scholarly journals Stepwise Global Error Control in an Explicit Runge-Kutta Method Using Local Extrapolation with High-Order Selective Quenching

2011 ◽  
Vol 3 (2) ◽  
Author(s):  
Justin Steven Prentice
Keyword(s):  
Error Control ◽  
High Order ◽  
Global Error ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Local Extrapolation

scholarly journals Nine-stage multi-derivative Runge-Kutta method of order 12

2009 ◽  
Vol 86 (100) ◽  
pp. 75-96 ◽  
Author(s):  
Truong Nguyen-Ba ◽  
Vladan Bozic ◽  
Emmanuel Kengne ◽  
Rémi Vaillancourt
Keyword(s):  
Order Conditions ◽  
High Order ◽  
Kutta Method ◽  
True Solution ◽  
First Order ◽  
Cpu Time ◽  
Runge Kutta Method ◽  
Higher Derivatives ◽  
Runge Kutta ◽  
Taylor Methods

A nine-stage multi-derivative Runge-Kutta method of order 12, called HBT(12)9, is constructed for solving nonstiff systems of first-order differential equations of the form y'= f(x, y), y(x0) = y0. The method uses y' and higher derivatives y(2) to y(6) as in Taylor methods and is combined with a 9-stage Runge-Kutta method. Forcing an expansion of the numerical solution to agree with a Taylor expansion of the true solution leads to order conditions which are reorganized into Vandermonde-type linear systems whose solutions are the coefficients of the method. The stepsize is controlled by means of the derivatives y(3) to y(6). The new method has a larger interval of absolute stability than Dormand-Prince's DP(8,7)13M and is superior to DP(8,7)13M and Taylor method of order 12 in solving several problems often used to test high-order ODE solvers on the basis of the number of steps, CPU time, maximum global error of position and energy. Numerical results show the benefits of adding high-order derivatives to Runge-Kutta methods.

The modified high-order Haar wavelet scheme with Runge–Kutta method in the generalized Burgers–Fisher equation and the generalized Burgers–Huxley equation

2021 ◽  
pp. 2150419
Author(s):  
Ming Zhong ◽  
Qi-Jun Yang ◽  
Shou-Fu Tian
Keyword(s):  
High Efficiency ◽  
Haar Wavelet ◽  
High Order ◽  
Fisher Equation ◽  
Kutta Method ◽  
Third Order ◽  
Time Layer ◽  
Runge Kutta Method ◽  
Huxley Equation ◽  
Runge Kutta

In this work, we focus on the modified high-order Haar wavelet numerical method, which introduces the third-order Runge–Kutta method in the time layer to improve the original numerical format. We apply the above scheme to two types of strong nonlinear solitary wave differential equations named as the generalized Burgers–Fisher equation and the generalized Burgers–Huxley equation. Numerical experiments verify the correctness of the scheme, which improves the speed of convergence while ensuring stability. We also compare the CPU time, and conclude that our scheme has high efficiency. Compared with the traditional wavelets method, the numerical results reflect the superiority of our format.

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