Dynamical Systems and Adaptive Timestepping in ODE Solvers
Dynamical Systems and Adaptive Timestepping in ODE Solvers
复制标题
ODE 求解器中的动力系统和自适应时间步长
DOI：
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
H. Lamba
中科院分区：
文献类型：
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作者：
H. Lamba
常微分方程的初值问题通常用自适应时间步长算法数值求解。这些算法由用户定义的公差控制，该公差从每个步骤的估计误差上方开始。我们制定了一大类这样的算法作为离散动力系统，这是不连续的和更高的维度比基本的常微分方程。通过假设足够强的有限时间收敛结果的吸引子的常微分方程的某些邻域，我们证明了存在性和上连续性结果附近的数值吸引子的公差趋于zero.This足够强的有限时间收敛结果的假设，然后检查自适应算法，使用一对显式的Runge-Kutta方法不同的顺序，以估计一步的错误。对于任意的龙格-库塔对，必要的有限时间收敛结果在包含常微分方程所有平衡点的相空间中的一组点上不成立。因此，在一般情况下，渐近收敛的结果不能适用于吸引子包含平衡。然而，对于一类特殊的龙格-库塔对，有限时间收敛的结果可以加强，包括附近的平衡点的雅可比矩阵是可逆的。
Initial value problems for ODEs are often solved numerically using adaptive timestepping algorithms. These algorithms are controlled by a user-defined tolerance which bounds from above the estimated error committed at each step. We formulate a large class of such algorithms as discrete dynamical systems which are discontinuous and of higher dimension than the underlying ODE. By assuming sufficiently strong finite-time convergence results on some neighbourhood of an attractor of the ODE we prove existence and upper semicontinuity results for a nearby numerical attractor as the tolerance tends to zero.This assumption of sufficiently strong finite-time convergence results is then examined for adaptive algorithms that use a pair of explicit Runge-Kutta methods of different order to estimate the one-step error. For arbitrary Runge-Kutta pairs the necessary finite-time convergence results fail to hold on a set of points in the phase space that includes all the equilibria of the ODE. Therefore, in general, the asymptotic convergence results cannot be applied to attractors containing equilibria. However, for a particular class of Runge-Kutta pairs, the finite-time convergence results can be strengthened to include neighbourhoods of equilibrium points for which the Jacobian is invertible.