• 论文 •

### 基于勒让德多项式逼近的4级4阶隐式Runge-Kutta方法

1. 西安理工大学理学院, 西安 710054
• 收稿日期:2014-04-28 出版日期:2015-03-15 发布日期:2015-03-05
• 基金资助:

陕西省教育厅科学研究计划(11JK0524)资助项目.

Liu Cuicui, Zhang Ruiping. A FOUR-STAGE FOURTH-ORDER IMPLICIT RUNGE-KUTTA METHOD BASED ON LEGENDRE POLYNOMIALS APPROXIMATION[J]. Journal of Numerical Methods and Computer Applications, 2015, 36(1): 22-30.

### A FOUR-STAGE FOURTH-ORDER IMPLICIT RUNGE-KUTTA METHOD BASED ON LEGENDRE POLYNOMIALS APPROXIMATION

Liu Cuicui, Zhang Ruiping

1. Schools of Sciences, Xi'an University of Technology, Xi'an 710054, China
• Received:2014-04-28 Online:2015-03-15 Published:2015-03-05

By using the Legendre polynomials approximation theory and Gauss-Lobatto quadrature formula, a four-stage fourth-order implicit Runge-Kutta method is presented. It is showed that the new algorithm has good stability properties in theoretical analysis, A(α)-stable and α is close to ninety degrees, and stiff stable and D is close to zero. It is almost A-stable and almost L-stable. The new method can solve stiff ordinary differential equations effectively. The numerical examples illustrate its effectiveness.

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 [1] Goeken D, Johnson O. Fifth-order Runge-Kutta with higher order derivative approximations[J]. Electronic Journal of Differential Equations, 1999, 2: 1-9.[2] Podisuk M. Open formula of Runge-Kutta method for solving autonomous ordinary differential equation[J]. Applied Mathematics and Computation. 2006, 181(1): 536-542.[3] Ramos H, Vigo-Aguiar J. A fourth-order Runge-Kutta method based on BDF-type Chebyshev approximations[J]. Journal of computational and applied mathematics, 2007, 204(1): 124-136.[4] Kulikov G Yu, Shindin S K. Adaptive nested implicit Runge-Kutta formulas of Gauss type[J]. Applied Numerical Mathematics, 2009, 59: 707-722.[5] Wu Xinyuan. A class of Runge-Kutta of order three and four with reduced evaluations of function[ J]. Applied Mathematics and Computation. 2003, 146: 417-432.[6] Niegemann J, Diehl R, Busch K. Efficient low-storage Runge-Kutta schemes with optimized stability regions[J]. Journal of Computational Physics, 2012, 231(2): 364-372.[7] Najafi-Yazdi A, Mongeau L. A low-dispersion and low-dissipation implicit Runge-Kutta scheme[J]. Journal of computational physics, 2013, 233: 315-323.[8] Kalogiratou Z. Diagonally implicit trigonometrically fitted symplectic Runge-Kutta methods[J]. Applied Mathematics and Computation, 2013, 219: 7406-7421.[9] 《现代应用数学手册》编委会. 现代应用数学手册-计算与数值分析 卷[M]. 北京: 清华大学出版社, 2005.[10] 李寿佛. 刚性常微分方程及刚性泛函微分方程数值分析[M]. 湖南: 湘潭 大学出版社, 2010.
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