Solving variable-coefficient burnup equations by exponential Rosenbrock methods

Abstract

Burnup calculation is an essential topic in reactor physics analysis, which is based on the neutronic-depletion coupling calculation. High order neutronic-depletion coupling methods have been demonstrated much superior to the traditional coupling methods in terms of accuracy, but introduce the variable-coefficient burnup equations. The straightforward and general method for solving variable-coefficient burnup equations is to discretize the burnup step into several substeps and then approximate variable-coefficient equations by a sequence of constant-coefficient ones. However, substep sizes must be rather small to obtain satisfactory accuracy, which causes numerous computational cost. In this work, a class of one-step methods called exponential Rosenbrock methods is proposed to solve variable-coefficient burnup equations. The major part of the implementation of exponential Rosenbrock methods is computing the matrix exponentials and related φ functions, so the matrix exponential methods for constant-coefficient burnup equations can be applied to exponential Rosenbrock methods directly. Moreover, each exponential Rosenbrock method has a corresponding embedded error estimator for adaptive time-stepping. A 4th-order exponential Rosenbrock method called EXPRB43 is used in this work, and two EXPRB43-based predictor–corrector coupling methods called EXPRB43-CE/LI and EXPRB43-LE/QI are further established by replacing the time step discretization method used in the conventional CE/LI and LE/QI with EXPRB43. To investigate the correctness and performance of EXPRB43 for solving variable-coefficient burnup equations, three artificial test Cases are first constructed by combining three different time-dependent fluxes and a specific cross-section library. The numerical results show EXPRB43 can improve the computational efficiency by 2 to 3 orders of magnitude compared to the straightforward method, and this novel method could easily obtain more accurate and reliable solutions by using an adaptive stepsize controller. Then, a test case for depleting a 2D UO2-Gd fuel pin is investigated to further demonstrate the performance of EXPRB43 in realistic neutronic-depletion coupling problems. By comparing to the conventional CE/LI and LE/QI, EXPRB43-based coupling methods can obtain slightly more accurate solutions especially for several important fission products, such as 155Gd, 157Gd, 135Xe and 149Sm. Although the overall computational errors dominated by the errors of polynomial approximation of the time dependent burnup matrix cover up the improvement of EXPRB43 to the naive substep method, EXPRB43-based coupling methods can be alternative to the conventional ones.