# Presentation: 209190

## Advances in Deterministic Methods for the First-Order Transport Equation

- Edward Larsen - University of Michigan
- Jim Morel - Los Alamos National Laboratory

## First-Order Particle Transport Problems:

- In these equations:

## Deterministic Particle Transport

- Discretization Methods
- Goal: Efficiently approximate the transport equation by a discrete linear system of algebraic equations.
- Iteration Methods
- Goal: Efficiently solve the resulting (large) discrete system
- Discussion Topics:
- Advances (since 1975) and current state-of-the-art.
- (In the following slides, * = new)

## Outline of Talk

- Discretization Methods
- Analysis of Discretization Methods
- Iteration Methods
- Analysis of Iterative Methods
- Summary & Conclusions

## Discretization Methods

- Time Discretization
- Fully Implicit
- Linear Discontinuous*
- Energy Discretization
- Multigroup
- Linear Discontinuous (for charged particles)*
- Angular (Direction-of-Flight) Discretization
- SN (Discrete Ordinates)
- PN (Spherical Harmonics)
- SPN (Simplified PN)*
- Spatial Discretization
- Finite Difference (Diamond & Weighted Diamond)
- Finite Element
- Characteristic*
- Nodal*
- Corner Balance, Multiple Balance, etc.*

## Time Discretization: Implicit

- (Pseudo
- steady-state form)

- Features:
- 	• Equivalent to steady-state transport problem
- 	• Solution is first-order accurate in
- 	• Solution is positive for all
- 	• Solution is unconditionally stable for all

## Time Discretization: LD*

-  Assume:

-  Take two moments:

- Features:
-  Second-order accurate with small error coefficient
-  Resistant to negativities and oscillations
-  Treat like another spatial derivative during sweep

## Energy Discretization: Multigroup

## Energy Discretization: LD*

-  Assume:

-  Take two moments:

- Features:
-  Second-Order accurate with small error coefficient
-  Resistant to negativities and oscillations
-  Treat like another spatial derivative during sweep
-  Boltzmann terms modified for compatibility with             multigroup cross-sections

## Angular Discretization: SN

## Angular Discretization: PN

|  |
| --- |

## Angular Discretization: Simplified PN*

- The SPN equations can also be derived by asymptotic and variational procedures.
- The SPN equations are simpler (diffusion) and fewer in number than the PN
- 	equations in 2D and 3D.

- Replace

- by

|  |
| --- |

## SN Spatial Discretization: Finite (Diamond & Weighted Diamond) Difference  

## SN Spatial Discretization: Discontinuous Finite Element

|  |
| --- |

|  |
| --- |

## SN Spatial Discretization: Characteristic Methods*

## SN Spatial Discretization: Nodal Methods*

## SN Spatial Discretization: Corner & Multiple Balance*

- Traditional Approach: Balance  and Auxiliary Equations

- Multiple Balance: Balance  and Subcell Balance

- For **

- For **

- Corner Balance: Balance on each cell “corner”-- total balance met by summation

- Right half

- Left half

## Outline of Talk

- Discretization Methods
- Analysis of Discretization Methods
- Iteration Methods
- Analysis of Iterative Methods
- Summary & Conclusions

## MathematicalAnalysis of Discretization Methods

- Truncation Error Analysis
- Preservation of Special Solutions*
- Constant, linear in space & angle
- Asymptotic Diffusion  Limit*

## Discretization Analysis: Truncation Error

## Discretization Analysis: Preservation of Polynomial Solutions*

## Discretization Analysis: Asymptotic Diffusion Limit*

## Outline of Talk

- Discretization Methods
- Analysis of Discretization Methods
- Iteration Methods
- Analysis of Iterative Methods
- Summary & Conclusions

## Iteration Methods

- Source Iteration
- Coarse and Fine-Mesh Rebalance
- Quasidiffusion
- Diffusion Synthetic Acceleration*
- Transport Synthetic Acceleration*
- Krylov Methods*

## Iteration Methods: Source Iteration

## Iteration Methods: Rebalance

## Iteration Methods: Quasidiffusion

## Iteration Methods: Diffusion Synthetic Acceleration*

## Iteration Methods: Transport Synthetic Acceleration*

## Iteration Methods: Krylov Methods*

- Given the system A**x**=**b**, Krylov space elements are “optimally” combined to obtain a solution.
- Given an initial solution guess **x****0**, the Krylov space is spanned by the vectors (**r****0**, A **r****0**, A2 **r****0****, **...An-1**r****0**), where **r****0**=**b**-A **x****0**, and n is the *dimension* of **x****0****.**
- Krylov algorithms require only the “action” of the matrix A on a set of vectors, i.e., they require that a set of vectors be multiplied by A.
- This makes more efficient options possible for transport solutions.  For instance, one could use a Krylov method to solve (L-S)**=Q, but it is better to solve (I-L-1S)**= L-1Q, which is obtained by multiplying the original equation by L-1.  The matrix associated with (I-L-1S) is dense, but its action can be computed with a sweep. Obtaining a “better” system by multiplying the equation with another operator is called *preconditioning*.
- One can do further preconditioning using variants of traditional acceleration schemes. The resulting hybrid schemes can be much more robust than accelerated source iteration.

## Outline of Talk

- Discretization Methods
- Analysis of Discretization Methods
- Iteration Methods
- Analysis of Iterative Methods
- Summary & Conclusions

## Analysis of Iteration Methods

- Fourier Analysis*
- Infinite homogeneous media
- Infinite spatially periodic media
- “Clustering” of Iteration Eigenvalues*
- Krylov methods

## Iterative Analysis: Fourier Analysis*

## Krylov Eigenvalue Analysis*

- Krylov methods such as the CG and GMRES methods converge in one iteration if the *condition number* is one. The condition number is largest eigenvalue magnitude divided by the smallest eigenvalue magnitude.
- Computing the condition number of a system gives an indication of the convergence rate one can expect with most Krylov solvers.
- However, given two linear systems with the same condition number, fewer iterations are required for the system with eigenvalues “clustered” about a small number of values.
- Thus, System B will require significantly fewer iterations than System A, even though they have the same condition number.

- System A:

- System B:

## Outline of Talk

- Discretization Methods
- Analysis of Discretization Methods
- Iteration Methods
- Analysis of Iterative Methods
- Conclusions

## Some Current Research Topics

- Spatial Discretization
- Greater accuracy and more efficient acceleration
- Structured (AMR) and unstructured spatial grids
- Angular Discretization
- Adaptation in angle
- Improved treatments of anisotropic scattering
- Suppression of ray effects in SN
- Krylov Methods
- Can reduce the need for consistent spatial discretization in DSA
- Can stabilize TSA and make it reasonably efficient
- Equations can be reformulated to minimize memory usage and achieve more clustering of eigenvalues

## Summary & Conclusions

- The computer-simulation of radiation interactions with matter has expanding applications in engineering, physics, & medicine.
- Deterministic methods for these problems have become much more sophisticated in the past 25 years.
- •	(Most of these advances have been reported at this series of meetings.)
- New methods often require a broader knowledge of mathematics (finite elements, asymptotics, Fourier analysis, linear algebra, etc.) than in the past.
- Computing and algorithm resources have greatly expanded, but the problems are now much more complicated and realistic (often 3-D).
- Radiation transport simulations will always be costly (7-D).
- The problem of producing “maximal” solutions, using “minimal” computing resources, remains as important and intellectually challenging as ever.