Almost Runge-Kutta Methods Of Orders Up To Five

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1 Almost Runge-Kutta Methods Of Orders Up To Five ABRAHAM OCHOCHE Department of Information Technology, Federal University of Technology, Minna, Niger State PMB 65, Nigeria PETER NDAJAH Graduate School of Science and Technology, Niigata University, 85 Ikarashi -no-cho, Nishi-ku Niigata, 95-8, Japan Abstract: In this paper, we have sought to investigate the viability of a type of general linear methods called Almost Runge-Kutta (ARK) methods, as a means of obtaining acceptable numerical approximations of the solution of problems in continuous mathematics We have outlined the derivation and implementation of this class of methods up to order five Extensive numerical experiments were carried out and the results clearly show that ARK methods are indeed a viable alternative to existing traditional methods Key Words: Almost, Order, Alternative, Euler, Runge Kutta, General Linear Methods Introduction Ordinary differential equations (ODEs) can be used to model many different types of physical behaviors, form chemical reactions to the notion of planets around the sun they are at the very heart of our understanding of the physical world today[see, 5] Unfortunately, as the systems being modeled get more complex, so do the solutions This can mean that we are left in the position of being unable to solve the problem analytically This is where the need for numerical methods become obvious Even though they do not give exactly solutions, they do enable us to find good approximations to the solution at any point where it is required General Linear Methods The name General linear methods applies to a large family of numerical methods for ordinary differential equations They were introduced in [9] to provide a unifying theory of the basic questions of consistency, stability and convergence Later they were used as a framework for studying accuracy questions and later the phenomena associated with nonlinear convergence They combine the multi-stage nature of Runge-Kutta methods with the notion of passing more than one piece of information between steps that is used in linear multi-step methods Their discovery opened the possibility of obtaining essentially new methods that were neither Runge-Kutta nor linear multi-step methods would exist which are practical and have advantages over the traditional methods These extremely broad class of methods, besides containing Runge-Kutta and linear multi-step methods as special cases also contain hybrid methods, cyclic composite linear multi-step methods and pseudo Runge-Kutta methods Examples of general linear methods As noted above, this class of methods is a large one It includes the traditional methods such as Runge Kutta methods and linear multi-step methods, along with methods that have been developed within the general linear methods framework, such as DIMSIMs, IRKS methods (with DESIRE and ESIRK as special cases) and Almost Runge-Kutta (ARK) methods 3 Almost Runge-Kutta Methods Almost Runge-Kutta (ARK) methods are a very special class of general linear methods introduced by Butcher [] The basic idea of these methods, is to retain the multi-stage nature of Runge-Kutta methods, while allowing more than one value to be passed from step to step Hence, they have a multi-value nature These methods have advantages over traditional methods, which are to be found in low-cost local estimation and dense output These latter features will be a consequence of the higher stage orders that are possible because of the multi-value nature of the new methods This multi-value nature brings its own difficulties The number of values passed between steps varies among general linear methods; the number of val- ISSN: Issue 5, Volume, May

2 ues is three for ARK methods Of the three input and output values in ARK methods, one approximates the solution value ie y(x n ) the second approximates the scaled first derivative (hy (x n )) while the third approximates second, derivatives respectively (h y (x n )) To simplify the starting procedure, the second derivative is required to be accurate only to within O(h 3 ), where h as usual is the In order to ensure that this low order does not adversely affect the solution value, the method has unbuilt annihilation conditions As a result of these extra input values, we are able to obtain stage order two, unlike explicit Runge-Kutta methods that are only able to obtain stage order one The advantage of this higher order is that we are able to interpolate and obtain an estimate at little extra cost [] 3 Motivation for the Methods When ARK methods were first proposed, it was hoped to develop a method that retained the spirit of R- K methods, but had advantages over the traditional methods The method was required to retain the simple stability functions of R-K methods as well as pass through very little information between steps The information that was passed from step to step was required to be easy to adjust for changes The main advantages that could be seen were a cheap estimator and dense output as well as the higher stage order that could be achieved [] 3 Structure of ARK Methods The general form of ARK methods is given by Y Y Y s 3 hf (Y ) hf (Y ) hf (Y s ) y [n ] y [n ] y [n ] r () where s is the number of internal stages A is an s s strictly lower triangular matrix, such that e T s A b T which implies that the last row of the A matrix is the same as the b vector in the first row of the B matrix s is the number of internal stages The vector e is a vector of ones, and is also of length s e s is a vector of length s consisting entirely of zeros except the sth component which is For an order p method the three output values are given by the equations below y(x n ) + O(h p+ ), hy (x n ) + O(h p+ ), () h y (x n ) + O(h 3 ) In choosing the coefficients of the method, we are careful to ensure that the simple stability properties of Runge-Kutta methods are retained In this introductory paper, we would focus on the case where the matrix A is strictly lower triangular by which we mean the methods are explicit Explicit ARK methods have the general form: Y Y Y s 3 a a 3 a 3 e c Ae c Ac a s, a s, a s,3 b b b 3 b s b b b 3 b s b β β β 3 β s β s β hf (Y ) hf (Y ) hf (Y s ) y [n ] y [n ] y [n ] r (3) As in traditional Runge-Kutta theory, b is a vector of length s representing the weights and c is a vector of length s representing the positions at which the function f is evaluated The vector e is a vector of ones, and is also of length s The form of the U matrix is given by a Taylor series expansion of the internal stages The internal stages of an ARK method is given by y(x + hc i ) u i y + u i hy + u i3 h y ISSN: Issue 5, Volume, May

