_n^2$ for the fixed pair $(m,\bar{k}) = (0,1)$, on a computational domain with $x_{\max} = 25$ and $N = 250$ grid points. The dashed horizontal line marks the threshold $-\bar{k}^2$. The inset shows the last ten eigenvalues.

File:MAES-SpectrumM0K1Xmax25N500.png

Emilio.S — Fri, 10 Apr 2026 13:59:27 GMT

Emilio.S reverted File:MAES-SpectrumM0K1Xmax25N500.png to an old version Reverted to version as of 15:52, 10 April 2026 (CEST)

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File:MAES-SpectrumM0K1Xmax25N500.png

Emilio.S — Fri, 10 Apr 2026 13:57:31 GMT

Emilio.S uploaded a new version of File:MAES-SpectrumM0K1Xmax25N500.png Inset Smaller

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← Older revision		Revision as of 16:12, 10 April 2026
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	In the following figure, \(\Delta b(\bar\sigma^2)\) is shown as a function of \(\bar\sigma^2\) for \((m,\bar k) = (0,1)\). The zeros of \(\Delta b(\bar\sigma^2)\) correspond exactly to the eigenvalues.		In the following figure, \(\Delta b(\bar\sigma^2)\) is shown as a function of \(\bar\sigma^2\) for \((m,\bar k) = (0,1)\). The zeros of \(\Delta b(\bar\sigma^2)\) correspond exactly to the eigenvalues.

	~~\colorbox{yellow}{Plot of \(\Delta b(\bar\sigma^2)\), caption~~: Shooting analysis for \((m,\bar k)=(0,1)\): ~~on the left,~~ \(\Delta b(\bar\sigma^2)\); on the ~~right,~~ \(\|\Delta b(\bar\sigma^2)\|\) on a logarithmic scale. The red points mark the zeros of \(\Delta b\), which correspond exactly to the eigenvalues. The dashed line indicates the threshold \(-\bar k^2=-1\).}		[[File:Deltab Sigma2 M0 K1 Xmax12.png\|800px\|thumb\|center\|Shooting analysis for \((m,\bar k)=(0,1)\): \(\Delta b(\bar\sigma^2)\) on the left and the absolute value of\(\Delta b(\bar\sigma^2)\) on the right on a logarithmic scale. The red points mark the zeros of \(\Delta b\), which correspond exactly to the eigenvalues. The dashed line indicates the threshold \(-\bar k^2=-1\).]]

	The numerical evaluation of the shooting method shows that \(\Delta b(\bar\sigma^2)\) has several zeros below the threshold \(-\bar k^2\), whereas only a single zero occurs above this threshold. This behavior is consistent with the spectrum discussed above, since several eigenvalues occur below the threshold, whereas only discrete eigenvalues are possible above the threshold. For the case considered here, \((m,\bar k) = (0,1)\), exactly one eigenvalue above \(-\bar k^2\) is obtained on the finite computational domain.		The numerical evaluation of the shooting method shows that \(\Delta b(\bar\sigma^2)\) has several zeros below the threshold \(-\bar k^2\), whereas only a single zero occurs above this threshold. This behavior is consistent with the spectrum discussed above, since several eigenvalues occur below the threshold, whereas only discrete eigenvalues are possible above the threshold. For the case considered here, \((m,\bar k) = (0,1)\), exactly one eigenvalue above \(-\bar k^2\) is obtained on the finite computational domain.
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	In the following figure, \(\|\psi(x_{\max})\|\) is shown as a function of \(x_{\max}\) for several values of \(\bar\sigma^2\) in the vicinity of the zero found above the threshold. For each value of \(\bar\sigma^2\), the value \(b_\psi\) was chosen such that \(u(x_{\max})=0\) is satisfied. According to the argument above, only for the actual eigenvalue should \(\psi(x_{\max})=0\) additionally hold, whereas for neighboring values of \(\bar\sigma^2\), \(u(x_{\max})\) generally vanishes, but \(\psi\) still contains a small, nonvanishing contribution of the asymptotically growing branch.		In the following figure, \(\|\psi(x_{\max})\|\) is shown as a function of \(x_{\max}\) for several values of \(\bar\sigma^2\) in the vicinity of the zero found above the threshold. For each value of \(\bar\sigma^2\), the value \(b_\psi\) was chosen such that \(u(x_{\max})=0\) is satisfied. According to the argument above, only for the actual eigenvalue should \(\psi(x_{\max})=0\) additionally hold, whereas for neighboring values of \(\bar\sigma^2\), \(u(x_{\max})\) generally vanishes, but \(\psi\) still contains a small, nonvanishing contribution of the asymptotically growing branch.

			[[File:MAES-PsiXmaxXmaxM0K1Xmax12.png\|400px\|thumb\|center\|Behavior of the absolute value of \(\psi(x_{\max})\) as a function of \(x_{\max}\) for different values of \(\bar\sigma^2\) in the vicinity of the eigenvalue found above the threshold \(-\bar k^2\), here for \((m,\bar k)=(0,1)\). For each \(\bar\sigma^2\), \(b_\psi\) was chosen such that \(u(x_{\max})=0\) holds.]]

	It turns out that \(\|\psi(x_{\max})\|\) remains very small as \(x_{\max}\) increases only for one distinguished value of \(\bar\sigma^2\), which indicates that the prefactor of the growing asymptotic branch vanishes there. By contrast, even small deviations from the actual eigenvalue lead to a strong growth of \(\|\psi(x_{\max})\|\).		It turns out that \(\|\psi(x_{\max})\|\) remains very small as \(x_{\max}\) increases only for one distinguished value of \(\bar\sigma^2\), which indicates that the prefactor of the growing asymptotic branch vanishes there. By contrast, even small deviations from the actual eigenvalue lead to a strong growth of \(\|\psi(x_{\max})\|\).