var callback = function (xhrj) { } ajaxj.get(url, true, callback, "try"); } // function SetNumTwo(item) { // alert("jinlia"); // var url = "../userInformation/PDFLogin.aspx"; // var refererrurl = document.referrer; // var downloadurl = window.location.href; // var args = "PaperID=" + item + "&RefererUrl=" + refererrurl + "&DownloadUrl="+downloadurl; // url = url + "?" + args + "&rand=" + RndNum(4); // //// window.setTimeout("show('" + url + "')", 500); // } // function pdfdownloadjudge() { // $("a").each(function(index) { // var rel = $(this).attr("rel"); // if (rel == "true") { // $(this).removeAttr("onclick"); // $(this).attr("href","#"); // //$(this).bind('click', function() { SetNumTwo(31701)}); // var url = "../userInformation/PDFLogin.aspx"; // var refererrurl = document.referrer; // var downloadurl = window.location.href; // var args = "PaperID=" + 31701 + "&RefererUrl=" + refererrurl + "&DownloadUrl=" + downloadurl; // url = url + "?" + args + "&rand=" + RndNum(4); // // $(this).bind('click', function() { ShowTwo(url)}); // } // }); // } // //获取下载pdf注册的cookie // function getcookie() { // var cookieName = "pdfddcookie"; // var cookieValue = null; //返回cookie的value值 // if (document.cookie != null && document.cookie != '') { // var cookies = document.cookie.split(';'); //将获得的所有cookie切割成数组 // for (var i = 0; i < cookies.length; i++) { // var cookie = cookies[i]; //得到某下标的cookies数组 // if (cookie.substring(0, cookieName.length + 2).trim() == cookieName.trim() + "=") {//如果存在该cookie的话就将cookie的值拿出来 // cookieValue = cookie.substring(cookieName.length + 2, cookie.length); // break // } // } // } // if (cookieValue != "" && cookieValue != null) {//如果存在指定的cookie值 // return false; // } // else { // // return true; // } // } // function ShowTwo(webUrl){ // alert("22"); // $.funkyUI({url:webUrl,css:{width:"600",height:"500"}}); // } //window.onload = pdfdownloadjudge;
ENG> Vol.5 No.5, May 2013
Share This Article:
Cite This Paper >>

Runge-Kutta Schemes Coefficients Simulation for Comparison and Visual Effects

Abstract Full-Text HTML Download Download as PDF (Size:1052KB) PP. 530-536
DOI: 10.4236/eng.2013.55063    3,606 Downloads   5,092 Views   Citations
Author(s)    Leave a comment
Tajudeen Abiola Ogunniyi Salau, Olusegun Olufemi Ajide


Department of Mechanical Engineering, University of Ibadan, Ibadan, Nigeria.


Runge-Kutta scheme is one of the versatile numerical tools for the simulation of engineering systems. Despite its wide and acceptable engineering use, there is dearth of relevant literature bordering on visual impression possibility among different schemes coefficients which is the strong motivation for the present investigation of the third and fourth order schemes. The present study capitalise on results of tedious computation involving Taylor series expansion equivalent supplemented with Butcher assumptions and constraint equations of well-known works which captures the essential relationship between the coefficients. The simulation proceeds from random but valid specification of two out of the total coefficients possible per scheme. However the remaining coefficients are evaluated with application of appropriate function relationship. Eight and thirteen unknown coefficients were simulated respectively for third and fourth schemes over a total of five thousand cases each for relevant distribution statistics and scatter plots analysis for the purpose of scheme comparison and visual import. The respective three and four coefficients of the slope estimate for the third and fourth schemes have mix sign for large number of simulated cases. However, none of the two schemes have above three of these coefficients lesser than zero. The percentages of simulation results with two coefficients lesser than zero dominate and are respectively 56.88 and 77.10 for third and fourth schemes. It was observed that both popular third and fourth schemes belong to none of the coefficients being zero classification with respective percentage of 0.72 and 3.28 intotal simulated cases. The comparisons of corresponding scatter plots are visually exciting. The overall difference between corresponding scatter plots and distribution results can be used to justify the accuracy of fourth scheme over its counterpart third scheme.


