Article Contents
Article Contents

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

• * Corresponding author: Florian Schneider
• Mixed-moment models, introduced in [8,44] for one space dimension, are a modification of the method of moments applied to a (linear) kinetic equation, by choosing mixtures of different partial moments. They are well-suited to handle equations where collisions of particles are modelled with a Laplace-Beltrami operator. We generalize the concept of mixed moments to two dimensions. In the context of minimum-entropy models, the resulting hyperbolic system of equations has desirable properties (entropy-diminishing, bounded eigenvalues), removing some drawbacks of the well-known M1 model. We furthermore provide a realizability theory for a first-order system of mixed moments by linking it to the corresponding quarter-moment theory. Additionally, we derive a type of Kershaw closures for mixed-and quarter-moment models, giving an efficient closure (compared to minimum-entropy models). The derived closures are investigated for different benchmark problems.

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

 Citation:

• 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

Table 1.  Absolute and relative running times (wrt. ${\rm{M}}{{\rm{M}}_1}$) of some models for the source-beam test

 Model $n$ Abs. runtime [s] Rel. runtime ${\rm{M}}{{\rm{M}}_1}$ 5 1.305e+05 3.885e-01 ${\rm{M}}{{\rm{M}}_2}$ 13 3.359e+05 1.0 ${\rm{M}}{{\rm{K}}_1}$ 5 3.408e+01 1.015e-04 ${{\rm{M}}_1}$ 3 9.418e+04 2.804e-01 ${{\rm{M}}_2}$ 6 2.269e+05 6.757e-01 $Q{K_1}$ 12 5.309e+01 1.581e-04

Table 2.  Absolute and relative running times (wrt. ${\rm{M}}{{\rm{M}}_1}$) of some models for the source-beam test.

 Model $n$ Abs. runtime [s] Rel. runtime ${\rm{M}}{{\rm{M}}_1}$ 5 2.489e+05 1.0 ${\rm{M}}{{\rm{M}}_2}$ 13 9.526e+04 3.828e-01 ${\rm{M}}{{\rm{K}}_1}$ poly 5 1.535e+02 6.169e-04 ${{\rm{M}}_1}$ 3 6.639e+04 2.668e-01 ${{\rm{M}}_2}$ 6 1.189e+05 4.777e-01 ${{\rm{M}}_3}$ 12 2.232e+05 8.967e-01

Table 3.  Absolute and relative running times (wrt. ${\rm{M}}{{\rm{M}}_3}$) of some models for the line-source test

 Model $n$ Abs. runtime [s] Rel. runtime ${\rm{M}}{{\rm{M}}_1}$ 5 1.638e+05 2.091e-01 ${\rm{M}}{{\rm{M}}_2}$ 13 3.491e+05 4.456e-01 ${\rm{M}}{{\rm{M}}_3}$ 25 7.834e+05 1.0 ${\rm{M}}{{\rm{K}}_1}$ poly 5 1.888e+02 2.409e-04 ${\rm{M}}{{\rm{K}}_1}$ 5 2.898e+04 3.700e-02 ${{\rm{M}}_1}$ 3 5.458e+04 6.968e-02 ${{\rm{M}}_2}$ 6 9.696e+04 1.238e-01 ${{\rm{M}}_3}$ 10 1.823e+05 2.327e-01

Figures(16)

Tables(3)