Study of Numerical Accuracy of Runge-Kutta Second, Third and Fourth Order Method

Transcription

1 Study of Numerical Accuracy of Runge-Kutta Second, Third and Fourth Order Method Najmuddin Ahmad, Shiv Charan,Vimal Partap Singh Department of Mathematics, Integral University, Lucknow Abstract : We have considered ten ordinary differential equation of first order with boundary condition. These equations have been solved by Heun s method, Runge Kutta third order method and by Runge Kutta fourth order method in the interval [ 0, 1 ] by dividing the interval into 200 parts with the help of MATLAB by us. At each point of the interval we have calculated the value of y and compared it with its exact value at that point. Error in the value of y is the difference between calculated and exact values. Percentage error has also been calculated at each point of the intervals. Comparison of the results of the of the Heun s method, Runge Kutta third order method and by Runge Kutta fourth order method shows that Runge Kutta fourth order method is better in all the cases. Mean of the results for all differential equations shows Runge Kutta fourth order method is times better than Runge Kutta second order method ( Heun s method ) and, Runge Kutta third order method. Introduction : In mathematics and computational science, Heun s method ( also called the modified Euler s method or the explicit trapezoidal rule ), named after Karl L. W. M. Heun, is a numerical procedure for solving ordinary differential equations (ODEs) with a given initial. It can be seen as an extension of the Euler method into a two stage second order Runge Kutta. Heun s Method is used to generate a numerical solution to an initial value problem of the form : =f ( x,y ) with the boundary conditions y( ) =. Heun s Method was an improvement over the rather simple Euler Method, and that though it uses Euler s method as a basic, it attempting to compensate for the Euler Method s failure to take the curvature of the solution curve into account. Heun s Method is one of the simplest of a class of method called predictor Corrector algorithms. One of the most powerful predictor Corrector algorithms of all one which is so accurate, that most computer packages designed to find numerical solutions for differential equations will use it by default the fourth order Runge Kutta. The Runge Kutta is a little hard to follow even when one only considers it from a geometric point of view. In reality the formula was not originally derived in this fashion, but with a purely analytical approach. After all, among other things, our geometric explanation doesn t even account for the weights that were used. Numerical ordinary differential is part of numerical analysis which studies the numerical solution of ordinary differential equations (ODEs).This field is also known under the name numerical integration, but some people reserve this term for the computation of integrals. Many differential equations cannot be solved analytically, in which case we have to satisfy ourselves with an approximation to the solution. The algorithms studied here can be used to compute such an approximation. An alternative method is to use techniques from calculus to obtain a series expansion of the solution. Ordinary differential occur in many scientific disciplines, for instance in mechanics, chemistry, biology and economics. In addition, some methods in numerical partial differential equation convert the partial differential equation into an ordinary differential equation, which must then be. We have restricted ourselves to first order differential equations. However, a higher order equation can easily be converted to a system of first order equations by introducing extra. 111

2 112 International Journal of Computer & Mathematical Sciences The proper numerical modeling method heavily depends on the situation, the available resources, and the desired accuracy of the result. If only a quick estimate of a differential equation is required, the Euler method may provide the simplest solution. If much higher accuracy is required, a fifth order Runge Kutta method may be. In all numerical models, as the step size is decreased, the accuracy of the model is increased. The tradeoff here is that smaller step sizes require more computation and therefore increase the amount of time to obtain a solution. A balance between desired accuracy and time required for producing an answer can be achieved by selecting an appropriate step size. One suggested algorithm for selecting a suitable step size is to produce models using two different. the steps size can then be systematically cut in half until the difference between both model is acceptably small creating an error tolerance ). ( effectively Objective : We have considered ten differential equations with initial condition. We have developed MATLAB to solve them to get exact value and approximated value by Heun s method and Runge Kutta fourth order method. We have then compared the accuracy of the results obtained by Heun s method and by Runge Kutta fourth order method. We have also studied that whether the accuracy obtained by Runge Kutta fourth order method can be achieved by Heun s method by increasing the number of intervals. Differential equations under study are given below 1. dy/dx = x + y ; y = 1 when x = 0 2. dy/dx = y - x ; y = 2 when x = 0 3. dy/dx = - 2x ; y = 1 when x = 0 4. dy/dx = x + y + 1 ; y = 1 when x = 0 5. dy/dx = -x + y + 1 ; y = 1 when x = 0 6. dy/dx = (1 + x) x ; y = 1 when x = 0 7. dy/dx = (1 + x) /2 ; y = 1 when x = 0 8. dy/dx = ; y = 1 when x = 0 9. dy/dx = - y ; y = 1 when x = dy/dx = 1 + y ; y = 1 when x = 0 With the help of computer programs, we have calculated the value of y, exact value of y, difference between calculated and exact value of y and percentage error in the value of y. We have calculated the values of y in the interval [ 0,1 ] by assuming the step size of Percentage error in the value of y is defined as = 100 We have also performed multi linear regression ( MLR ) analysis between the value of x and the percentage error in the value y with the help of CAChe Software of Fujitsu. MLR equation is of the form = m * x + n, rcv 2=p, r 2 =q where is the percentage error in the value of y, rcv 2 is the cross validation coefficient, r 2 is the regression coefficient, m, n, p, q R. Predictive power of the regression equation depends on the value of cross validation and regression coefficients. A regression equation is said to possess a good predictive power if the value of cross validation coefficient is greater than 0.2 and the value of regression coefficient is greater than 0.5. As the value of regression coefficient increase, the predictive power of the regression equation increase. The maximum value of regression coefficient may by unity and in this case the regression equation possesses 100% predictive power. Methodology : Second order Runge Kutta Method - the fourth order Runge Kutta method is one of the standard algorithm to solve differential equations. Before we give the algorithm of the fourth order Runge Kutta

