Second-order mixed-moment model with differentiable ansatz function in slab geometry

Florian Schneider

doi:10.3934/krm.2018049

Article Contents

2018, Volume 11, Issue 5: 1255-1276. Doi: 10.3934/krm.2018049

This issue Previous Article Stability of traveling waves for nonlocal time-delayed reaction-diffusion equations Next Article

Second-order mixed-moment model with differentiable ansatz function in slab geometry

Florian Schneider

Fachbereich Mathematik, TU Kaiserslautern, Erwin-Schrödinger-Str., 67663 Kaiserslautern, Germany

Received: December 31, 2016

Revised: September 30, 2017

Published: May 2018

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Abstract

Mixed-moment minimum-entropy models (${\rm{M}}{{\rm{M}}_N}$ models) are known to overcome the zero net-flux problem of full-moment minimum entropy ${{\rm{M}}_N}$ models but lack regularity. We study differentiable mixed-moment models (full zeroth and first moment, half higher moments, called ${\rm{DM}}{{\rm{M}}_N}$ models) for a Fokker-Planck equation in one space dimension. Realizability theory for these modification of mixed moments is derived for second order. Numerical tests are performed with a kinetic first-order finite volume scheme and compared with ${{\rm{M}}_N}$, classical ${\rm{M}}{{\rm{M}}_N}$ and a ${{\rm{P}}_N}$ reference scheme.

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Mathematics Subject Classification: Primary: 35L40, 35Q84, 65M70; Secondary: 65M08.

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Figure 1. Two ansatz functions and their derivatives for the ${\rm{M}}{{\rm{M}}_2}$ and ${\rm{DM}}{{\rm{M}}_2}$ model, respectively.
Left: ${\hat \psi }(\mu ) = \exp\left(\left(\mu -\mu ^2\right)1_{[0, 1]}-2\mu 1_{[-1, 0]}\right)$,
Right: ${\hat \psi }(\mu ) = \exp\left(-2\mu -\mu ^21_{[0, 1]}\right)$

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Figure 2. The normalized realizable set for the differentiable mixed-moment basis of order $N = 2$.
Online version: Press to activate 3D view ($x$-axis (red): ${\phi _{2 + }}$, $y$-axis (green): ${\phi _{2 - }}$, $z$-axis (blue): ${{\phi }_{1}}$)

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Figure 3. Eigenvalues of the ${\rm{DM}}{{\rm{M}}_2}$ flux Jacobian $\frac{\partial \mathbf{F}(\mathbf{u} )}{\partial \mathbf{u}}$ along ${{\phi }_{1}} = \frac{1}{2}\left({\phi _{2 + }} - \sqrt{{\phi _{2 - }}\, \left(1-{\phi _{2 + }}\right)} +\sqrt{{\phi _{2 + }}\, \left(1-{\phi _{2 - }}\right)} - {\phi _{2 - }}\right)$

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Figure 4. Minimal and maximal distance between adjacent eigenvalues of the ${\rm{DM}}{{\rm{M}}_2}$ flux Jacobian $\frac{\partial \mathbf{F}(\mathbf{u})}{\partial \mathbf{u}}$ along the cut ${{\phi }_{1}} = \frac{1}{2}\left({\phi _{2 + }} - \sqrt{{\phi _{2 - }}\, \left(1-{\phi _{2 + }}\right)} +\sqrt{{\phi _{2 + }}\, \left(1-{\phi _{2 - }}\right)} - {\phi _{2 - }}\right)$

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Figure 5. Eigenvalues at $5\%$ regularized boundary moments.
Online version: Press to activate 3D view ($x$-axis (red): ${\phi _{2 + }}$, $y$-axis (green): ${\phi _{2 - }}$, $z$-axis (blue): ${{\phi }_{1}}$)

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Figure 6. Minimal eigenvalue distance for $5\%$ regularized boundary moments.
Online version: Press to activate 3D view ($x$-axis (red): ${\phi _{2 + }}$, $y$-axis (green): ${\phi _{2 - }}$, $z$-axis (blue): ${{\phi }_{1}}$)