3 a ij y (x + hc j ) (4) i +h j By carrying out the Taylor series expansion on both sides of equation (4) and equate the coefficients of y, y and y, we find u i, i,,, s u i c i j a ij u i3 c i j a ijc j (5) We must note some very important features of ARK methods: The first row of the B matrix is the same as the last row of the A matrix, we denote this as b T and the first row of the V matrix is the same as the last row of the U matrix, because we wish the final internal stage to give us the same quantity that is to be exported as the first outgoing approximation It also means that c s We also wish the second outgoing approximations to be h times the derivative of the final stage This means that the second row of the B and V matrices consists of zeros, with the exception of a in the (, s) position of B that is B is e T s The last row of B isβ T There are three types of conditions to consider when deriving Almost Runge-Kutta methods: Order conditions Annihilation conditions 3 Runge-Kutta stability conditions Detailed explanation of how these conditions are implemented, can be found in [], [] Theorem 3: An ARK method of order p with p stages has RK- stability if and only if β T (I + β s A) β s e T s (6) ( + β sc )b T A s c s! c exp s( β s ) β s exp s ( β s ) (7) (8) Proof: A proof of this theorem can be found in a number of literature including [], [3] 33 Methods with s p This section, considers the derivation procedure for methods which have the same number of stages as the order of the method Methods with this property are considered so as to minimize computational costs, and because, it is not possible to satisfy all the order conditions for s < p Also, the stability function is always a polynomial of degree s but the stability function is an approximation to exp(z) with an of (o{z p+ })as z This means that the stability function must have degree at least p and therefore s p 33 Derivation of Methods With s p 4 The general form of a fourth order, four stage ARK method is [ ] c c a c a c a c a 3 a 3 c 3 a 3 a 3 c 3 a 3c a 3 c b b b 3 b b b b 3 b β β β 3 β 4 β (9) The order conditions for a fourth order method are: b + b T e () b T c b T c 3 b T c 3 4 () () (3) b T Ac 6 b T Ac (4) (5) β T e + β (6) β T (I + β 4 A) β 4 e T 4 (7) c exp 4( β 4 ) β 4 exp 3 ( β 4 ) (8) ISSN: Issue 5, Volume, May

4 ( + β 4c )b T A c 4! (9) and (5) Provided c c, These methods have three free parameters, and we will take them to be c, c 3 and β 4 Once we assign values for these parameters, the method can then be uniquely defined using equations ()-(9) c can be calculated from equation (8), the b T vector can be found from the quadrature conditions (), () and (3), from where we get b 3 4c 4c 3 + 6c c 3 c (c c )(c c 3 ) b 3 4c 4c 3 + 6c c 3 c (c c )(c c 3 ) b 3 3 4c 4c + 6c c c 3 (c 3 c )(c 3 c ) () () () The next step is to obtain b from equation (), and a 3 from the linear combination of equations (4) a 3 c b 3 c (c c ) From conditions (4) and (9), a a 3 4b 3 a 3 c ( + β 4c ) (3) (4) 6 b 3a 3 c b a c b 3 c (5) Finally, β T can be found from equation (7) Based on the complete classification of fourth order methods with four stages given in [4], we present two methods based on Case 4 (ark4a) and Case 5 (ark4b) c T [,,, ], b ct [,,, ], β 4 [ ] , (6) 33 Derivation of Methods with s p 5 A fifth order, five stage ARK method takes the form c c a c a c a c a 3 a 3 c 3 a 3 a 3 c 3 a 3c a 3 c a 4 a 4 a 43 c 4 a 4 a 4 a 43 c 4 a 4c a 4 c a 43 c 3 b b b 3 b 4 b b b b 3 b 4 b β β β 3 β 4 β 5 β (7) with c [c, c, c 3 c 4, ] T The order conditions for a fifth order five stage method are: b + b T e (8) b T c (9) b T c 4 5 (3) c 4 (33) b T A b T (I C) (34) b T c 3 (3) b T (I C)Ac 4 (35) b T c 3 4 (3) b T (I C)Ac 6 (36) ISSN: Issue 5, Volume, May

