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Article Contents

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

• 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.

 Citation:

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

 $S$ $\min\limits_{\mathbf{u} \in S} \lambda _{\min}$ $\max\limits_{\mathbf{u} \in S} \lambda _{\max}$ $\mathop {{\rm{mean}}{\lambda _{{\rm{min}}}}}\limits_{\mathbf{u} \in S}$ $\mathop {{\rm{mean}}{\lambda _{{\rm{max}}}}}\limits_{\mathbf{u} \in S}$ $\min\limits_{\mathbf{u} \in S} \lambda _{1}$ $\max\limits_{\mathbf{u} \in S} \lambda _{1}$ $\min\limits_{\mathbf{u} \in S} \lambda _{4}$ $\max\limits_{\mathbf{u} \in S} \lambda _{4}$ $\mathcal{R}_\mathbf{b}{}$ 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

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

Figures(9)

Tables(2)