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Figure 7. Results for the plane-source test at the final time ${t_f} = 1$

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Figure 8. Results for the source-beam test at the final time ${t_f} = 2.5$

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Figure 9. DMM₂ approximation of the non-differentiable MM₂ ansatz (27)

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Table 1. Quantitative investigation of the minimal and maximal gaps of the eigenvalues at selected parts $S$ of the realizable set for fixed ${{\phi }_{1}} = \frac{1}{2}\left({\phi _{2 + }} - \sqrt{{\phi _{2 - }}\, \left(1-{\phi _{2 + }}\right)} +\sqrt{{\phi _{2 + }}\, \left(1-{\phi _{2 - }}\right)} - {\phi _{2 - }}\right)$

$S$	$\min\limits_{\mathbf{u} \in S} \lambda _{\min}$	$\max\limits_{\mathbf{u} \in S} \lambda _{\max}$	$\mathop {{\rm{mean}}{\lambda _{{\rm{min}}}}}\limits_{\mathbf{u} \in S} $	$\mathop {{\rm{mean}}{\lambda _{{\rm{max}}}}}\limits_{\mathbf{u} \in S} $	$\min\limits_{\mathbf{u} \in S} \lambda _{1}$	$\max\limits_{\mathbf{u} \in S} \lambda _{1}$	$\min\limits_{\mathbf{u} \in S} \lambda _{4}$	$\max\limits_{\mathbf{u} \in S} \lambda _{4}$
$\mathcal{R}_\mathbf{b}{}$	0.0260	1.9058	0.3287	1.0283	-0.9948	-0.1326	0.1326	0.9948
${\phi _{2 + }}+{\phi _{2 - }}\geq \frac{19}{20}$	0.0260	1.9058	0.0438	1.7811	-0.9948	-0.3778	0.3778	0.9948
${\phi _{2 + }}\leq \frac{1}{120}$	0.0314	1.2696	0.2066	0.5884	-0.9943	-0.1326	0.1326	0.4376
${\phi _{2 - }}\leq \frac{1}{120}$	0.0314	1.2696	0.2066	0.5884	-0.4376	-0.1326	0.1326	0.9943
${\phi _{2 + }}={\phi _{2 - }}$	0.0392	1.9058	0.3949	0.8791	-0.9921	-0.1326	0.1326	0.9921

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Table 2. Quantitative investigation of the minimal and maximal gaps of the eigenvalues at selected parts $S$ of the realizable set for the $5\%$-regularized boundary moments

$S$	$\min\limits_{\mathbf{u} \in S} \lambda _{\min}$	$\max\limits_{\mathbf{u} \in S} \lambda _{\max}$	$\mathop {{\rm{mean}}{\lambda _{{\rm{min}}}}}\limits_{\mathbf{u} \in S} $	$\mathop {{\rm{mean}}{\lambda _{{\rm{max}}}}}\limits_{\mathbf{u} \in S} $	$\min\limits_{\mathbf{u} \in S} \lambda _{1}$	$\max\limits_{\mathbf{u} \in S} \lambda _{1}$	$\min\limits_{\mathbf{u} \in S} \lambda _{4}$	$\max\limits_{\mathbf{u} \in S} \lambda _{4}$
$\mathcal{R}_\mathbf{b}{}$	0.0334	1.8812	0.1921	1.0945	-0.9936	-0.2315	0.2315	0.9936
${\phi _{2 + }}+{\phi _{2 - }}\geq \frac{19}{20}$	0.0334	1.8812	0.0393	1.8611	-0.9936	-0.9709	0.9709	0.9936
${\phi _{2 + }}\leq \frac{1}{120}$	0.0603	1.4064	0.2291	0.8726	-0.9917	-0.2806	0.2315	0.9855
${\phi _{2 - }}\leq \frac{1}{120}$	0.0603	1.4064	0.2291	0.8726	-0.9855	-0.2315	0.2806	0.9917
${\phi _{2 + }}={\phi _{2 - }}$	0.0527	1.8138	0.1792	1.0526	-0.9897	-0.2789	0.2789	0.9897

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