基于勒让德多项式逼近的4级4阶隐式Runge-Kutta方法
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摘要
利用勒让德多项式逼近理论和高斯-洛巴托求积公式,构造了一个4级4阶的隐式RungeKutta方法.理论分析发现,该算法具有良好的稳定性-是A(α)稳定的且α接近于90~0,是刚性稳定的且D值接近于0,几乎是A稳定的和L稳定的,并能有效求解刚性常微分方程初值问题,数值算例显示了该算法的有效性.
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 andα 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.
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 andα 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.
引文
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[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.
[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.