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.

A High Order Time Advancement Scheme for Prediction of Solidification Processes

2010 ◽  
Vol 297-301 ◽  
pp. 779-784 ◽  
Author(s):  
A. Abbasnejad ◽  
M.J. Maghrebi ◽  
H. Basirat Tabrizi
Keyword(s):  
Analytical Solutions ◽  
Second Order ◽  
High Order ◽  
Kutta Method ◽  
Third Order ◽  
Runge Kutta Method ◽  
Solidification Processes ◽  
Order Scheme ◽  
High Order Time ◽  
Good Agreement

The aim of this study is the simulation of alloys and pure materials solidification. A third order compact Runge-Kutta method and second order scheme are used for time advancement and space derivative modeling. The results are compared with analytical and semi-analytical solutions and show very good agreement.

Numerical Simulations of a Polydisperse Sedimentation Model by Using Spectral WENO Method with Adaptive Multiresolution

2016 ◽  
Vol 13 (06) ◽  
pp. 1650037
Author(s):  
Carlos A. Vega ◽  
Francisco Arias
Keyword(s):  
Convergence Rates ◽  
Time Discretization ◽  
Strong Stability ◽  
Kutta Method ◽  
Strong Stability Preserving ◽  
Third Order ◽  
Runge Kutta Method ◽  
Sedimentation Model ◽  
Runge Kutta ◽  
Fifth Order

In this work, we apply adaptive multiresolution (Harten’s approach) characteristic-wise fifth-order Weighted Essentially Non-Oscillatory (WENO) for computing the numerical solution of a polydisperse sedimentation model, namely, the Höfler and Schwarzer model. In comparison to other related works, time discretization is carried out with the ten-stage fourth-order strong stability preserving Runge–Kutta method which is more efficient than the widely used optimal third-order TVD Runge–Kutta method. Numerical results with errors, convergence rates and CPU times are included for four and 11 species.

scholarly journals ANN-based modeling of third order runge kutta method

2015 ◽  
Vol 4 (1) ◽  
pp. 180
Author(s):  
M. Dehghanpour ◽  
A. Rahati ◽  
E. Dehghanian
Keyword(s):  
Differential Equations ◽  
Quantum Physics ◽  
Numerical Approach ◽  
Mathematical Explanation ◽  
Kutta Method ◽  
Specific System ◽  
Third Order ◽  
Runge Kutta Method ◽  
Numerical Solution Methods ◽  
Runge Kutta

<p>The world's common rules (Quantum Physics, Electronics, Computational Chemistry and Astronomy) find their normal mathematical explanation in language of differential equations, so finding optimum numerical solution methods for these equations are very important. In this paper, using an artificial neural network (ANN) a numerical approach is designed to solve a specific system of differential equations such that the training process of the ANN  calculates the  optimal values for the coefficients of third order Runge Kutta method. To validate our approach, we performed some experiments by solving two body problem using coefficients obtained by ANN and also two other well-known coefficients namely Classical and Heun. The results show that the ANN approach has a better performance in compare with two other approaches.</p>

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.

scholarly journals Numerical Scheme Based on the Implicit Runge-Kutta Method and Spectral Method for Calculating Nonlinear Hyperbolic Evolution Equations

Axioms ◽  
2022 ◽  
Vol 11 (1) ◽  
pp. 28
Author(s):  
Yasuhiro Takei ◽  
Yoritaka Iwata
Keyword(s):  
Spectral Method ◽  
Numerical Scheme ◽  
Evolution Equations ◽  
Unit Size ◽  
Kutta Method ◽  
Third Order ◽  
Runge Kutta Method ◽  
Klein Gordon ◽  
Runge Kutta ◽  
The Stability

A numerical scheme for nonlinear hyperbolic evolution equations is made based on the implicit Runge-Kutta method and the Fourier spectral method. The detailed discretization processes are discussed in the case of one-dimensional Klein-Gordon equations. In conclusion, a numerical scheme with third-order accuracy is presented. The order of total calculation cost is O(Nlog2N). As a benchmark, the relations between numerical accuracy and discretization unit size and that between the stability of calculation and discretization unit size are demonstrated for both linear and nonlinear cases.

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