对角隐式龙格库塔法在点堆动力学中的应用
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摘要
点堆动力学方程刚性比较强,采用常规全隐式龙格库塔方法则求解耗时多。对角隐式龙格库塔方法保留全隐式龙格库塔善于求解刚性方程的特点,同时又大大降低计算量。通过嵌入低阶龙格库塔方法,实现自适应时间步技术,提高计算效率。通过计算阶跃、线性、正弦3种反应性变化基准题,计算结果表明该方法和其他方法结果符合很好,而且相对于θ方法能够在相同的计算时间内给出更加精确的解,特别是在快速插入线性反应性的情况下。
The stiffness of point kinetics is very strong, while it usually takes a lot of time to solve by using full implicit Runge Kutta(FIRK). Diagonally Implicit Runge Kutta(DIRK) is a useful tool like FIRK to solve the stiff differential equations, while it greatly reduces the computation compared with FIRK. By embedded lower order Runge Kutta, the time step-size adaptation technique is implemented, which improves the computation efficiency of DIRK. Through three typical cases with step, ramp and sinusoidal reactivity insertions, it shows that the results obtained by DIRK are in good agreement with other available results and DIRK can give far more accurate results than θ method at the same computation cost, especially in the case of fast ramp reactivity insertion.
The stiffness of point kinetics is very strong, while it usually takes a lot of time to solve by using full implicit Runge Kutta(FIRK). Diagonally Implicit Runge Kutta(DIRK) is a useful tool like FIRK to solve the stiff differential equations, while it greatly reduces the computation compared with FIRK. By embedded lower order Runge Kutta, the time step-size adaptation technique is implemented, which improves the computation efficiency of DIRK. Through three typical cases with step, ramp and sinusoidal reactivity insertions, it shows that the results obtained by DIRK are in good agreement with other available results and DIRK can give far more accurate results than θ method at the same computation cost, especially in the case of fast ramp reactivity insertion.
引文
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[4]Patra A,Ray S S.A numerical approach based on Haar wavelet operational method to solve neutron point kinetics equation involving imposed reactivity insertions[J].Annals of Nuclear Energy,2014,68(3):112-117.
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[2]Picca P,Furfaro R,Ganapol B D.A highly accurate technique for the solution of the non-linear point kinetics equations[J].Annals of Nuclear Energy,2013,58(8):43-53.
[3]Hak Tae Kim,Yujin Park,Kazantzis N,et al.A numerical solution to the point kinetic equations using Taylor–Lie series combined with a scaling and squaring technique[J].Nuclear Engineering and Design,2014,272:1-10.
[4]Patra A,Ray S S.A numerical approach based on Haar wavelet operational method to solve neutron point kinetics equation involving imposed reactivity insertions[J].Annals of Nuclear Energy,2014,68(3):112-117.
[5]Ganapol B D.A highly accurate algorithm for the solution of the point kinetics equations[J].Annals of Nuclear Energy,2013,62:564-571.
[6]Sánchez,J.On the numerical solution of the point reactor kinetics equations by generalized Runge–Kutta methods[J].Nucl.Sci.Eng,1989,103,94-99.
[7]廖茶清.优化时间步长的数值方法解核反应堆点动态学方程[J].核动力工程,2007,28(2):8-12.
[8]Hairer E,Wanner G.Solving Ordinary Differential Equation II[M].北京:科学出版社,2006.