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A Study of Second and Third Order Stiffly Accurate Generalized Runge-Kutta Integration Methods

机译：二阶和三阶僵硬准确的贯通跑步 - 库特拉集成方法研究

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GRK methods are semi-explicit integration methods not requiring any iteration in the solution stages, yet have stability properties similar to implicit Runge-Kutta methods. In this study eleven new sets of coefficients for second and third order stiffly accurate generalized Runge-Kutta (SGRK) methods, with appended or embedded methods for error estimation, are tested for suitability to integrate systems of stiff differential equations. Kaps and Renthrop’s classic fourth order GRK method-GRK4T and Harrier and Wanner’s six stage fourth order SGRK-RODAS, are also compared with these new methods. Results of numerical experiments on four test problems are presented as work-precision diagrams. Numerical experiments on two small test problems: Robertson’s reaction and HIRES, and two large test problems: Burgers’ equation and CUSP are preformed to demonstrate the effectiveness of these methods for solving stiff differential equations. The best three point approximation for (e)u2/2)/(e)x in Burgers’ equation was developed during this study. Two stage second order stiffly accurate GRK with embedded or appended methods for error estimation are integration methods that have not been studied previously. The excellent stability properties of these methods make them good candidates for stiff and differential algebraic equations (DAE). Stiffly accurate three stage third order methods with embedded or appended methods for error estimation have not been described previously either. Results of numerical experiments using five different second order methods and six different third order stiffly accurate methods are presented and compared to two GRK from the literature.
机译：GRK方法是半显式集成方法，不需要在解决方案阶段中的任何迭代，但具有类似于隐式跳闸-Kutta方法的稳定性属性。在本研究中，对二阶和三阶精确准确的漫长 - Kutta（SGRK）方法的11个新的系数进行了应用的误差估计，以适当地集成僵硬微分方程的系统。与这些新方法相比，KAPS和Renthrop的经典四阶GRK方法-GK4T和Harrier和Wanner的六阶段四阶SGRK-Rodas。四个测试问题的数值实验结果作为工作精度图呈现。两个小型测试问题的数值实验：Robertson的反应和雇用，以及两个大型测试问题：预先形成汉堡方程和尖端，以证明这些方法求解凝固方程的有效性。在本研究期间开发了汉堡方程中（e）U2 / 2）/（e）x的最佳三点近似。两个阶段二阶准确GRK与嵌入或附加的误差估计方法是尚未先前研究的集成方法。这些方法的优异稳定性特性使其成为刚性和差分代数方程（DAE）的良好候选者。毫无疑问，僵硬准确的三阶段三阶方法，尚未描述误差估计的嵌入或附加方法。使用五种不同二阶方法的数值实验和六种不同的三阶精确方法的数值实验结果显示，并与来自文献的两种Grk相比。

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