ODE Solving via Automatic Differentiation and Rational Prediction
ODE Solving via Automatic Differentiation and Rational Prediction
复制标题
通过自动微分和理性预测进行 ODE 求解
DOI：
--
复制
发表时间：
1996
期刊：
影响因子：
--
通讯作者：
A. Griewank
中科院分区：
文献类型：
--
作者：
A. Griewank
我们考虑普通微分方程初值问题解的经典泰勒级数逼近，并研究静态常微分方程数值解的隐式变量。发现状态向量的泰勒系数与解轨迹沿着右侧的雅可比矩阵的泰勒系数密切相关。这些国家和雅可比coeecients之间的联系，利用他们的eecient评估自动dierientiation与少量的前向和反向扫描。它示出了如何这些coeecients可以利用在一个新的合理的预测的Hermite-Obreshkov-Pad e（HOP）的方法，一个家庭的高阶数值积分器，最后检查Wanner在六十年代。线性隐式预报和全HOP方法在常数系数情况下产生矩阵指数的Pade逼近。A-和L-稳定性实现的对角线和第一二次对角线的选择的Pad e参数对（q; p）。初步的数值结果表明，在stii和高振荡问题，大的步骤可以实现与一个单一的修正迭代和可接受的离散误差。
We consider the classical Taylor series approximation to the solution of initial value problems in ordinary diierential equations and examine implicit variants for the numerical solution of stii ODEs. The Taylor coeecients of the state vector are found to be closely related to those of the Jacobian of the right hand side along the solution trajectory. These connections between state and Jacobian coeecients are exploited for their eecient evaluation by automatic diierentiation with a small number of forward and reverse sweeps. It is shown how these coeecients can be utilized in a new rational predictor for the Hermite-Obreshkov-Pad e (HOP) methods, a family of high order numerical integrators, last examined by Wanner in the sixties. The linearly implicit predictor and the full HOP methods yield in the constant coeecient case Pad e approximants of the matrix exponential. A-and L-stability is achieved for the diagonal and rst two subdiagonal choices of the Pad e parameter pair (q; p). Preliminary numerical results demonstrate that on stii and highly oscillatory problems large steps can be realized with a single correction iteration and acceptable discretization error.