A new family of time integration methods for heat conduction problems using numerical green’s functions
A new family of time integration methods for heat conduction problems using numerical green’s functions
复制标题
使用数值格林函数解决热传导问题的一系列新的时间积分方法
DOI：
10.1007/s00466-009-0389-0
复制
发表时间：
2009
影响因子：
4.1
通讯作者：
W. Mansur
中科院分区：
文献类型：
--
作者：
F. Loureiro;W. Mansur
本文涉及应用于线性热传导问题的一类新型时间积分方案的制定和数值实现。通过考虑解析时间积分方程，根据模型问题的数值格林函数矩阵计算任何时间水平的温度场。通过有限元方法进行空间离散后，使用高效的隐式和显式龙格-库塔方法在节点坐标中显式计算将解从 t 转移到 t + Δt 的格林函数矩阵。结果表明，当在第一个时间步结束时使用子步程序递归计算格林函数矩阵时，该方法的稳定性和准确性得到了很大提高。因此，通过适当选择子步数，可以使用大时间步长而不会使数值解退化。最后，通过分析两个数值例子证明了本方法的有效性。
This paper is concerned with the formulation and numerical implementation of a new class of time integration schemes applied to linear heat conduction problems. The temperature field at any time level is calculated in terms of the numerical Green’s function matrix of the model problem by considering an analytical time integral equation. After spatial discretization by the finite element method, the Green’s function matrix which transfers solution from t to t + Δt is explicitly computed in nodal coordinates using efficient implicit and explicit Runge-Kutta methods. It is shown that the stability and the accuracy of the proposed method are highly improved when a sub-step procedure is used to calculate recursively the Green’s function matrix at the end of the first time step. As a result, with a suitable choice of the number of sub-steps, large time steps can be used without degenerating the numerical solution. Finally, the effectiveness of the present methodology is demonstrated by analyzing two numerical examples.