5 and β T e + β (37) β T (I + β 5 A) β 5 e T 5 (38) c exp 5( β 5 ) β 5 exp 4 ( β 5 ) (39) ( + β 5c )b T A 3 c 5! (4) 333 Some Derived Methods Not much is known about the best choice of parameters for order five methods; however we will attempt to present some methods ARK5a and ARK5a respectively c T [ 5 65, 4,,, ], β c T [ 5 65,, 4 5,, ], β (4) (4) 34 Methods with s p+ From traditional Runge-Kutta theory, we know that we can achieve enhanced performance if there are more stages than are required for a particular order; the same applies to ARK methods The focus is on the case where there exists one more stage than is required to attain the required order 34 Methods with s 5, p 4 A fourth order method with five stages takes the form [ with stability function ] a a 3 a 3 e c Ae a 4 a 4 a 43 c Ae b b b 3 b 4 b b b 3 b 4 b β β β 3 β 4 β 5 β R(z) + z + z + z3 3! + z4 4! + Kz5 (43) ISSN: Issue 5, Volume, May

6 34 Order Conditions The following conditions must be satisfied by an ARK method with five stages to have order four b b T e (44) b T c b T c 3 (45) (46) b T c 3 4 (47) c 4 (48) b T Ac 6 b T Ac (49) (5) β β T e (5) β T c (5) β T Ac β 5(θ + ) φ θβ 5 (53) β 5 e T 5 (I + θa) β T (I + φa + β 5 θa ) (54) K( β 5c θα 4 α 5 ) ( + β 5c )(+α + α + α 3 3! + α 4 4! ) (55) where the values of α i are determined by expanding and + (φ β 5 )z + φz + β 5 θz α i z i (56) i b T A c 4 θ(bt A 3 c K) (57) β 5 (b T A c K) (β 5 φ)(b T A 3 c K) (58) 343 Some Examples Two methods are presented here; ark45a and ark45b respectively We have chosen L 5 and K for both methods, ensuring zero for both the bushy tree and the tall tree c T [ 4,, 3 4,, ], φ 4, µ 8, L 5, K c T [ 4, 3, 3 4,, ], φ 4, µ 8, L 5, K Implementation of ARK Methods (59) (6) The use of an ARK method is very similar to that of an RK method, with the main difference being that three pieces of information is now passed between steps The first two starting values are y(x )and hf(y(x ))respectively and the third starting value is obtained by taking a single Euler step forward and then taking the difference between the derivatives at these two points Therefore, the starting vector is ISSN: Issue 5, Volume, May

7 given by [y(x ), hf(y(x )), hf (y(x ) + hf(y(x ))) hf(y(x ))] (6) This choice of starting method was chosen for its simplicity, but it is adequate The method for computing the three starting approximations can be written in the form of the generalized Runge Kutta tableau where the zero in the first column of the last two rows indicates the fact that the term y n is absent from the output approximation This can be interpreted in the same way as a Runge Kutta method, but with three output approximations To change the we simply scale the vector in the same way we would scale a Nordsieck vector [5] Setting r h j /h j means the y vector needs to be scaled by [, r, r ] Once the starting vector is obtained from equation (6), the next thing is to calculate the internal stages; Y [n] i s j (hf (Y n i )a ij ) + Uy [n ] i (6) as well as the output approximations; i s j (hf (Y n i )B ij ) + V y [n ] i (63) 5 Numerical Experiments The problems that have been chosen are part of the DETest set of problems [6]; a set of standard test problems were suggested for testing ODE solvers Solutions have all been done with variable A: Logistics Curve: Single Equations y y 4 ( y ); y() Exact Solution: y E (t) e t 4 9+e t 4 B: The Radioactive Decay Chain: Moderate Systems y y y 9 y 9 9 with y() [,,,, ] T y y y 9 y C: A Non-linear Chemical Reaction Problem y y, y (), y y y, y (), y 3 y, y 3() ; (64) (65) D: A Problem Derived From Van der Pol s Equation y y, y (), y ( y )y y, y () 6 Discussions of Results (66) Figures to (see, next pages) show the results when the ARK methods we have derived in this paper, are used to solve problems A D From the s given by these methods, they may be considered as very good approximations of the exact results and we can say that they performed very well 7 Conclusion In this paper an effort has been made to present this unique class of methods and show that they are quite viable and reliable for solving not just single ODEs but large system as well For a more detailed analysis of Ark methods see [7] ISSN: Issue 5, Volume, May

