Positivity Limiters for Filtered Spectral Approximations of Linear Kinetic Transport Equations

Abstract

We analyze the properties and compare the performance of several positivity limiters for spectral approximations with respect to the angular variable of linear transport equations. It is well-known that spectral methods suffer from the occurrence of (unphysical) negative spatial particle concentrations due to the fact that the underlying polynomial approximations are not always positive at the kinetic level. Positivity limiters address this defect by enforcing positivity of the polynomial approximation on a finite set of preselected points. With a proper PDE solver, they ensure positivity of the particle concentration at each step in a time integration scheme. We review several known positivity limiters proposed in other contexts and also introduce a modification for one of them. We give error estimates for the consistency of the positive approximations produced by these limiters and compare the theoretical estimates to numerical results. We then solve two benchmark problems with these limiters, make qualitative and quantitative observations about the solutions, and then compare the efficiency of the different limiters.

Appendix A: Implementation Details

1.1 Appendix A.1: The Tensor Product Quadrature

We use a tensor product quadrature to define the positivity limiters and evaluate the numerical flux in the PDE solver. For functions g on $\mathbb {S}^2$, we write $g(\varOmega )$ in the polar coordinate as $g(\mu ,\phi )$, where $\mu =\varOmega _3\in [-1,1]$ and $\phi =(0,2\pi ]$ is the azimuthal angle on the sphere. This quadrature rule integrates any integrable g by

$$\begin{aligned} \int _{\mathbb {S}^2} g(\varOmega )d\varOmega = \int _{-1}^1\int _0^{2\pi } g(\mu ,\phi )d\phi d\mu \simeq \frac{\pi }{N_\mathcal {Q}}\sum _{k=1}^{N_\mathcal {Q}}\sum _{j=1}^{2N_\mathcal {Q}}{w}_k g(\mu _k,\phi _j). \end{aligned}$$

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Here $\{\mu _k\}_{k=1}^{N_\mathcal {Q}}$ and $\{{w}_k\}_{k=1}^{N_\mathcal {Q}}$ are the Gauss–Legendre abscissas and weights, $\{\phi _j\}_{j=1}^{2N_\mathcal {Q}}$ are equally spaced points from 0 to $2\pi$, and $N_\mathcal {Q}$ denotes the degree of precision of $\mathcal {Q}$. As discussed in Sect. 2.2, the quadrature is required to have degree of precision $2N+1$ when the approximation is of order N. Thus, a grid of at least $N+1$ (or $\lceil (N+1)/2 \rceil$ for even functions on $\mu$) Gauss–Legendre points in the $\mu$ direction and $2(N+1)$ equally spaced points in the $\phi$ direction is needed.

1.2 Appendix A.2: The Sweeping Permutation

As mentioned in Sect. 3.2.1, the swp approximation depends on the permutation $\varPi$. In this appendix, we give the details on the choices of $\varPi$ used in the numerical tests in Sects. 4.5 and 5.

For functions $\varphi$ on $[-1,1]$ tested in Sect. 4.5, the regular swp limiter simply chooses $\varPi$ to be the identity map, i.e., $\varPi (i)=i$ for all $i=1,\ldots ,|{Q_N}|$. This choice of $\varPi$ implies that the Gauss–Legendre quadrature points are sorted as $\mu _{\varPi (1)}\le \mu _{\varPi (2)}\le \cdots \le \mu _{\varPi (N+1)}$. For the swp-e limiter tested in Sect. 4.5, $\varPi$ is chosen so that all negative nodal values are first visited in the forward sweeping procedure, followed by the nonnegative nodal values in the ascending order with respect to the associated quadrature weights. Specifically,

$$\begin{aligned} \varPi (i)\in \{1,\ldots ,|{Q_N^-}|\}, \, \forall i\in {Q_N^-}\quad \text{ and } \quad w_{\varPi (i)}\le w_{\varPi (j)} \text { iff } {\varPi (i)}\le {\varPi (j)}, \, \forall i,j\in {Q_N^+}. \end{aligned}$$

