Transport problems with highly forward-peaked scattering are prevalent in a variety of areas, including astrophysics, medical physics, and plasma physics HGK ; aristova ; multiphysics . For these problems, solutions of the transport equation converge slowly when using conventional methods such as source iteration (SI) adamslarsen and the generalized minimal residual method (GMRES) gmres . Moreover, diffusion-based acceleration techniques like diffusion synthetic acceleration (DSA) alcouffe and nonlinear diffusion acceleration (NDA) smithetall are generally inefficient when tackling these problems, as they only accelerate up to the first moment of the angular flux JapanFPSA . In fact, higher-order moments carry important information in problems with highly forward-peaked scattering and can be used to further accelerate convergence japanDiss .
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 equationparentequation
|with boundary conditions|
Here, represents the angular flux at position and direction , is the macroscopic total cross section, is the differential scattering cross section, and 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 equations and modified scattering cross section moments to accelerate convergence of anisotropic neutron transport problems khattab . In order to speed up the convergence of radiative transfer in clouds, a quasi-diffusion method has been developed aristova . In addition, the DSA-multigrid method was developed to solve problems in electron transport more efficiently trucksin .
One of the most recent convergence methods developed is Fokker-Planck Synthetic Acceleration (FPSA) JapanFPSA ; japanDiss . FPSA accelerates up to 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 JapanFPSA .
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 multiphysics , 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.
2 Fokker-Planck Acceleration
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.
Here, is the Legendre moment of the angular flux, is the Legendre coefficient of the differential scattering cross section, and is the -order Legendre polynomial. For simplicity, we will drop the notation in the remainder of this section.
where is the momentum transfer cross section and is the macroscopic absorption cross section.
2.1 FPSA: Fokker-Planck Synthetic Acceleration
In the FPSA scheme JapanFPSA ; japanDiss , the FP approximation is used as a preconditioner to synthetically accelerate convergence when solving Eq. 2.1 (cf. adamslarsen for a detailed description of synthetic acceleration). When solving Eq. 2.4, the angular flux at each iteration has an error associated with it. FPSA systematically follows a predict, correct, iterate scheme. A transport sweep, one iteration in Eq. 2.4, is made for a prediction. The FP approximation is used to correct the error in the prediction, and this iteration is performed until a convergence criterion is met. The equations used are: equationparentequation
where we define as
In this synthetic acceleration method, the FP approximation is used to correct the error in each iteration of the high-order (HO) equation (2.6a). Therefore, there is no consistency between the angular moments of the flux in the HO and low-order (LO) equations.
2.2 NFPA: Nonlinear Fokker-Planck Acceleration
Similar to FPSA, NFPA uses the FP approximation to accelerate the convergence of the solution. We introduce the additive term to Eq. 2.3, obtaining the modified FP equation
The NFPA method is given by the following equations: equationparentequation
where is the angular flux obtained from the HO equation and is the angular flux obtained from the LO equation. The nonlinear HOLO-plus-consistency system given by Eq. 2.10 can be solved using any nonlinear solution technique kelley . Note that the NFPA scheme returns a FP equation that is consistent with HO transport. Moreover, this modified FP equation accounts for large-angle scattering which the standard FP equation does not. The LO equation (2.3) can then be integrated into multiphysics models in a similar fashion to standard HOLO schemes patelFBR . To solve the HOLO-plus-consistency system above, we use Picard iteration kelley : equationparentequation
|Transport Sweep for HO||(2.11a)|
|Evaluate Consistency Term||(2.11b)|
|Solve LO Equation||(2.11c)|
3 Numerical Experiments
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 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 matlab17 , 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.
The Transport and FP equations were discretized using linear discontinuous finite element discretization in space mpd1 , and discrete ordinates (S) in angle landm . The Fokker-Planck operator was discretized using moment preserving discretization (MPD) mpd1 . Details of the derivation of the linear discontinuous finite element discretization can be seen in japanDiss ; martin . The finite element discretization for the Fokker-Planck equation follows the same derivation.
A brief review for the angular discretization used for the FP equation is given below. First, we use Gauss-Legendre quadrature to discretize the FP equation in angle:
for . Here, term is the discrete form of the angular Laplacian operator evaluated at angle .
