Nonlinear Fokker-Planck Acceleration for Forward-Peaked Transport Problems in Slab Geometry

Transport problems with highly forward-peaked scattering are prevalent in a variety of areas, including astrophysics, medical physics, and plasma physics [1, 2, 3]. For these problems, solutions of the transport equation converge slowly when using conventional methods such as source iteration (SI) [4] and the generalized minimal residual method (GMRES) [5]. Moreover, diffusion-based acceleration techniques like diffusion synthetic acceleration (DSA) [6] and nonlinear diffusion acceleration (NDA) [7] are generally inefficient when tackling these problems, as they only accelerate up to the first moment of the angular flux [8]. In fact, higher-order moments carry important information in problems with highly forward-peaked scattering and can be used to further accelerate convergence [9].

This paper focuses on solution methods for the monoenergetic, steady-state transport equation in homogeneous slab geometry. Under these conditions, the transport equation is given by

	$$\mu\frac{\partial}{\partial x}\psi(x,\mu)+\sigma_{t}\psi(x,\mu)=\int_{-1}^{1}d\mu^{\prime}\sigma_{s}(\mu,\mu^{\prime})\psi(x,\mu^{\prime})+Q(x,\mu),\,\,\,x\in[0,X],-1\leq\mu\leq 1,\\$$			(1.1a)
with boundary conditions
	$\displaystyle\psi(0,\mu)$	$\displaystyle=\psi_{L}(\mu),\quad\mu>0,$		(1.1b)
	$\displaystyle\psi(X,\mu)$	$\displaystyle=\psi_{R}(\mu),\quad\mu<0.$		(1.1c)

Here, $\psi(x,\mu)$ represents the angular flux at position $x$ and direction $\mu$, $\sigma_{t}$ is the macroscopic total cross section, $\sigma_{s}(\mu,\mu^{\prime})$ is the differential scattering cross section, and $Q$ is an internal source.

New innovations have paved the way to better solve this equation in systems with highly forward-peaked scattering. For instance, work has been done on modified $P_{L}$ equations and modified scattering cross section moments to accelerate convergence of anisotropic neutron transport problems [10]. In order to speed up the convergence of radiative transfer in clouds, a quasi-diffusion method has been developed [2]. In addition, the DSA-multigrid method was developed to solve problems in electron transport more efficiently [11].

One of the most recent convergence methods developed is Fokker-Planck Synthetic Acceleration (FPSA) [8, 9]. FPSA accelerates up to $N$ moments of the angular flux and has shown significant improvement in the convergence rate for the types of problems described above. The method returns a speed-up of several orders of magnitude with respect to wall-clock time when compared to DSA [8].

In this paper, we introduce a new acceleration technique, called Nonlinear Fokker-Planck Acceleration (NFPA). This method returns a modified Fokker-Planck (FP) equation that preserves the angular moments of the flux given by the transport equation. This preservation of moments is particularly appealing for applications to multiphysics problems [3], in which the coupling between the transport physics and the other physics can be done through the (lower-order) FP equation. To our knowledge, this is the first implementation of a numerical method that returns a Fokker-Planck-like equation that is discretely consistent with the linear Boltzmann equation.

This paper is organized as follows. Section 2 starts with a brief description of FPSA. Then, we derive the NFPA scheme. In Section 3, we discuss the discretization schemes used in this work and present numerical results. These are compared against standard acceleration techniques. We conclude with a discussion in Section 4.

In this section we briefly outline the theory behind FPSA, describe NFPA for monoenergetic, steady-state transport problems in slab geometry, and present the numerical methodology behind NFPA. The theory given here can be easily extended to higher-dimensional problems. Moreover, extending the method to energy-dependence shall not lead to significant additional theoretical difficulties.

To solve the transport problem given by Eq. 1.1 we approximate the in-scattering term in Eq. 1.1a with a Legendre moment expansion:

$$\mu\frac{\partial}{\partial x}\psi(x,\mu)+\sigma_{t}\psi(x,\mu)=\sum_{l=0}^{L}\frac{(2l+1)}{2}P_{l}(\mu)\sigma_{s,l}\phi_{l}(x)+Q(x,\mu),$$

(2.1)

with

$$\phi_{l}(x)=\int_{-1}^{1}d\mu P_{l}(\mu)\psi(x,\mu).$$

(2.2)

Here, $\phi_{l}$ is the $l^{th}$ Legendre moment of the angular flux, $\sigma_{s,l}$ is the $l^{th}$ Legendre coefficient of the differential scattering cross section, and $P_{l}$ is the $l^{th}$-order Legendre polynomial. For simplicity, we will drop the notation $(x,\mu)$ in the remainder of this section.

