Maria Pusa

Computing the Matrix Exponential in Burnup Calculations

Abstract The topic of this paper is the computation of the matrix exponential in the context of burnup equations. The established matrix exponential methods are introduced briefly. The eigenvalues of the burnup matrix are important in choosing the matrix exponential method, and their characterization is considered. Based on the characteristics of the burnup matrix, the Chebyshev rational approximation method (CRAM) and its interpretation as a numeric contour integral are discussed in detail. The introduced matrix exponential methods are applied to two test cases representing an infinite pressurized water reactor pin-cell lattice, and the numerical results are presented. The results suggest that CRAM is capable of providing a robust and accurate solution to the burnup equations with a very short computation time.

Rational Approximations to the Matrix Exponential in Burnup Calculations

Abstract The topic of this paper is solving the burnup equations using dedicated matrix exponential methods that are based on two different types of rational approximation near the negative real axis. The previously introduced Chebyshev Rational Approximation Method (CRAM) is now analyzed in detail for its accuracy and convergence, and correct partial fraction coefficients for approximation orders 14 and 16 are given to facilitate its implementation and improve the accuracy. As a new approach, rational approximation based on quadrature formulas derived from complex contour integrals is proposed, which forms an attractive alternative to CRAM, as its coefficients are easy to compute for any order of approximation. This gives the user the option to routinely choose between computational efficiency and accuracy all the way up to the level permitted by the available arithmetic precision. The presented results for two test cases are validated against reference solutions computed using high-precision arithmetics. The observed behavior of the methods confirms the previous conclusions of CRAM’s excellent suitability for burnup calculations and establishes the quadrature-based approximation as a viable and flexible alternative that, like CRAM, has its foundation in the specific eigenvalue properties of burnup matrices.

Solving Linear Systems with Sparse Gaussian Elimination in the Chebyshev Rational Approximation Method

Abstract The Chebyshev Rational Approximation Method (CRAM) has recently been introduced by the authors to solve burnup equations, and the results have been excellent. This method has been shown to be capable of simultaneously solving an entire burnup system with thousands of nuclides both accurately and efficiently. The method was prompted by an analysis of the spectral properties of burnup matrices, and it can be characterized as the best rational approximation on the negative real axis. The coefficients of the rational approximation are fixed and have been reported for various approximation orders. In addition to these coefficients, implementing the method requires only a linear solver. This paper describes an efficient method for solving the linear systems associated with the CRAM approximation. The introduced direct method is based on sparse Gaussian elimination, where the sparsity pattern of the resulting upper triangular matrix is determined before the numerical elimination phase. The stability of the proposed Gaussian elimination method is discussed based on consideration of the numerical properties of burnup matrices. Suitable algorithms are presented for computing the symbolic factorization and numerical elimination in order to facilitate the implementation of CRAM and its adoption into routine use. The accuracy and efficiency of the described technique are demonstrated by computing the CRAM approximations for a large test case with 1606 nuclides.

Perturbation-Theory-Based Sensitivity and Uncertainty Analysis with CASMO-4

The topic of this paper is the development of sensitivity and uncertainty analysis capability to the reactor physics code CASMO-4 in the context of the UAM (Uncertainty Analysis in Best-Estimate Modelling for Design, Operation, and Safety Analysis of LWRs) benchmark. The sensitivity analysis implementation is based on generalized perturbation theory, which enables computing the sensitivity profiles of reaction rate ratios efficiently by solving one generalized adjoint system for each response. Both the theoretical background and the practical guidelines for modifying a deterministic transport code to compute the generalized adjoint solutions and sensitivity coefficients are reviewed. The implementation to CASMO-4 is described in detail. The developed uncertainty analysis methodology is deterministic, meaning that the uncertainties are computed based on the sensitivity profiles and covariance matrices for the uncertain nuclear data parameters. The main conclusions related to the approach used for creating a covariance library compatible with the cross-section libraries of CASMO-4 are presented. Numerical results are given for a lattice physics test problem representing a BWR, and the results are compared to the TSUNAMI-2D sequence in SCALE 6.1.

Improving the Accuracy of the Chebyshev Rational Approximation Method Using Substeps

Abstract The Chebyshev rational approximation method (CRAM) for solving the decay and depletion of nuclides is shown to have a remarkable decrease in error when advancing the system with the same time step and microscopic reaction rates as the previous step. This property is exploited here to achieve high accuracy in any end-of-step solution by dividing a step into equidistant substeps. The computational cost of identical substeps can be reduced significantly below that of an equal number of regular steps, as the lower-upper decompositions for the linear solutions required in CRAM need to be formed only on the first substep. The improved accuracy provided by substeps is most relevant in decay calculations, where there have previously been concerns about the accuracy and generality of CRAM. With substeps, CRAM can solve any decay or depletion problem with constant microscopic reaction rates to an extremely high accuracy for all nuclides with concentrations above an arbitrary limit.