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Second Order Closures for the Radiative Transfer Equation: Some Are Unstable

Nickolay Y. Gnedin, Harley Katz

Abstract

The largest existing simulations of cosmic reionization model radiative transfer with moment methods that require a closure relation. The two most commonly used closure relations are M1 and OTVET; both close the moment hierarchy at the first moment. We explore the properties of a higher, second-order closure. We show that direct generalizations of M1 and OTVET to one higher order are physically unstable - i.e., the closure equations themselves result in unstable solutions, not just their numerical implementation. In fact, a generalization of OTVET to any order higher than the first one is unstable. We are also able to show that any local (i.e., depending only on the local moments of the radiation field, like M1) second-order closure that depends only on the radiation intensity and radiation flux, but does not explicitly depend on the radiation pressure, is physically unstable. This result restricts the choice of possible second-order closure relations.

Second Order Closures for the Radiative Transfer Equation: Some Are Unstable

Abstract

The largest existing simulations of cosmic reionization model radiative transfer with moment methods that require a closure relation. The two most commonly used closure relations are M1 and OTVET; both close the moment hierarchy at the first moment. We explore the properties of a higher, second-order closure. We show that direct generalizations of M1 and OTVET to one higher order are physically unstable - i.e., the closure equations themselves result in unstable solutions, not just their numerical implementation. In fact, a generalization of OTVET to any order higher than the first one is unstable. We are also able to show that any local (i.e., depending only on the local moments of the radiation field, like M1) second-order closure that depends only on the radiation intensity and radiation flux, but does not explicitly depend on the radiation pressure, is physically unstable. This result restricts the choice of possible second-order closure relations.
Paper Structure (9 sections, 45 equations, 4 figures)

This paper contains 9 sections, 45 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Well-known Problems with First Order Closures
  3. Stability Analysis for Closures at the Second Order
  4. OTVET-like Closures at the Second Order
  5. Restricted Local Closures at the Second Order
  6. Special Case $A(f)\equiv 1$
  7. Conclusions
  8. A. Characteristic Equation for the Restricted Local Second-Order Closure
  9. B. Full Linear Stability Analysis for the Special Case $A\equiv 1$.

Figures (4)

  • Figure 1: Top: radiation field in M1 closure (in arbitrary units) from two equal-luminosity sources at three different snapshots. Bottom: ionized bubbles in OTVET closure around two unequal sources; the right source is 10 times more luminous than the left one. In both rows, there are noticeable artifacts.
  • Figure 2: Radiation field (in arbitrary units) from a single isotropic source in the empty space at four snapshots. The top row shows the second-order closure with $A(f)=f$ and the bottom row is for $A(f)=f^3$. The instability develops faster in the bottom row than in the top row, as expected.
  • Figure 3: Radiation field (in arbitrary units) from a single isotropic source in the empty space at four snapshots linearly spaced in time for the special case $A(f)\equiv 1$ closure. Simulations start with the exact solution $E=L/(4\pi c r^2)$ to avoid any dynamical effect of a light front propagating from an instantly switched-on source. The top and middle panels show tests with point sources and effective resolution of $512^3$ and $1024^3$, respectively (only 1/6 of the simulation volume is shown in the image). The bottom row shows the same test as in the middle row, with the source being spread in a Gaussian cloud with $\sigma$ equal to 3 cell sizes. Despite being linearly stable, the closure does develop non-spherical structures in the solution.
  • Figure 4: Evolution of non-trivial multipoles for the numerical solution of the $A(f)\equiv 1$ test at a distance of 0.1 of the box size from the source. Initially, the spherical symmetry is preserved, but the deviation from the exact solution increases linearly with time until it enters the non-linear regime, at which point the hexadecapole and later higher overtones appear.