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For functions $\varPhi$ on $\mathbb {S}^2$ in Sects. 4.5 and 5, the tensor-product quadrature points $\{\varOmega _i=(\mu _k,\phi _j)\}$ are given in “Appendix A.1”. Suppose that $\{\mu _k\}$ and $\{\phi _j\}$ are both sorted in the ascending order, i.e., $\mu _1\le \cdots \le \mu _{\lceil (N+1)/2 \rceil }$, $\phi _1\le \cdots \le \phi _{2(N+1)}$, and $\{\varOmega _i=(\mu _k,\phi _j)\}$ is ordered such that $i=2(k-1)(N+1)+j$. Then the regular swp limiter chooses $\varPi$ such that for $\varOmega _i = (\mu _k,\phi _j)$, $\varPi (i)=i=2(k-1)(N+1)+j$. Such choice of $\varPi$ sorts $\{\varOmega _i\}$ first in the ascending order of $\mu _k$, and then in the ascending order of the $\phi _j$. On the other hand, the swp-s limiter utilizes the rotational invariance of the line source solution and chooses $\varPi$ based on the spatial position. Specifically, at position $(x_1,x_2)$, let

$$\begin{aligned} {\hat{j}}\in {\mathop {{{\mathrm{argmin}}}}\limits _{j=1,\ldots ,2(N+1)}} |\phi _j - \arctan (\frac{x_2}{x_1})|. \end{aligned}$$

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In the swp-s limiter, $\varPi$ is chosen such that for $\varOmega _i = (\mu _k,\phi _j)$, if $x_1x_2\ge 0$,

$$\begin{aligned} \varPi (i)={\left\{ \begin{array}{ll} 2k(N+1)+j-\hat{j}+1, &{} \text {if } j<\hat{j},\\ 2(k-1)(N+1)+j-\hat{j}+1,&{} \text {otherwise}, \end{array}\right. } \end{aligned}$$

(69)

and if $x_1x_2<0$,

$$\begin{aligned} \varPi (i)={\left\{ \begin{array}{ll} 2(k-1)(N+1)-j+\hat{j}+1, &{} \text {if } j\le \hat{j},\\ 2k(N+1)-j+\hat{j}+1,&{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

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This choice of $\varPi$ is designed to produce solutions that are symmetric with respect to the quadrants, as discussed in Sect. 5.1.

1.3 Appendix A.3: The Hierarchical Structure

The implementation of h-clp limiter requires a hierarchy of cells on the angular space $\mathcal {S}$. When $\mathcal {S}=[-1,1]$, we construct the hierarchy to satisfy (17) via uniform mesh refinements with $K=3$. Specifically, we start with $C_{0,1}:=[-1,1]$, and decompose $C_{0,1}$ evenly into three cells $C_{1,1}:=[-1, -\frac{1}{3})$, $C_{1,2}:=[-\frac{1}{3}, \frac{1}{3})$, and $C_{1,3}:=[\frac{1}{3}, 1]$. Then we continue the refinement by evenly decomposing $C_{\ell ,k}$ into $C_{\ell +1, 3k-2}$, $C_{\ell +1, 3k-1}$, and $C_{\ell +1, 3k}$, until there exists some $C_{\ell ,k}$ that contains at most one quadrature point. Similarly, when $\mathcal {S}=\mathbb {S}^2$, we use the tensor product mesh by applying the uniform mesh for $[-1,1]$ on both the $\mu$ and $\phi$ directions, and continue refining the tensor product mesh until some cells contain at most one quadrature point. Specifically, given $C_{0,1}:=\mathbb {S}^2$, at level $\ell =1$, $C_{0,1}$ is decomposed into $C_{1,1}:=[-1, -\frac{1}{3}) \times [0, \frac{2\pi }{3})$, $C_{1,2}:=[-1, -\frac{1}{3}) \times [\frac{2\pi }{3}, \frac{4\pi }{3}), \ldots , C_{1,9}:=[\frac{1}{3}, 1] \times [\frac{4\pi }{3}, 2\pi ]$. For level $\ell >1$, the cells are defined analogously via this refinement.