The MPD scheme is then shown as
where is the MPD discretized operator defined by equationparentequation
for . Here, are the Legendre polynomials evaluated at each angle and are the respective weights.
is defined as a (N x N) operator for a vector ofangular fluxes , at spatial location .
In summary, if we write the FP equation as
then is Diag for , is a vector of source terms , and is represented by .
3.2 Numerical Results
It is shown that for slowly converging problems, typical convergence methods like suffer from false convergence adamslarsen . To work around this issue, the criterion is modified to use information about the current and previous iteration:
Two problems were tested using 200 spatial cells, = 400, , = 15, and = 16. Problem 1 has vacuum boundaries and a homogeneous isotropic source for . Problem 2 has no internal source and an incoming beam at the left boundary. The source and boundary conditions used are shown in Table 1.
We consider three scattering kernels in this paper: Screened Rutherford pomraning1 , Exponential pomraning2 , and Henyey-Greenstein HGK . Three cases for each kernel were tested. The results obtained with NFPA are compared with those obtained using GMRES, DSA, and FPSA with the MPD scheme.
3.2.1 SRK: Screened Rutherford Kernel
The SRK has a valid FP limit as approaches 0 patelFBR . Three different values of were used to generate the scattering kernels shown in Fig. 2. GMRES, DSA, FPSA, and NFPA all converged to the same solution for problems 1 and 2. Figure 3 shows the solutions for SRK with .
The results of all solvers are shown in Tables 3 and 2. We see that NFPA and FPSA tremendously outperform GMRES and DSA in runtime for all cases. FPSA is a simpler method than NFPA, requiring less calculations per iteration; therefore, it is expected that it outperforms NFPA in runtime. We see a reduction in runtime and iterations for FPSA and NFPA as the FP limit is approached, with DSA and GMRES requiring many more iterations by comparison as approaches 0.
An advantage that NFPA offers is that the angular moments of the flux in the LO equation will remain consistent with those of the transport equation even as a problem becomes less forward-peaked. On the other hand, the moments found using only the FP equation and source iteration lose accuracy. To illustrate this, Problem 1 was tested using different Screened Rutherford Kernels with increasing parameters. The percent errors (relative to the transport solution) for the scalar flux obtained with the LO equation and with the standard FP equation at the center of the slab are shown in Fig. 4. It can be seen that the percent relative errors in the scalar flux of the FP solution is orders of magnitude larger than the error produced using the LO equation. The same trend can be seen when using the exponential and Henyey-Greenstein kernels.
3.2.2 EK: Exponential Kernel
The exponential kernel pomraning2 ; JapanFPSA is a fictitious kernel made for problems that have a valid Fokker-Planck limit pomraning1 . The zero moment, , is chosen arbitrarily; we define as the same zero moment from the SRK. The parameter determines the kernel: the first and second moments are given by equationparentequation
|and the relationship for is|
As is reduced, the scattering kernel becomes more forward-peaked.
The EK has a valid FP limit as approaches 0 patelFBR . Three different values of were used to generate the scattering kernels shown in Fig. 5. The generated scattering kernels are shown in Fig. 5. GMRES, DSA, FPSA, and NFPA all converged to the same solution for problems 1 and 2. Figure 6 shows the solutions for EK with .
The runtimes and iterations for GMRES, DSA, FPSA, and NFPA are shown in Tables 5 and 4. We see a similar trend with the EK as seen with SRK. Smaller values lead to a reduction in runtime and iterations for NFPA and FPSA, which greatly outperform DSA and GMRES in both categories.
3.2.3 HGK: Henyey-Greenstein Kernel
As goes from zero to unity, the scattering shifts from isotropic to highly anisotropic.
The HGK does not have a valid FP limit patelFBR . The three kernels tested are shown in Fig. 7. GMRES, DSA, FPSA, and NFPA all converged to the same solution for problems 1 and 2. Figure 8 shows the solutions for HGK with . The results of each solver are shown in Tables 7 and 6.
Here we see that NFPA and FPSA do not perform as well compared to their results for the SRK and EK. Contrary to what happened in those cases, both solvers require more time and iterations as the problem becomes more anisotropic. This is somewhat expected, due to HGK not having a valid Fokker-Planck limit. However, both NFPA and FPSA continue to greatly outperform GMRES and DSA. Moreover, NFPA outperforms FPSA in iteration count for problem 1.
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 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.
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