First-order quarter-and mixed-moment realizability theory and Kershaw closures for a Fokker-Planck equation in two space dimensions

Figure 1.  Interpolation between two realizability boundaries in the projected three-space $(\phi _{\mathcal{S}_x^ + }^{\left( {1,0} \right)},\phi _{\mathcal{S}_x^ - }^{\left( {1,0} \right)},\phi _{\mathcal{S}_y^ + }^{\left( {0,1} \right)})$ along an isoline of $\mathcal{N}\left( {{\phi ^{\left| 1 \right|}}} \right)$. The realizable set with respect to the quadrant ${\mathcal{S}_{ - + }}$ is plotted in grey

Figure 2.  The eigenvector ${\bf{v}}_1({\phi _{{\mathcal{S}_{ + + }}}})$ (blue arrows) of the ${\rm{Q}}{{\rm{M}}_1}$ second moment $\phi _{{\mathcal{S}_{ + + }}}^{\left| 2 \right|}$ (Figure (a)) and its minimal deviation from $\phi _{{\mathcal{S}_{+ +}}}^{\left| 1 \right|}$ in degrees (Figure (b))

Figure 3.  Components of the $Q{K_1}$ second moment on ${\mathcal{S}_{ + + }}$

Figure 4.  Deviation of $\phi _{{\mathcal{S}_{ + + }}}^{\left( {2,0} \right)}$ and $\phi _{{\mathcal{S}_{ + + }}}^{\left( {2,0} \right)}$ for ${\rm{Q}}{{\rm{M}}_1}$ and $Q{K_1}$, respectively

Figure 5.  Eigenvalues of the flux Jacobian in $x$-direction ${\bf{F}}_1^\prime$

Figure 6.  Minimal and maximal distance of adjacent eigenvalues of the flux Jacobian ${\bf{F}}_1^\prime$

Figure 7.  Visualization of the coefficients $\rho _{\iota \pm}$ for the example $\phi _{{\mathcal{S}_{+ +}}}^{\left| 1 \right|}=\left(0.4,0.6\right)^T$. The length of the arrows represent the maximal value for the respective coefficient

Figure 8.  Second moment $\phi _{\mathcal{S}_y^ + }^{\left( {0,2} \right)}$ for the ${\rm{M}}{{\rm{M}}_1}$ and Kershaw model for $\phi _{\mathcal{S}_y^ + }^{\left( {0,1} \right)} = -\phi _{\mathcal{S}_y^ - }^{\left( {0,1} \right)} = \frac14$

Figure 9.  Local particle density ${u^{\left( {0,0} \right)}}$ of some models at $t = 1.2$ in the two-beams test case

Figure 10.  Local particle density ${u^{\left( {0,0} \right)}}$ at $t = 1.2$ in the opposing two-beams test case

Figure 11.  Local particle density ${u^{\left( {0,0} \right)}}$ at $t = 1.2$ in the rotated two-beams test case

Figure 12.  Local particle density ${u^{\left( {0,0} \right)}}$ of some models at $t = 1$ in the source-beam test case

Figure 13.  Local particle density ${u^{\left( {0,0} \right)}}$ of some models at $t = 2.5$ in the source-beam test case

Figure 14.  Horizontal cut ($y = 1.5$) of the local particle density ${u^{\left( {0,0} \right)}}$ of some models at $t = 1$ and $t = 2.5$, respectively, in the source-beam test case

Figure 15.  Local particle density ${u^{\left( {0,0} \right)}}$ of some models at $t = 0.45$ in the line-source test case

Figure 16.  Local particle density ${u^{\left( {0,0} \right)}}$ of some models at $t = 0.45$ in the line-source test case, horizontal and diagonal cuts