Comparison of some runge kutta methods for solving differential algebraic equations

VIETNAM NATIONAL UNIVERSITY, HANOI UNIVERSITY OF SCIENCE - VNU NGUYEN THI HONG THAM COMPARISON OF SOME RUNGE-KUTTA METHODS FOR SOLVING DIFFERENTIAL-ALGEBRAIC EQUATIONS MASTER OF SCIENCE THESIS Hanoi - 2017 VIETNAM NATIONAL UNIVERSITY, HANOI UNIVERSITY OF SCIENCE - VNU - - - - - - - - - o0o - - - - - - - - - Nguyen Thi Hong Tham COMPARISON OF SOME RUNGE-KUTTA METHODS FOR SOLVING DIFFERENTIAL-ALGEBRAIC EQUATIONS Major: Applied Mathematics Code: 60460112 MASTER OF SCIENCE THESIS THESIS SUPERVISOR: Assoc Prof Dr VU HOANG LINH Hanoi - 2017 ACKNOWLEDGEMENT I would like toIt thank the people havebut helped make thesis possible is not all possible to listwho all here I will me name justthis a few First of all, I am very grateful to my supervisor Assoc Prof Dr Vu Hoang Linh, who has spent a lot of time guiding and encouraging me I would like to express my deepest gratitude to him for his enormous help, critical comment, advice and for providing inspiration which cannot expressed by words I wish to thank all the otherand lectures and professors at Faculty of Mathematics, Mechanics Informatics of University of Science for their teaching, continuous support, tremendous research and study environment they have created I also thank to my classmates for their friendship and support I will never forget their care and kindness I express deep appreciation to mysupport family forand all the Finally, wonderful, my never-ending, unlimited encouragement I thank my parents, who have sacrificed so much for my education and have encouraged me to- ward master degree Without their emotional support, I am sure I would not have been able to finish my study and to complete this thesis Hanoi, April 28th 2017 Student Nguyen Thi Hong Tham Contents Introduction1 1.1 Differential-algebraic equations .1 1.1.1 Definition of DAEs 1.1.2 Index of a DAE 1.1.3 Classification of DAEs .4 1.1.4 Special DAE Forms 1.2 Runge-Kutta methods .8 1.2.1 Formulation of Runge-Kutta methods 1.2.2 Classes of Runge-Kutta methods 10 1.2.3 Simplifying assumptions 12 Implicit RK methods and half-explicit RK methods for semiexplicit DAEs of index 213 2.1 Introduction 14 2.2 Implicit Runge-Kutta methods 15 2.2.1 Formula of implicit Runge-Kutta methods .15 2.2.2 Convergence of implicit Rung-Kutta methods 16 2.2.3 Order conditions 18 2.2.4 Numerical experiment 21 2.3 Half-explicit Rung-Kutta methods 22 2.3.1 Formula of half-explicit Runge-Kutta methods 23 2.3.2 Discussion of the convergence .23 2.3.3 Order conditions .24 2.3.4 Numerical experiment 28 2.4 Discussion 30 Partitioned HERK methods for semi-explicit DAEs of index 231 3.1 Introduction .32 3.2 Partitioned half-explicit Runge-Kutta methods 34 3.2.1 Definition of partitioned half-explicit RK method 34 3.2.2 Existence and influence of perturbations .35 3.2.3 Convergence Kutta methodsof partitioned half-explicit Runge 39 3.3 Construction of partitioned half-explicit Runge-Kutta methods42 3.3.1 Methods of order up to .43 3.3.2 Methods of order and .47 3.4 Numerical experiment 48 3.5 Discussion 50 Bibliography53 Abstract recent years, the use of equations in connection withInalgebraic constraints ondifferential the variables, for example due to laws of conserva- tion or position constraints, has become a widely accepted tool for modeling the dynamical behaviour of physical processes Such combinations of both differential and algebraic equations are called differential-algebraic equations (DAEs) Differential-algebraic equations arise in a variety of applications such as modeling constrained multibody systems, electrical networks, aerospace engineering, chemical processes, computational fluid dynamics, gas transport networks Therefore, their analysis and numerical treatment plays an impor- tant role in modern applied mathematics Fast and efficient numerical solvers for DAEs are highly desirable for finding solutions Many numerical meth- ods have been developed for DAEs Most numerical methods for differential algebraic equations based on standard methods from the theory of ordinary differential equations It is well known that the robust and numerically stable application of these ODE methods to higher index DAEs has to be based on the structure of the DAE Numerical methods for differential-algebraic equa- tions of index-1 have already discussed in my undergraduate thesis Therefore, thesis concentrates on numerical methods for semi-explicitthis DAEs of index Here, concerned one-step methods for efficient index DAEs in we Hes-are senberg form.with These methods combine integrators for ODE theory with a method to handle algebraic part We aim to present three classes of Rung-Kutta methods and give a comparison We introduce primarily about implicit Rung-Kutta methods we also introduce half-explicit Runge-Kutta methods (HERK)Then, that allows to solve more certain problems of the semi-explicit DAEs of index efficiently form arising in the simulation of multi-body systems in (index 2) descriptor form For half-explicit Rung-Kutta methods, although they are efficient, robust, and easy to implement, they suffer from order reduction To reestablish su- perconvergence, we also pay a particular attention to partitioned half-explicit RungKutta methods (PHERK) A detailed analysis of these methods is also presented in this thesis We examine the existence and uniqueness of