Non-linear S2 Acceleration for Multidimensional Problems with Unstructured Meshes

The SN transport equation is popularly used to describe the distribution of neutrons in many applications including nuclear reactors. The topic of this research is a non-linear acceleration method for accelerating convergence of the scalar flux when the SN equation is solved iteratively. The SN angular flux iterate is used to compute average direction cosines in each octant. These direction cosines define a vector in each octant that may not have a unit length. Nonetheless, these eight average directions are used to form an S2-like equation that serves as the low-order equation in a nonlinear acceleration scheme. The acronym NL-S2 will be used to denote this non-linear S2-like equation. This method is investigated for use accelerating k-eigenvalue calculations and in this case, a k-eigenvalue can be converged on the low order system. NL-S2 is simple to discretize consistently with the SN equation and when this is done the scalar flux solution for the NL-S2 equation is the same as that for the SN equation. A primary motivation for this investigation of NL-S2 acceleration is that an SN style sweeper might be effective for inverting the NL-S2 “streaming plus collision” operator. However, the NL-S2 system, while looking similar to an S2 equation, has some significant differences. For any mesh other than one consisting entirely of rectangles or rectangular cuboids, the NL-S2 system will have many cyclic dependencies coupling cells. The NL-S2 method has been investigated in a number of other works, however all previous investigations focused either on one-dimensional problems or two-dimensional problems using a structured mesh. In this work, several methods for using an SN style sweeper were investigated for the NL-S2 system. It is found that modifications can be made to the NL-S2 linear system that drastically reduce the amount of off-diagonal matrix coefficients. The modified NL-S2 system is equivalent to the original at convergence of the scalar flux solution. An SN style sweeper is shown to be effective for this modified NL-S2 streaming plus collision operator. Acceleration of k-eigenvalue calculations is investigated for the well known two-dimensional C5G7 benchmark as well as a C5G7 like three-dimensional problem. A pincell problem containing a large void in the center is also investigated and NL-S2 acceleration is found to not be significantly impacted by the void. Our results indicate that NL-S2 acceleration is an effective alternative to traditional diffusion-based methods

GMRES And Integral Operators

In this paper we show how the properties of integral operators and their approximations are reflected in the performance of the GMRES iteration and how these properties can be used to smooth the GMRES iterates, thereby strengthening the norm in which convergence takes place. The smoothed iteration has very similar properties to Broyden's method and we present some comparisons of the two methods with the standard (unsmoothed) implementation of GMRES