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首页> 外文期刊>Calcolo >Three-stage Hermite-Birkhoff solver of order 8 and 9 with variable step size for stiff ODEs
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Three-stage Hermite-Birkhoff solver of order 8 and 9 with variable step size for stiff ODEs

机译：8阶和9阶三级Hermite-Birkhoff求解器，具有可变步长，适用于刚性ODE

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摘要

Variable-step (VS) -stage Hermite-Birkhoff (HB) methods HB of order and 9 are constructed as a combination of linear -step methods of order and a diagonally implicit one-step -stage Runge-Kutta method of order 3 (DIRK3) for solving stiff ordinary differential equations. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to multistep and Runge-Kutta type order conditions which are reorganized into linear confluent Vandermonde-type systems. This approach allows us to develop A-stable methods of order up to 5 and A()-stable methods of order up to 10. Fast algorithms are developed for solving these systems in O operations to obtain HB interpolation polynomials in terms of generalized Lagrange basis functions. The stepsize of these methods are controlled by a local error estimator. When programmed in C++, HB() of order and 9 compare favorably with existing Cash modified extended backward differentiation formulae of order 7 and 8, MEBDF(7-8), in solving problems often used to test higher order stiff ODE solvers on the basis of CPU time and error at the endpoint of the integration interval.
机译：阶数为9的可变步阶（VS）级Hermite-Birkhoff（HB）方法构造为阶数的线性步阶方法和阶数为3的对角隐式一步阶Runge-Kutta方法（DIRK3） ）求解刚性常微分方程。强迫泰勒展开数值解使其与真实解的展开相一致会导致多步和Runge-Kutta型有序条件，这些条件条件被重组为线性融合的Vandermonde型系统。这种方法使我们能够开发阶数为5的A稳定方法和阶数为10的A（）稳定方法。开发了快速算法来求解这些系统的O运算，从而根据广义Lagrange基获得HB插值多项式功能。这些方法的步长由本地误差估计器控制。当用C ++编程时，阶数9的HB（）与现有的阶数7和8的Cash修改后的扩展后向微分公式MEBDF（7-8）相比，在解决经常用于测试基于高阶刚性ODE求解器的问题时，具有优势积分间隔的端点处的CPU时间和错误。

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