8 Results for The Logistics Curve Problem ARK4a Results for The x Radioactive Decay Chain Problem ARK4a Figure : Solving Problem A using ark4a and ark4b Results for The Logistics Curve Problem ARK45a 6 3 Figure 4: Solving Problem B using ARK4a and 3 Results for The x Radioactive Decay Chain Problem ARK45a Figure : Solving Problem A using ark45a and ark45b 6 3 Results for The Logistics Curve Problem ARK5a Figure 5: Solving Problem B using ARK45a and 3 Results for The x Radioactive Decay Chain Problem ARK5a Figure 3: Solving Problem A using ARK5a and Figure 6: Solving Problem B using ARK5a and ISSN: Issue 5, Volume, May

9 6 8 Result for The Non linear Chemical Reaction Problem ARK4a Result for The Problem Derived From Van der Pol Equation ARK4a Figure 7: Solving Problem C using ARK4a and Figure : Solving Problem D using ARK4a and 7 8 Result for The Non linear Chemical Reaction Problem ARK45a 4 Result for The Problem Derived From Van der Pol Equation ARK45a Figure 8: Solving Problem C using ARK45a and Figure : Solving Problem D using ARK45a and 8 9 Result for The Non linear Chemical Reaction Problem ARK5a 4 Result for The Problem Derived From Van der Pol Equation ARK5a Figure 9: Solving Problem C using ARK5a and Figure : Solving Problem D using ARK5a and ISSN: Issue 5, Volume, May

10 Special Acknowledgement The authors wish to thank the ISAT fund of New Zealand for funding the recent visit of Abraham Ochoche, to the Department of Mathematics, The University of Auckland, Auckland, New Zealand where he had a one month stimulating exposure under Prof J C Butcher, who facilitated the visit Also, to acknowledge the immense contributions and cooperation received from Dr Nicolette Rattenbury, of the Manchester Metropolitan University, UK and Dr Helmut Podhaisky of the Institute of Mathematics, Martin-Luther-University, Halle- Wittenberg, Germany We also want to acknowledge Dr Allison Heard of the Department of Mathematics, University of Auckland, New Zealand for her immense contributions that led to the success of the Auckland visit References: [] Abraham, O, Improving the Improved Modified Euler Method for Better Performance on Autonomous Initial Value Problems, Leonardo J of Sciences, Issue, January June 8, [] Euler H, Institutiones Calculi Integralis Volumen Primum (768), Opera Omnia, Vol XI, B G Teubneri Lipsiae et Berolini MCMXIII [3] Butcher, J C, General Linear Methods: A Survey, Appl Numer Math (985) [4] Runge, C, Uber die Numerische Auflosung von Differentialgleichungen, Math Ann,46 (895), [5] Heun, K Neue Methoden zur Approximativen Integration Der Differentialgleichungen Einer UnabhAngigen VerAnderlichen, Z Math Phys, 45 (9), 3 38 [6] Kutta, W, Beitrag fur NAherungsweisen Integration Totaler Differentialgleichungen, Z Math Phys, 46 (9), [7] Burrage, K & Butcher, J C Non-Linear Stability for a General Class of Differential Equation Methods, BIT, (98), 85-3 [8] Bashforth, F and Adams, J C, An attempt to Test the Theories of Capillary Action by Comparing the Theoretical and Measured Forms of Drops of Fluid, with an Explanation of the Method of Integration Employed in Constructing the Tables which Give the Theoretical Forms of Such Drops, Cambridge University Press, Cambridge (883) [9] Butcher,J C, On the Convergence of Numerical Solutions of Ordinary Differential Equations, Math Comp, (966), [] Butcher, J C, An Introduction to Almost Runge Kutta Methods, Appl Numer Math, 4 (997),33 34 [] Rattenbury, N, Almost Runge-Kutta Methods for Stiff and Non-Stiff Problems, PhD Thesis, The University of Auckland, Department of Mathematics, 5 [] Rattenbury, N, Almost Runge-Kutta Methods: A Theoretical and Practical Investigation, MSc Thesis, The University of Auckland, department of Mathematics, (998) [3] Butcher, J C, Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, Second Edition (8) [4] Butcher, J C, ARK Method Up to Order Five,Numer Algorithm, 7 (998), 93- [5] Nordseick, A, On Numerical Integration of Ordinary Differential Equations, Math Comp, 6 (96), 49 [6] Hull, T E, Enright, W H, Fellen, B M and Sedgwick, A E, Comparing Numerical Methods for Ordinary Differential Equations, SIAM J Numer Anal, 9 (97), [7] Abraham, O, The Dynamics of Almost Runge- Kutta Methods, Abacus: J of the Math Assoc of Nig, Vol 33, Num A (6) 6 7 ISSN: Issue 5, Volume, May