3 method we will drive the second order Runge Kutta method. WE start with the original differential equation and integrate it formally = f ( t, y ) y ( t ) = = + = + We essentially changed the task at hand from performing a differentiation to an integration. To do this we expend f( t ) in a second order Taylor series around the midpoint of the integration subinterval f ( t, y ) f (, ) + ( t - ).. (1) Yet since the integral of ( t - ) vanishes when evaluate about the midpoint, we automatically get improved precision using only the first term in (1) f ( t, y ) f (, ) + h f (, ) = + h f (, ) This algorithm cannot be applied immediately since it requires a knowledge of which is not in the scheme of things. We thus approximate with Euler s algorithm. + t = + + h f + = h f = h f The second order Runge Kutta algorithm (2) requires the known derivative function f at the endpoint and midpoint of the interval, and the unknown function y at the previous point. Since we start with initial condition, the algorithm is self starting. Note too that it is applicable with a general function f ( for example nonlinear ), and simple to. In numerical analysis, the Runge Kutta methods are an important family of implicit and explicit iterative methods for the approximation of solution of ordinary differential. These techniques were developed around 1900 by the German mathematicians Runge and Kutta. The Classical Fourth Order Runge Kutta Method - One member of family of Runge Kutta methods is so commonly used, that it is often referred to as RK4 or simply as the Runge Kutta. Let an initial value problem be specified as follows 113

4 = f ( t, y ), y =. Then, the RK4 method for this problem is given by the following equation : = + where = f (, ), = f, = f, = f (, ). Thus, the next value ( ) is determined by the present value ( plus the product of the size of the interval (h) and an estimated slope. The slope is a weighted average of slopes : is the slope at the beginning of the interval ; is the slope at the midpoint of the interval, using slope to determine the value of y at the point using Euler s method ; is again the slope at the midpoint, but now using the slope to determine the y value ; is the slope at the end of the interval, with its y value determined using. When the four slopes are averaged, more weight is given to the slope at the slope = the RK4 method is a fourth order method, meaning that the error per step is on the order of, while the total accumulated error has order. Above formulas are valid for both scalar and vector valued function ( y can be a vector ). Result and Discussions : Solution of Differential equation = x + y with boundary condition x = 0, y = 1 by Heun s method Differential equation is given by = x + y ; x = 0, y = 1. Exact solution of this differential equation is given by x + y + 1 = 2 We have solved this differential equation by Heun s method with the computer program given in the methodology at different values in the interval [ 0, 1 ] by talking the step size of Value of x, calculated value of y, exact value of y, difference between calculated and exact value of y and percentage error in the value of y are shown in the Table 1. This Table indicates that the percentage error in the value of y increases as the value of x increases. The maximum percentage error is at x = 1. Graph between calculated value of y by Heun s method and exact value of y is shown in the graph 1. As the maximum error in the value of y is only , the curves of calculated and exact values of y coincide. Difference between calculated and exact values of y at different values of x for the differential equation is shown in the Graph 2. The shape of this curve is similar to a parabola. Graph 3 shows the percentage error in the value of y at different values of x for the different equation. We have also performed multilinear regression ( MLR ) analysis between the values of x and the percentage error in the value y with the help of CAChe Software of Fujitsu. MLR equation is given below P = *x e

