Additive Runge-Kutta schemes for convection-diffusion-reaction equations
Additive Runge-Kutta schemes for convection-diffusion-reaction equations
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DOI：
10.1016/s0168-9274(02)00138-1
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发表时间：
2003-01-01
影响因子：
2.8
通讯作者：
Carpenter, MH
中科院分区：
文献类型：
--
作者：
Kennedy, CA;Carpenter, MH
研究了加性Runge-Kutta(ARK)方法在空间离散的一维对流扩散反应(CDR)方程中的应用。对于将N个不同的龙格-库塔方法组合成一个组合方法的一般情况，首先考虑了精度、稳定性、守恒性和稠密输出。然后，提出了三阶到五阶的隐显(N=2)加性龙格-库塔(ARK(2))方法，它允许用L稳定的、刚性精确的显式单对角隐式龙格-库塔方法积分刚性项，而非刚性项与传统的显式龙格-库塔方法积分。分块方法的耦合误差项与单元方法的耦合误差项具有相同的量级。导出的Ark(2)方法对于很大的刚性尺度本征值z([i])-无穷大具有零稳定性函数，并且在没有刚度z([i])-gt；0的情况下保持了高的稳定性效率。基于密集输出公式，给出了外推型级值预报器。优化方法最小化了前序Ark(2)误差项和布彻系数的大小，同时最大化了守恒性。对CDR问题的数值试验表明，新格式的刚度泄漏可以忽略不计，并且具有接近经典阶的收敛速度。然而，对三个简单的奇异摄动问题的测试表明，降阶通常是可以预测的。差错控制最好由一个PID控制器来管理。虽然五阶方法的结果令人失望，但新的三阶和四阶方法至少与现有的ARK(2)方法一样有效。(C)2002个iMACs。爱思唯尔科学公司出版。版权所有。
Additive Runge-Kutta (ARK) methods are investigated for application to the spatially discretized one-dimensional convection-diffusion-reaction (CDR) equations. Accuracy, stability, conservation, and dense-output are first considered for the general case when N different Runge-Kutta methods are grouped into a single composite method. Then, implicit-explicit, (N = 2), additive Runge-Kutta (ARK(2)) methods from third- to fifth-order are presented that allow for integration of stiff terms by an L-stable, stiffly-accurate explicit, singly diagonally implicit Runge-Kutta (ESDIRK) method while the nonstiff terms are integrated with a traditional explicit Runge-Kutta method (ERK). Coupling error terms of the partitioned method are of equal order to those of the elemental methods. Derived ARK(2) methods have vanishing stability functions for very large values of the stiff scaled eigenvalue, z([I]) --> -infinity, and retain high stability efficiency in the absence of stiffness, z([I]) --> 0. Extrapolation-type stage-value predictors are provided based on dense-output formulae. Optimized methods minimize both leading order ARK(2) error terms and Butcher coefficient magnitudes as well as maximize conservation properties. Numerical tests of the new schemes on a CDR problem show negligible stiffness leakage and near classical order convergence rates. However, tests on three simple singular-perturbation problems reveal generally predictable order reduction. Error control is best managed with a PID-controller. While results for the fifth-order method are disappointing, both the new third- and fourth-order methods are at least as efficient as existing ARK(2) methods. (C) 2002 IMACS. Published by Elsevier Science B.V. All rights reserved.