Practical Implementation of High-Order Multiple Precision Fully Implicit Runge-Kutta Methods with Step Size Control Using Embedded Formula
Practical Implementation of High-Order Multiple Precision Fully Implicit Runge-Kutta Methods with Step Size Control Using Embedded Formula
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使用嵌入式公式控制步长的高阶多精度全隐式龙格库塔方法的实际实现
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发表时间：
2013
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通讯作者：
T. Kouya
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作者：
T. Kouya
提出了一种在多精度浮点环境下实现高阶全隐式Runge-Kutta(IRK)方法的方法。尽管在IEEE754双精度环境中基于IRK方法的实现已被报道为由海尔开发的RADAU5和由Jay开发的SPARK3，但它们仅支持3级IRK系列。为了减小截断误差，必须采用更多的级数和更高阶的IRK公式。在多精度环境中，截断误差比舍入误差大得多。结果表明，基于W变换的SPARK3型降阶算法比RADAU5型降阶算法能更有效地减少高阶IRK过程的内迭代计算时间，并且混合精度迭代求精方法在多精度浮点运算环境中是非常有效的。最后，我们证明了基于嵌入公式的高阶IRK方法的实现可以得到一些常微分方程解的精确数值解。
We propose a practical implementation of high-order fully implicit Runge-Kutta(IRK) methods in a multiple precision flo atingpoint environment. Although implementations based on IRK methods in an IEEE754 double precision environment have been reported as RADAU5 developed by Hairer and SPARK3 developed by Jay, they support only 3-stage IRK families. More stages and higher-order IRK formulas must be adopted in order to decrease truncation errors, which become relatively larger than round-off errors in a multiple precision environment. We show that SPARK3 type reduction based on the so-called W-transformation is more effective than the RADAU5 type one for reduction in computational time of inner iteration of a high-order IRK process, and that the mixed precision iterative refinement method is very effi cient in a multiple precision floating-point environment. Finally, we show that our implementation based on high-order IRK methods with embedded formulas can derive precise numerical solutions of some ordinary differential equations.