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

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  • 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.

    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: ${\hat \psi }(\mu ) = \exp\left(-2\mu -\mu ^21_{[0, 1]}\right)$

    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): ${{\phi }_{1}}$)

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

    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 ${{\phi }_{1}} = \frac{1}{2}\left({\phi _{2 + }} - \sqrt{{\phi _{2 - }}\, \left(1-{\phi _{2 + }}\right)} +\sqrt{{\phi _{2 + }}\, \left(1-{\phi _{2 - }}\right)} - {\phi _{2 - }}\right)$

    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): ${{\phi }_{1}}$)

    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): ${{\phi }_{1}}$)

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

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

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

    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} $ $\min\limits_{\mathbf{u} \in S} \lambda _{1}$ $\max\limits_{\mathbf{u} \in S} \lambda _{1}$ $\min\limits_{\mathbf{u} \in S} \lambda _{4}$ $\max\limits_{\mathbf{u} \in S} \lambda _{4}$
    ${\phi _{2 + }}+{\phi _{2 - }}\geq \frac{19}{20}$0.02601.90580.04381.7811-0.9948-0.37780.37780.9948
    ${\phi _{2 + }}\leq \frac{1}{120}$0.03141.26960.20660.5884-0.9943-0.13260.13260.4376
    ${\phi _{2 - }}\leq \frac{1}{120}$0.03141.26960.20660.5884-0.4376-0.13260.13260.9943
    ${\phi _{2 + }}={\phi _{2 - }}$0.03921.90580.39490.8791-0.9921-0.13260.13260.9921
     | Show Table
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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} $ $\min\limits_{\mathbf{u} \in S} \lambda _{1}$ $\max\limits_{\mathbf{u} \in S} \lambda _{1}$ $\min\limits_{\mathbf{u} \in S} \lambda _{4}$ $\max\limits_{\mathbf{u} \in S} \lambda _{4}$
    ${\phi _{2 + }}+{\phi _{2 - }}\geq \frac{19}{20}$0.03341.88120.03931.8611-0.9936-0.97090.97090.9936
    ${\phi _{2 + }}\leq \frac{1}{120}$0.06031.40640.22910.8726-0.9917-0.28060.23150.9855
    ${\phi _{2 - }}\leq \frac{1}{120}$0.06031.40640.22910.8726-0.9855-0.23150.28060.9917
    ${\phi _{2 + }}={\phi _{2 - }}$0.05271.81380.17921.0526-0.9897-0.27890.27890.9897
     | Show Table
    DownLoad: CSV
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