5 115 International Journal of Computer & Mathematical Sciences rcv 2 = r 2 = where P is the percentage error in the value of y, rcv 2 is the cross validation coefficient, r 2 is the regression coefficient. Predictive power of the regression equation depends on the value of cross validation and regression coefficient. A regression equation is said to possess a good predictive power if the value of cross validation coefficient is greater than 0.2 and the value of regression coefficient is greater than 0.5 A the value of regression coefficient increase the predictive power of the regression equation increases. The maximum value of regression coefficient may by unity and in this case the regression equation possesses 100% predictive power. Here, the value of regression coefficient is which indicates that the regression equation has very good predictive power. Conclusions : We have calculated the percentage error at each point of the interval [ 0, 1 ] by dividing it in 200 equal parts for each differential equation using RUnge Kutta second order method and Runge Kutta fourth order method. Average percentage error has been calculated by the formula. Average percentage error = Sum of percentage error at each point of the interval / 200. In each differential equation, the average percentage error in Runge Kutta fourth order method is very low as compared to Runge Kutta second method. Accuracy of Runge Kutta fourth order method over Runge Kutta second order method has been defined by the factor Ä define as Ä = average percentage error in Runge Kutta second order method / average percentage error in Runge Kutta fourth order method. Table 3 includes average percentage error in Runge Kutta second order method ( Heun s method ), average percentage error in Runge Kutta fourth order method and accuracy factor ( Ä ) for each differential equation. A references to Table 3 indicades that In differential equation = x + y ; y = 1, x = 0 ; the average percentage error in Heun s method is ,, the average percentage error in Runge Kutta third order method is and the average percentage error in Runge Kutta fourth order method is In differential equation = y - x ; y = 2, x = 0 ; the average percentage error in Heun s method is ,, the average percentage error in Runge Kutta third order method is and the average percentage error in Runge Kutta fourth order method is In differential equation = -2x ; y = 1, x = 0 ; the average percentage error in Heun s method is ,, the average percentage error in Runge Kutta third order method is and the average percentage error in Runge Kutta fourth order method is In differential equation = x + y + 1 ; y = 1, x = 0 ; the average percentage error in Heun s method is ,, the average percentage error in Runge Kutta third order method is and the average percentage error in Runge Kutta fourth order method is In differential equation =- x + y + 1 ; y = 1, x = 0 ; the average percentage error in Heun s method is ,, the average percentage error in Runge Kutta third order method is and the average percentage error in Runge Kutta fourth order method is

6 In differential equation = ( 1 + x )x ; y = 1, x = 0 ; the average percentage error in Heun s method is , the average percentage error in Runge Kutta third order method is and the average percentage error in Runge Kutta fourth order method is In differential equation = ( 1 + x ) /2 ; y = 1, x = 0 ; the average percentage error in Heun s method is ,, the average percentage error in Runge Kutta third order method is and the average percentage error in Runge Kutta fourth order method is In differential equation = ; y = -1, x = 0 ; the average percentage error in Heun s method is ,, the average percentage error in Runge Kutta third order method is and the average percentage error in Runge Kutta fourth order method is In differential equation = - y ; y = 1, x = 0 ; the average percentage error in Heun s method is ,, the average percentage error in Runge Kutta third order method is and the average percentage error in Runge Kutta fourth order method is In differential equation = 1 + y ; y = 1, x = 0 ; the average percentage error in Heun s method is ,, the average percentage error in Runge Kutta third order method is and the average percentage error in Runge Kutta fourth order method is Table 1 : Average Percentage Error in Runge Kutta Second Order Method ( Heun s method ), Average Percentage Error in Runge Kutta Fourth Order Method and Factor ( Ä ) for Each Differential Equation. S. No. Differential Equation Average Percentage Error in Heun s Metod Average Percetage Error in Runge Kutta third Order Method Average Percetage Error in Runge Kutta third Order Method 1 dy/dx = x + y ; y = 1, x = dy/dx = y - x ; y = 2, x = dy/dx = - 2x ; y = 1, x = dy/dx = x+y+1 ; y = 1, x = dy / dx = -x+y+1; y = 1, x = (1 + x) x ; y = 1, x = (1 + x) /2; y = 1, x = dy/dx = ; y = -1, x = dy / dx = - y ; y = 1, x = dy / dx = 1+ y; y = 1, x =

