点堆中子动力学方程组曲率权重法的解

针对核反应堆安全工程对某些数值计算结果要求较高的精度和正的误差, 以及舰船核反应堆机动性对计算速度的要求, 需要从数学上寻找一种新的数值计算方法, 以满足实际曲线向上凸或向下凹时计算值总是略高于真实值, 且误差不大于欧拉法和改进的欧拉法所得值. 本文研究曲率权重法求解点堆中子动力学方程组, 该方法是在曲率圆法的基础上引入权重的思想来衡量间隔步长上两个曲率对该步长曲率平均值的贡献. 与欧拉法和改进的欧拉法比较, 曲率权重法的计算结果总是能够高于真实值或有正的误差, 且精度和计算速度得到明显提升. 将该方法用于次临界堆阶跃和线性引入反应性时中子密度的求解, 能够快速得到满足计算要求和高精度的数值结果.

Curvature weight method of solving the point reactor neutron kinetic equations

The point kinetic equations are the system of a couple stiff ordinary differential equations. Many studies have focused on the development of more advanced and efficient methods of solving the equations, such as the high order Taylor polynomials method, the Haar wavelet operational method, the fractional point-neutron kinetic model method, the basis function method, the homotopy analysis method, and other methods. Most of these methods are successful in some specific problems, but still have, more or less, disadvantages. For example, the accuracy of the Haar wavelet operational method is limited by the collocation points, and it needs more computing time for a high precision. Aiming at the requirements that some numerical calculation results must have the higher precision and only the positive error in the nuclear reactor safety engineering and ship reactor for the maneuverability, in this paper we try to look for a new numerical method to satisfy that the calculation value is slightly higher than the real value when the actual curve is upward convex or downward concave, and the error is not greater than that by the Euler and improved Euler method. The new method is so-called the curvature weight (CW) method, which is based on the curvature circle method and considers the contributions of two curvatures at the interval step point to the average curvature inside the interval step. Using the decoupling method to remove the stiffness of equations and the instantaneous jump approximation to derive the neutron differential equations, the first and second derivative of neutron density are obtained. Then the CW method is used to solve the point reactor neutron kinetic equations, and thus obtaining the numerical solution. Compared with the results by the Euler and improved Euler method, the numerical calculation results by the CW method are always higher than the real value, and the calculation accuracy and speed are improved significantly. When this new method is used to solve the point reactor neutron differential equations with the step and linear reactivity inserted into the subcritical reactor, the numerical results which satisfy the requirements of positive calculation error and high precision can be obtained quickly. After improving the calculation step length, the precision reduction by the CW method is significantly lower than that by the Euler and improved Euler method. So the CW method can greatly shorten the total computing time, and it is also effective for most of differential equation systems.