The solution to Eq. 2.1 converges asymptotically to the solution of the following Fokker-Planck equation in the forward-peaked limit [12]:

$$\mu\frac{\partial\psi}{\partial x}+\sigma_{a}\psi=\frac{\sigma_{tr}}{2}\frac{\partial}{\partial\mu}(1-\mu^{2})\frac{\partial\psi}{\partial\mu}+Q\,,$$

(2.3)

where $\sigma_{tr}=\sigma_{s,0}-\sigma_{s,1}$ is the momentum transfer cross section and $\sigma_{a}=\sigma_{t}-\sigma_{s,0}$ is the macroscopic absorption cross section.

Source Iteration [4] is generally used to solve Eq. 2.1, which can be rewritten in operator notation:

$$\mathcal{L}\psi^{m+1}=\mathcal{S}\psi^{m}+Q\,,$$

(2.4)

where

$$\mathcal{L}=\mu\frac{\partial}{\partial x}+\sigma_{t},\quad\mathcal{S}=\sum_{l=0}^{L}\frac{(2l+1)}{2}P_{l}(\mu)\sigma_{s,l}\int_{-1}^{1}d\mu P_{l}(\mu),$$

(2.5)

and $m$ is the iteration index. This equation is solved iteratively until a tolerance criterion is met. The FP approximation shown in Eq. 2.3 can be used to accelerate the convergence of Eq. 2.1.

In Section 3.1 we describe the discretization methods used to implement the algorithms. In Section 3.2 we provide numerical results for 2 different choices of source $Q$ and boundary conditions. For each choice we solve the problem using 3 different scattering kernels, applying 3 different choices of parameters for each kernel. We provide NFPA numerical results for these 18 cases and compare them against those obtained from FPSA and other standard methods.

All numerical experiments were performed using MATLAB. Runtime was tracked using the tic-toc functionality [15], with only the solver runtime being taken into consideration in the comparisons. A 2017 MacBook Pro with a 2.8 GHz Quad-Core Intel Core i7 and 16 GB of RAM was used for all simulations.

This paper introduced the Nonlinear Fokker-Planck Acceleration technique for steady-state, monoenergetic transport in homogeneous slab geometry. To our knowledge, this is the first nonlinear HOLO method that accelerates all $L$ moments of the angular flux. Upon convergence, the LO and HO models are consistent; in other words, the (lower-order) modified Fokker-Planck equation preserves the same angular moments of the flux obtained with the (higher-order) transport equation.

NFPA was tested on a homogeneous medium with an isotropic internal source with vacuum boundaries, and in a homogeneous medium with no internal source and an incoming beam boundary. For both problems, three different scattering kernels were used. The runtime and iterations of NFPA and FPSA were shown to be similar. They both vastly outperformed DSA and GMRES for all cases by orders of magnitude. However, NFPA has the feature of preserving the angular moments of the flux in both the HO and LO equations, which offers the advantage of integrating the LO model into multiphysics models.

In the future, we intend to test NFPA capabilities for a variety of multiphysics problems and analyze its performance. To apply NFPA to more realistic problems, it needs to be extended to include time and energy dependence. Additionally, the method needs to be adapted to address geometries with higher-order spatial dimensions. Finally, for the NFPA method to become mathematically “complete”, a full convergence examination using Fourier analysis must be performed. However, this is beyond the scope of this paper and must be left for future work.

The authors acknowledge support under award number NRC-HQ-84-15-G-0024 from the Nuclear Regulatory Commission. The statements, findings, conclusions, and recommendations are those of the authors and do not necessarily reflect the view of the U.S. Nuclear Regulatory Commission.

J. K. Patel would like to thank Dr. James Warsa for his wonderful transport class at UNM, as well as his synthetic acceleration codes. The authors would also like to thank Dr. Anil Prinja for discussions involving Fokker-Planck acceleration.