Discontinuous diffusion synthetic acceleration for transport on 2D arbitrary polygonal meshes
Introduction
In this paper, we present a Diffusion Synthetic Acceleration (DSA) scheme that employs the same discontinuous finite element discretization used in the transport equations. Specifically, we employ for both the transport solve and the diffusion acceleration Piece-Wise Linear Discontinuous (PWLD) finite elements and test the scheme on arbitrary polygonal cells.
Arbitrary polygonal (polyhedral in 3D) cells provide a natural transition for locally adapted meshes, may arise, for instance, when arbitrary cut lines are used to split an existing mesh, and are now more frequently found, e.g., in fluid simulations (e.g., StarCCM+ [1]).
Because analytical solutions are unavailable for most radiation transport problems of practical interest, one typically employs iterative techniques to solve the large system of equations that results from the spatial and angular discretizations of the transport equation. Standard iterative techniques for the first-order form of the discrete-ordinate () transport equation include Source Iteration (SI) and Krylov subspace techniques (usually GMRes [2]). For highly diffusive materials (i.e., with scattering ratios close to 1) and optically thick configurations (i.e., problems that are not leakage-dominated), these iterative techniques can become quite ineffective, requiring high iteration counts and possibly leading to false convergence. To mitigate these issues, SI and GMRes-based transport solves can be effectively accelerated (preconditioned) using Diffusion Synthetic Acceleration (DSA) [3], [4], [5], [6], [7], [8].
The spatial discretization of the DSA equations must be somewhat “consistent” with the one used for the transport equations in order to yield unconditionally stable and efficient DSA schemes [3], [4], [5], [6], [7], [8]. However, the search for full consistency between the discretized transport equations and the discretized diffusion may not be computationally practical (especially for unstructured arbitrary meshes, [3]). For instance, Warsa, Wareing, and Morel [5] derived a fully consistent DSA scheme for linear discontinuous finite elements on unstructured tetrahedral meshes; their DSA scheme yields a system of equations that is computationally more expensive than partially consistent DSA schemes that are based upon discretizations of a standard diffusion equation. Several partially consistent schemes have been analyzed for discontinuous finite element discretizations of the transport equation on unstructured meshes, for example, the modified-four-step (M4S) scheme [6], the Wareing–Larsen–Adams (WLA) scheme [7], and the Modified Interior Penalty (MIP) scheme [8].
To the authors' knowledge, no work is currently ongoing to adapt the M4S technique to polygonal meshes. This is very likely due to the fact that the M4S scheme does not yield a Symmetric Positive Definite (SPD) matrix and was found to be divergent for 3D tetrahedral meshes with linear discontinuous elements [5]. Recent work to develop a DSA scheme for polygonal cells has mainly focused on adapting the WLA scheme to polygonal meshes [9], [10]. The WLA scheme is a two-stage process, where first a diffusion solution is obtained using a continuous finite element discretization and then a discontinuous update is performed cell-by-cell in order to provide an appropriate discontinuous scalar flux correction to the discontinuous finite element transport solver. In [5], the WLA scheme was found to be a stable and effective DSA technique, though its efficiency degraded as the problem became optically thick and highly diffusive. In this paper, we extend the MIP technique to the PWLD discretization technique for arbitrary polygonal meshes. The MIP scheme is based on the standard Interior Penalty (IP) method for the discontinuous finite element discretization of diffusion equations. MIP was first derived in [8], where it was applied to triangular unstructured meshes (with locally adapted cells). MIP does not suffer from the same issues as the WLA technique when the problem becomes optically thick and highly diffusive and, therefore, can be a useful alternate DSA to accelerate DFE transport solves. In [8], a Preconditioned Conjugate Gradient (PCG) technique (with Symmetric Gauss–Seidel, SGS, as preconditioner) was used to solve the MIP-DSA equation. In this paper, we also analyze the effectiveness of algebraic multigrid methods (AMG) [11], [12] as a preconditioner.
The remainder of this paper is organized as follows. In Section 2, we briefly review the PWLD discontinuous finite element discretization and the iterative solution techniques applied to the transport equation. The MIP-DSA scheme is extended to the PWLD discretization for arbitrary polygons in Section 3. In Section 4, we describe the Algebraic MultiGrid (AMG) approaches used here: the ML package of Trilinos [13] and the AGMG (AGgregation-based algebraic MultiGrid) technique of [14], [15], [16], [17]. In Section 5, we present a Fourier analysis for the MIP-DSA scheme discretized with PWLD, and we compare the different preconditioned CG approaches. Conclusions are given in Section 6.
Section snippets
Discretization and solution techniques for the transport equation
In this section, we review the transport equation and the iterative solution techniques typically employed to solve it. We then describe the PWLD discontinuous spatial discretization for the transport equation with an emphasis on arbitrary polygonal grids.
DSA solution principle
As noted earlier, standard iterative techniques applied to transport solves can be slowly converging in thick diffusive configurations. A DSA scheme must be employed to accelerate their convergence. The idea behind synthetic acceleration is that the error between the (yet unknown) transport solution and the current iterate can be estimated from a computationally less expensive process, yielding a corrective term to be added to the current iterate in order to improve the next iterate. In DSA, a
AMG principles
A common way to solve an SPD system of equations is to use a Conjugate Gradient (CG) technique preconditioned with Symmetric Gauss–Seidel (PCG-SGS) or SSOR (PCG-SSOR); SGS is simply SSOR with a damping factor of one. Here, we compare CG preconditioned with Symmetric Gauss–Seidel (PCG-SGS) with CG preconditioned with an algebraic multigrid method. PCG-SGS was chosen because little difference was noted when employing other damping factors for MIP-DSA on triangular grids [25]. In our
Numerical results
In this section, we present two series of results. First, Fourier analyses are carried out to analyze the performance of the MIP-DSA acceleration scheme for a homogeneous infinite medium meshed with rectangular cells and discretized with PWLD finite elements. The effects of the order and the cell aspect ratio on the spectral radius of the iterative scheme are also studied. Second, the MIP-DSA technique is implemented in a 2D code that uses arbitrary polygonal grids with a PWLD spatial
Conclusions
We have extended the Modified Interior Penalty (MIP) form of Diffusion Synthetic Acceleration (DSA) to radiation transport solves performed on arbitrary polygonal grids discretized with Piece-Wise Linear Discontinuous finite elements. The MIP-DSA equation employs the same discontinuous finite element trial spaces as the spatial discretization of the transport equation. As such, only a few additional elementary matrices need to be implemented to an existing code.
Fourier analyses show
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Current address: Department of Mathematics, Texas A&M University, College Station, TX 77843, USA.