Runge-Kutta; Visual Effects; Simulation Coefficients; Taylor Series and Butcher Assumptions

Cite this paper

T. Salau and O. Ajide, "Runge-Kutta Schemes Coefficients Simulation for Comparison and Visual Effects," Engineering, Vol. 5 No. 5, 2013, pp. 530-536. doi: 10.4236/eng.2013.55063.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] R. Noorhelyna and R. A. Rokial, “Solving Lorenz System by Using Runge-Kutta Method,” European Journal of Scientific Research, Vol. 32, No. 2, 2009, pp. 241-251.
[2] B. Stéphane and M. Fabrice, “Embedded Runge-Kutta Scheme for Step-Size Control in the Interaction Picture Method,” Computer Physics Communications, Vol. 184, No. 4, 2013, pp. 1211-1219.
[3] A. Farahani, “Accelerated Runge-Kutta Methods and NonSlip Rolling,” Ph.D. Thesis, University of Southern California, Los Angeles, 2010.
[4] N. A. Salih, “Parametric Study of Nonlinear Beam Vibration Resting on Linear Elastic Foundation,” Journal of Mechanical Engineering and Automation, Vol. 2, No. 6, 2012, pp. 114-134.
[5] A. Najafi-Yazdi and L. Mongeau, “A Low-Dispersion and Low Dissipation Implicit Runge-Kutta Scheme,” Journal of Computational Physics, Vol. 233, 2013, pp. 315-323. doi:10.1016/
[6] M. T. Rosenstein and J. J. Collins, “Visualizing the Effects of Filtering Chaotic Signals,” Boston University, Boston, 1993.
[7] J.-K. Uk and K. Hoon, “Chaotic Behaviours in Fuzzy Dynamic Systems: ‘Fuzzy Cubic Map’,” Proceedings of the 1996 Asian Fuzzy Systems Symposium, Kenting, 1114 December 1996, pp. 314-319.
[8] S. Hermann, “Exploring Sitting Posture and Discomfort Using Nonlinear Analysis Methods,” IEEE Transactions on Information Technology in Biomedicine, Vol. 9, No. 3, 2005, pp. 392-401. doi:10.1109/TITB.2005.854513
[9] J. Cheng, J. R. Tan, and C. B. Gan, “Visualization of Chaotic Dynamic Systems Based on Mandelbrot Set Methodology,” Fractals, Vol. 16, No. 1, 2008, p. 89. doi:10.1142/s0218348X08003752
[10] M. Sun, C. Y. Zeng and L. Li, “Chaos Control and Chaotification for a Three-Dimensional Autonomous System,” Journal of Nonlinear Science, Vol. 7, No. 2, 2009, pp. 220-225.
[11] C. Stegemann, H. A. Albuquerque, R. M. Rubinger and P. C. Rech, “Lyapunov Exponent Diagram of a 4-Dimensional Chua System,” Chaos, Vol. 21, No. 3, 2011, Article ID: 033105. doi:10.1063/1.3615232
[12] X. Wei, M. F. Randrianandrasana, M. Ward and D. Lowe, “Nonlinear Dynamics of a Periodically Driven Duffing Resonator Coupled to a Van der Pol Oscillator,” Mathematical Problems in Engineering, Vol. 2011, 2011, Article ID: 248328.
[13] I. M. Anthanasios, “Simulation and Visualization of Chaotic System,” Computer and Information Science, Vol. 5, No. 4, 2012, pp. 25-52.
[14] S. C. Chapra and R. Canale, “Numerical Methods for Engineers,” 5th Edition, McGraw-Hill (International Edition), New York, 2006.
[15] H. Musa, S. Ibrahim and M. Y. Waziri, “A Simplified Derivation and Analysis of Fourth Order Runge-Kutta Method,” International Journal of Computer Applications, Vol. 9, No. 8, 2010, pp. 51-55.

comments powered by Disqus
ENG Subscription
E-Mail Alert
ENG Most popular papers
Publication Ethics & OA Statement
ENG News
Frequently Asked Questions
Recommend to Peers
Recommend to Library
Contact Us

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.