7 References : 1. Bogacki, P. and Shampine, L. F. (1996) : Efficient Runge Kutta ( 4,5 ) Pair Source :Computers and Mathematics with Application, Vol. 32, No. 6, pp 15-28, Compendex. 2. Butcher. J. C. (1965) : A Modified Multistep Method for the Numerical Integration of Ordinary Differential Equation. Journal of the ACM, Vol. 12 (1),pp Calvo, M.; Montijano, J. I. and Randeg, L. (1990) : New Continuous Extensions for Fifth Order RK Formulas. Rev. Acad. Cienc. Zaragoza (2) 45, pp 69-81, Math. Sci. Net. 4. Cockburn. B. and Shu, C.W. ( 1989) : TVB Runge Kutta Local Projection Discontinuous Galerkinnite Element Method for Conservation Laws. ΙΙ.General Framework. Math. Comp., Vol. 52 No. 186, pp Cotiu, A. (1962) : Delimitation of the error in Fehlberg s Procedure of Numerical Integration of first Order Differential Equations. ( Romanian ). Studia Univ. Babes Bolyai Ser. Math. Phys. Vol. 7, No. 2, pp 37 43, Math. Sci. Net. 6. Daniel Okunbor and Robert D. Skeel ( 1992) : An Explicit Runge Kutta Nystrom Method is Canonical If and Only If Its Adjoint is Explicit. SIAM Journal on Numerical Analysis, Vol. 29, No.2, pp , Jstor. Dekker, K. and Verwer, J. G. ( 1984) : Stability of Runge Kutta Methods for Nonlinear Differential Equations, Vol. 2 of CWI Monografhas. North Holland Publishing Co. ( Amsterdam ). England, R. (1968 ) : Error Estimates for Runge Kutta Solutions to Systems of Ordinary Differential Equations. Research and Development Department, Pressed Steel Fisher Ltd., Cowley, Oxford, U. K. Ernst Hairer, Syvert Paul Norsett and Gerhard Wanner (1993 ) : Solving Ordinary Differential Equations Ι : Nonstiff Problems, Second Edition, Springer Verlag, Berlin, ISBN Fredebeul, C. ; Kornmair, D. and Muller, M. W. ( 2002 ) : Multiple Order Double Output Runge Kutta Fehlberg Formulae : Strategies for Efficient Application. Journal of Computational and Applied Mathematics, Vol. 144, No. 1 2, pp George D. Byrne and Robert J. Lambert ( 1966 ) : Pseudo Runge Kutta Methods Involving two points. Journal of the ACM, Vol. 13 ( 1 ), pp George, E. Forsythe ; Michael A. Malcolm and Cleve B. ( 1977 ) Moler Computer Methods for Mathematical Computations. Englewood Cliffs, NJ : Prentice Hall.Gerisch, A. and Weiner R. ( 2003 ) : The Positivity of Low Order Explicit Runge Kutta Schemes Applied in Splitting Methods. Comput. Math.Appl., Vol. 45 No. 1 3, pp Hairer, E. ; Narsett, S. P. and Wanner, G. ( 1993 ) : Solving Ordinary Differential Equations. Ι. Nonstiff problems, Vol. 8 of Springer Series in Computational Mathematics. Springer Verlag ( Berlin ), Second Ed. 11. Hundsdofer, W. and Verwer, J. G. ( 2003 ) : Numerical Solution of Timedependent Advection Diffusion Reaction Equations, Vol. 33 of Springer Series in Computational Mathematics. Springer ( Berlin ). 12. Jain, R. K. and Jain, M. K. ( 1971 ) : Optimum Runge Kutta Fehlberg Methods for First Order Differential Equations. J. Inst. Math. Appl., Vol. 8, pp , Math. Sci. Net. John, W. ( 1958 ) : Carr, ΙΙΙ, Error Bounds for the Runge Kutta Single Step Integration Process, Journal of the ACM ( JACM ), Vol. 5 No. 1, pp Kaw, Autar ; Kalu, Egwu ( 2008 ) : Numerical Methods with Applications ( 1 st Ed. ), Kendall, E. Atkinson ( 1998 ) : An Introduction to Numerical Analysis. John Wiley & Sons. 15. Kharab, A. (1994 ) : Spreadsheet Solution to an initial Value Problem using the Runge Kutta Fehlberg Method. Computer Application in Engineering Education, Vol. 2, No. 2, pp , Compendex. 16. Kraaijevanger J. F. B. M. ( 1991 ) : Contractivity of Runge Kutta Methods. BIT, Vol. 31, No. 3, pp Ling, Xiang; Tu, Shan Tung ; Gong and Jain Ming ( 2000 ) : Application of Runge Kutta Merson Algorithm for Creep Damage Analysis. International Journal of Pressure Vessels and Piping, Vol. 77, No. 5, pp , Compendex. 18. Murugesan, K; Dhayabaran, D. Paul; Amiritharaj, E. C. Henry ; Evans, David J.( 2002 ) : A Fourth Order Embedded Runge Kutta RKACeM( 4, 4 ) Method Based on Arithmetic and Centroidal Means with Error Control. Int. J. Comput. Math., Vol. 79, No. 2, pp , Math. Sci. Net. 19. Numerical Methods in Physics, Chemistry and Engineering. Gottlieb S., Shu C. W. ( 1998 ) : Total Variation Diminishing Runge Kutta Schemes. Math. Comp., Vol. 67, No. 221, pp Pimenov, V. G. ( 2001 ) : General Linear Methods for the Numerical Solution of Functional Differential Equations, Differential Equations, Vol. 37, No Procedures of Runge Kutta Fehlberg Type with Error of Order 0, for the Approximation of First Order Integrodifferential Equations of Volterra Type. ( Romanian ) Micula, Maria Stud. Cerc. Mat., 25 (1973), pp 33 40, Math. Sci. Net. 117

