非齐次燃耗方程数值解法

Preliminary study on numerical solver of inhomogeneous burnup equations

非齐次燃耗方程常用于描述具有显著核素迁移效应的核能系统中核素含量随时间的变化规律.国内外许多燃耗计算程序无法求解方程非齐次项含时的情况.本文在方程非齐次项能够被有限阶关于时间的泰勒展开逼近这一前提下,研究了非齐次项含时情况下方程的两种解法.首先通过Laplace变换推导出了方程基于线性子链方法的解析解形式,然后使用Carathéodory-Fejér方法计算出了方程矩阵级数解的近最佳Chebyshev有理逼近式.将两种方法在燃耗计算程序JBURN中实现,并进行了数值计算,绝大部分计算结果符合很好,部分结果在较长有效数字内仍能保持一致,验证了方法的正确性和精度.同时为求解具有其他非齐次项形式的燃耗方程提供了一种思路.

The inhomogeneous burnup equation is often used for describing the time evolution of nuclides＇ depletion in nuclear systems which have a significant nuclide migration effect. However, lots of burnup calculations codes only deal with the homogeneous cases instead of the inhomogeneous ones, among them there are a few codes that can work only when the inhomogeneous term of the equation is constant. Based on the condition that the inhomogeneous term can be approximated by finite-order Taylor expansion, two methods are introduced to solve the inhomogeneous burnup equation whose inhomogeneous term is time dependent. For the first method, the transmutation trajectory analysis method is used to decompose the connections between nuclides into linear chains, for one chain the analytical solution is derived strictly by using the Laplace transform. For the second method, a solution of the inhomogeneous equation in the form of summation of infinite matrix series is first derived, and then the sum function of the series is found. Furthermore, the different-order nearly-best rational approximation function of the sum function is found by using Carathéodory-Fejér method. The error between the sum function and the rational function fluctuates in a certain range without exceeding a limit value, while the maximum error decreases exponentially with the order of rational function increasing. By adopting the nearly-best rational approximation, the summation of infinite matrix series converts into a finite expansion of matrix fraction, which is much easier to deal with. These two methods are implemented in the burnup calculation code JBURN and numerical tests are done through using two examples. The first example is a small-scale matrix example and the result shows that the results from the two methods agree well in at least 6 decimal precision together with the results from the reference solution. The second example is a large-scale problem based on real nuclides＇ reaction database, and the result shows that less than 1% among all nuclides have a deviation larger than 10% between two methods, while about 8% nuclides have a deviation larger than 0.01% and the remaining ones have a deviation smaller than 0.01%. These results validate the correctness and accuracy for each of the two methods. Finally, this paper provides a possible implementation process for solving inhomogeneous burnup equations which have other time-dependent forms of inhomogeneous term.