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On Nonlinear Closures for Moment Equations Based on Orthogonal Polynomials

Eda Yilmaz, Georgii Oblapenko, Manuel Torrilhon

Abstract

In the present work, an approach to the moment closure problem on the basis of orthogonal polynomials derived from Gram matrices is proposed. Its properties are studied in the context of the moment closure problem arising in gas kinetic theory, for which the proposed approach is proven to have multiple attractive mathematical properties. Numerical studies are carried out for model gas particle distributions and the approach is compared to other moment closure methods, such as Grad's closure and the maximum-entropy method. The proposed ``Gramian'' closure is shown to provide very accurate results for a wide range of distribution functions.

On Nonlinear Closures for Moment Equations Based on Orthogonal Polynomials

Abstract

In the present work, an approach to the moment closure problem on the basis of orthogonal polynomials derived from Gram matrices is proposed. Its properties are studied in the context of the moment closure problem arising in gas kinetic theory, for which the proposed approach is proven to have multiple attractive mathematical properties. Numerical studies are carried out for model gas particle distributions and the approach is compared to other moment closure methods, such as Grad's closure and the maximum-entropy method. The proposed ``Gramian'' closure is shown to provide very accurate results for a wide range of distribution functions.
Paper Structure (30 sections, 11 theorems, 67 equations, 7 figures, 1 table)

This paper contains 30 sections, 11 theorems, 67 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Moment Closures
  3. Structure-preserving Moment Closures
  4. Hyperbolicity
  5. Gauge-Invariance
  6. Equilibrium
  7. Orthogonal Polynomials
  8. Gramian Closure
  9. Definition and Hyperbolicity
  10. Invariance
  11. Extended Gramian Closure: The Even Case
  12. Definition and Hyperbolicity
  13. Invariance
  14. Equilibrium Preservation
  15. Extended Gramian Closure: The Odd Case
  16. ...and 15 more sections

Key Result

Theorem 3.1

For a moment closure $u_{M+1}=\mathcal{C}\left(u_{0},\cdots, u_{M}\right)$ the following statements are equivalent.

Figures (7)

  • Figure 1: Family of distribution functions for test problems in subsection \ref{['subsec:mottsmith']}, \ref{['subsec:electronhole']}, and \ref{['subsec:bimodal']}. The Mott-Smith distribution (a) for shock waves cite_6 is set up as a bimodal type for shock waves and uses parameters related by the Rankine-Hugoniot conditions. The electron-hole distribution (b) from bujarbarua1981theory with $\beta=-0.05$ is a piece-wise function, one part is a shifted Gaussian and the other one is close to a flat-top distribution. The distribution function in (c) is modeled by a superposition of two Gaussian to analyze the closures near the realizability boundary. Apart from some fixed parameters, the change in the distributions is with respect to (a): position, (b): potential, (c): width of two Gaussian. Some example distributions from these families are displayed.
  • Figure 2: Relative error $e_r(u_{\text{closure}})$ of the next higher moment calculated for different number of moments using different closure techniques is shown for the Mott-Smith shock wave distribution cite_6. We consider $\text{Ma}=4$, $\gamma=5/3$ and compute moments with the distribution $f_{\text{MS}}$ for each $x$ values from -10 to 10 with step size 0.25. For the maximum entropy closure, we discretize velocity domain $[-6,9]$ with 1000 points. Note, that the Gramian and extended Gramian closure are defined differently and marked by different color in the plots.
  • Figure 3: Relative error between $u_{\text{closure}}$ and $u_{M+1}$ with fixed parameters $v_{0}=1.5$, $\beta=-0.05$ for the electron hole distribution bujarbarua1981theory with different electrostatic potentials $0\leq\phi\leq2$ with step size $0.04$. Even (left) and odd (right) case scenarios were examined and are shown separately due to differences in the definition of the Gramian and extended Gramian closures. For the maximum entropy closure, the velocity domain $[-6,8]$ is discretized with 1000 grid points.
  • Figure 4: Relative error $e_r(u_{\text{closure}})$ of the next higher moment for a range of widths $w$ in the test of the realizability boundaries. The left part of the figure shows the relative error for even numbers of moments $M=4,6$ while the right one shows odd cases $M=5,7$. For the maximum entropy closure, computational domain is taken with $c\in[-4,5]$ with 1000 grid points.
  • Figure 5: Condition numbers of different closure methods as a function of the order of the moment theory for the bimodal test problem of subsection \ref{['subsec:bimodal']}. The x-axis represents the width of the distribution function, and the y-axis displays the corresponding condition number of the Gramian matrix. Each colored line represents a different closure method. Only even and odd Gramian closure results are shown in the figure. The condition numbers of the extended Gramian closures are quite similar to the Gramian closures. Therefore, these results are excluded from the figure.
  • ...and 2 more figures

Theorems & Definitions (20)

  • Definition 2.1: Moment Closure
  • Definition 3.1: Characteristic Polynomial
  • Definition 3.2: Strict Hyperbolicity
  • Definition 3.3: Gauge-Invariant Closure
  • Theorem 3.1
  • Definition 3.4: Equilibrium Preservation
  • Definition 4.1: Gramian Matrix
  • Definition 5.1: Gramian Closure -- even
  • Theorem 5.1
  • Corollary 5.1
  • ...and 10 more