Comparison of Chebyshev and Anderson accelerations for the neutron transport equation
Résumé
This work investigates the solution of the multigroup neutron transport equation with a discrete ordinates method. More specifically, it focuses on the k-eigenvalue problem of the equation. In this case, the variables of interest are the largest eigenvalue (keff) and the corresponding eigenmode is called the fundamental mode. Mathematically, this problem is usually solved using the power iteration method. However, the convergence of this algorithm can be very slow, especially if the dominance ratio (ratio of the first two eigenvalues) is high as is the case in some reactor physics applications. For thermal reactors, the dominance ratio is usually above 0.95, and the power converges slowly. Thus, the power iteration method has to be accelerated in some ways to improve its convergence. Over the years, several acceleration methods have been conceived among which are rebalance techniques, Chebyshev acceleration and more recent nonlinear methods.
The Chebyshev acceleration method has been applied to legacy codes. This consists of modifying the initial problem into an equivalent one with a smaller dominance ratio to improve the convergence of the power iteration algorithm. The Chebyshev acceleration is equivalent to applying the power iteration method on a polynomial of the initial matrix such that the dominance ratio is as small as possible without changing the eigenvectors of the latter. The main drawback of this method is that it requires a priori knowledge the dominance ratio κ; its efficiency depends on an appropriate estimate of the latter.
In recent years, nonlinear methods have been applied to solve the k-eigenvalue problem. One such acceleration method is the Anderson acceleration. This method has been proposed by Anderson in 1965 and is employed to accelerate fixed-point iterations. It has been applied to the neutron transport equation in several works since 2011. Nevertheless, these are often compared to the unaccelerated power iteration method or without the Chebyshev acceleration which has been historically applied to several codes. Hence, the goal of this paper is to apply the Anderson acceleration method to the power iteration method, and compare its performance to the Chebyshev acceleration method.
The case considered for the numerical test is monogroup 1D slab problem, with a fuel zone of nclosed between two reflector regions. The problem is solved using the unaccelerated PI, and PI accelerated by the previously described acceleration methods. The Anderson acceleration method has been successfully applied to the 1D problem considered in this work. During this work, ultigroup cases have also been tested, and the results remain still very similar to the observations made with the monogroup case. The aim is to extend it to further multidimensional problems and compare its behaviour with respect to the Chebyshev acceleration.