8 22. Runge Kutta Fehlberg Type Procedure on Two Nodes for Numerical Integration of Systems Differential Equations. Dumitras, Daria Elena Automat. Comput. Appl. Math., Vol. 2, pp , Math. Sci. Net. 23. Ruuth, S. J. ( 2004 ) : Global optimization of Explicit Strong Stability preserving Runge Kutta Methods. Tech. Rep., Department of Mathematics Simon Fraser University. 24. Sideridis, A. B. and Simos, T. E. ( 1992 ) : A Low Order Embedded Runge Kutta Method for Periodic Initial Value Problems. J. Comput. Appl. Math., Vol. 44. No. 2, pp , Math. Sci. Net. 25. Simos, T. E., ( 1993 ) : A Runge Kutta Fehlberg Method with Phase Leg of Order Infinity for initial Value Problem with Oscillating Solution. Computers and Mathematics with Applicatons, Vol. 25, No. 6 pp Simos, T. E., ( 1995 ) : Modified Runge Kutta _ Fehlberg Methods for Periodic Initial Value Problems. Japan J. Indust. Appl. Math., Vol. 12, No. 1, pp , Math. Sci, Net. 27. Spiteri, R. J. and Ruuth, S. J. ( 2003 ) : Non Linear Evolution using Optimal Fourth Order Strong Stability Preserving Runge - Kutta Methods. Math. Comput. Simulation, Vol. 62, No. 1 2, pp Strogatz, S. H. ( 1994 ) : Nonlinear Dynamics and Chaos with Applications in Physics, Biology, Chemistry and Engineering. Perseus Books ( Reading, M. A. ). 29. Suchitra Gupta ( 1985 ) : An Adaptive Boundary Value Runge Kutta Solver for First Order Boundary Value Problems SIAM Journal on Numerical Analysis, Vol. 22, No. 1., pp Tavernini, L. ( 1971 ) : One step Methods for the Numerical Solution of Volterra Functional Differential Equations, SIAM J. Numer. Anal., Vol. 8, No Test results on Initial Value Methods for Non Stiff Ordinary Differential Equations W. H. Enright ; T. E. Hull SIAM Journal on Numerical Analysis, Vol. 13, No. 6. ( Dec., 1976 ), pp , Jstor. 32. The Runge Kutta Theory in a Nutshell Peter Albrecht SIAM Journal on Numerical Analysis, Vol. 33, No. 5. ( Oct., 1996 ), pp , Jstor. 33. Toro, E. F. ( 1999 ) : Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introducing. Springer Verlag ( Berlin ), Second Ed. 34. Verner, J. H. ( 1990 ) : A Contrast of Some Runge Kutta Formula Pairs SIAM Journal on Numerical Analysis, Vol. 27, No. 5, pp , Jstor. 35. Verner, J. H. ( 1991 ) : Some Runge Kutta Formula Pairs. SIAM J. Numer. Anal. Vol. 28, No. 2, pp , Math. Sci. Net. 36. Verner, J. H. ( 1996 ) : Higher Order Explicit Runge Kutta Pairs with Low Stage Order Applied Numerical Mathematics, Vol. 22, No. 1 3, pp , Compendex. William B. Gruttke ( 1970 ) : Pseudo Runge Kutta Methods of the Fifth order. Journal of the ACM, Vol. 17 ( 4 ), pp