Numerical Integration of Systems of Stiff Ordinary Differential Equations with Special Structure
Numerical Integration of Systems of Stiff Ordinary Differential Equations with Special Structure
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特殊结构刚性常微分方程组的数值积分
DOI：
10.1093/imamat/18.2.249
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发表时间：
1976
影响因子：
1.2
通讯作者：
H. Robertson
中科院分区：
文献类型：
--
作者：
H. Robertson
一般刚性微分方程组的求解算法通常使用隐式积分公式。积分每一步的相关非线性方程都可以通过迭代（例如平行弦法）有效求解，其中矩阵是解的计算点处的雅可比行列式的近似值。在需要更新和重新求逆矩阵之前，这种迭代经常在多个积分步骤上提供足够快的收敛。当微分方程具有特殊结构时，可以通过比其余部分更频繁地更新雅可比行列式的分区来保持令人满意的收敛性，并且有效的计算过程在于计算逆矩阵的相应更新。局部收敛的充分条件可以用迭代矩阵与解处的导数之间的差或用相应逆矩阵的差来表达。类似地，渐近收敛率是根据这些扰动的范数来估计的。为了评估更新雅可比矩阵或其逆矩阵的划分的有效性，我们将相应的扰动设置为零并评估收敛速度的估计。方程的变量变换和“加权”可用于获得更准确的可计算收敛速度估计和更严格的收敛条件。这对于物理、化学和工程应用中出现的、与快运动和慢运动相关的非线性刚性系统尤其重要。此类系统在雅可比行列式和与系统对快速运动分量的敏感性相关的更高导数中表现出特殊结构，这使得它们特别适合矩阵更新技术。引用了文献中的许多说明性问题。已经使用逆矩阵对一个工作示例进行了数值求解，该逆矩阵的分区根据收敛的需要不规则地更新，并且将迭代计数和逆矩阵统计与完整逆矩阵的更新进行了比较。计算节省有时可能很大。
Algorithms for the solution of general systems of stiff differential equations commonly use implicit integration formulae. The associated non-linear equations at each step of the integration are efficiently solved by an iteration such as the parallel chord method, where the matrix is an approximation for the Jacobian at a calculated point of the solution. This iteration frequently gives sufficiently rapid convergence over a number of integration steps before updating and re-inversion of the matrix is required. When the differential equations have a special structure, satisfactory convergence may be maintained by updating a partition of the Jacobian less frequently than the remainder and an efficient computational procedure consists in calculating the corresponding update of the inverse. Sufficient conditions for local convergence may be expressed in terms of the difference between the iteration matrix and the derivative at the solution or in terms of the difference of the corresponding inverses. Similarly the asymptotic rate of convergence is estimated in terms of the norms of these perturbations. To assess the effectiveness of updating a partition of the Jacobian or its inverse we set the corresponding perturbation to zero and evaluate the estimate of the rate of convergence. Variable transformation and “weighting” of equations may be used to obtain more accurate computable estimates of convergence rates and tighter conditions for convergence. This is particularly relevant in non-linear stiff systems arising in applications from physics, chemistry and engineering and associated with fast and slow motions. Such systems exhibit special structure in the Jacobian and higher derivatives related to the sensitivity of the system to the components of the fast motion which makes them particularly amenable to matrix updating techniques. A number of illustrative problems from the literature are cited. A worked example has been solved numerically using an inverse whose partitions are updated irregularly as required for convergence and a comparison is made of iteration counts and inversion statistics with updating of the full inverse. Computational savings may sometimes belarge.