the proposed numerical solutions, the influence of perturbations, the local error and global convergence and order conditions of the methods Furthermore, we use MATLAB for numerical experiments on the Radau IIA, HERK and PHERK methods for DAEs of index are presented The thesis material is organized follows Chapter equations provides some background on as differential-algebraic and Runge-Kutta methods Im- plicit Runge-Kutta and half-explicit Runge-Kutta methods applied to semi- explicit DAEs of index and the characteristic properties of each method are presented in chapter Chapter is the main part of the thesis, in which we pay particular attention to PHERK for approximating the numerical solution of non-stiff semi-explicit DAEs of index and their numerical experiments Finally, we discuss the pros and cons of each family of the methods Chapter Introduction Differential-algebraic equations (DAEs) physical arise in process a varietyand of applications such as chemical process, electrical networks and mod- eling constrained multi-body system Therefore, their analysis and numerical treatment play an important role in modern applied mathematics This chap- ter gives an introduction to the theory of DAEs Some background material on DAEs and Runge-Kutta methods will be provided 1.1 Differential-algebraic equations 1.1.1 Definition of DAEs differential-algebraic equation (DAE) is an equation involving an A unknown function and its derivatives A first order DAE is a system of equations of the form F (t, x, x˙ ) = 0, where t ∈ R (1.1) is the independent variable (generally referred to as the ” time” d t (t) = variable), x(t) ∈ Rn is the unknown function, and x˙ dx (t) The function F: R × R n × R n → R n is assumed to be differentiable The systemonly (1.1) is avalue very problem, general form of DAEs.ofWe in this thesis initial i.e., system theconsider form (1.1) subject to the ad- ditional initial ncondition x(t0) = x0 for some initial time t0 ∈ R and value x0 ∈ R Remark 1.1.1 • In general, if the Jacobian matrix ∂ F ∂ x˙ is non-singular (invertible) then the system F (t, x, x˙ ) = can be transformed into an ordinary differential equation (ODE) of the form x˙ = f (t, x) Numerical methods for ODE models have been already well discussed Therefore, the most interesting case is ∂ is singular when ∂F • The method for solving of a DAE will depend on its structure A special but important class of DAEs of theequation form (1.1) is the semi- explicit DAE or ordinary differential (ODE) with constraints y˙ = f (t, y, z), = g(t, y, z), which appear frequently in applications Example 1.1.1 The system x1 − x˙ + = 0, x˙ x2 + = and (3.7)5 to satisfied (for given p, m,for and k) is developed in Section of be [10] Sufficient condition a PHERK method to have order up to are also obtained with the help of these simplifying assumptions, which are reported in the present section Let us first fix some notation i−1 ci = Σ j= a ij , c¯i = i Σ j= a¯ij , ≤ i ≤ s + 1, bi = as+1,i = a¯si , ≤ i ≤ s, ≤ i ≤ s Σ Note that the last equality above is equivalent to j ¯bj aji = bi (1−ci ), (2 ≤ i ≤ s) We will consider PHERK methods whose underlying RK method satisfies the standard simplifying assumption j=Σsi+1 bjaji = bi(1 − ci), (1 ≤ i ≤ s) ¯ Since the Ys = Y s−1 = y1 , cs and c¯s−1 must be equal to Let us consider following simplifying assumptions c¯l i Σ C¯ (q) : a¯ij j j=1 c l−1 li , ≤ i ≤ s, ≤ l ≤ q = Remark 3.3.1 C¯ (1) is just the definition of the coefficient c¯i The condition As the PHERK methods satisfy asi = bi for all i = 1, 2, , s, (3.7) holds at least for k = 1,for and therefore, if only according convergence of order is required algebraic variable, to ¯ Theorem of [10], the condition C (2) need to be satisfied We have the following results 3.3.1 Methods of order up to Theorem 3.3.1 Let us consider a s-stage PHERK method (3.2) such that the underlying (s − 1)-stage explicit Runge-Kutta method is of order for ODEs If its coefficients satisfy C¯ (2), and |ws1 | < 1, then (3.11) holds with p = and m = If in addition s Σ i=1 i w c¯ then (3.11) is fulfilled with m = = 3,si Using this result, we have constructed a 4-stage PHERK method (3 effective stages) of order (both y and z) based on the below third order Σ method, determining the remaining parameters so that s C¯ (2) and i i= wsi c¯ = hold, and choosing the free parameters w41 ƒ= and c¯2 , c¯4 in such a way that the local error coefficients |w41 | are reasonably small Construction of the 3thand order PHERK method divide in two steps: First, choose a standard order 3-stage IRK as follows Hence, we determines the coefficients aij = for j ≥ i, 4a21 = 1, a31 0 = , a32 = , c1 = 0, c2 = 1, c3 = 1 1 4 , b1 =1 b2 1= 66 2, 3 b3 = It is then straightforward to see that a 4-stage PHERK method have the coefficients satisfy a¯31 = a¯32 = (= b1 = b2 ), a¯33 = b3 = Secondly, we will determine the remaining parameters C¯ (2) and Σ so that i= i w3i a¯ = hold, and choose free parameters w41 ƒ= and w41 ∈ (−1, 1); c¯2 , c¯4 in such way that local error coefficients are reasonably small So, we choose w41 = , c¯2 = and c¯4 = Then we need to find a¯21 , a¯22 , a¯41 , a¯42 , a¯43 , a¯44 satisfy the system of equations  a¯21 + i  a¯41 + a¯42 + a¯43 a¯ij cj =   w4i1 c¯i =  a¯ 22 = i= j=1 + a¯44 = c¯2 2 ≤ i ≤ 4, Solving this systems (with c¯1 = 0,2c¯2 = a¯21 =8 , a¯228 = 4541 = , a¯ 296 , a42 = , c¯3 = 1, c¯4 = 45 , a43 = − 184 , a44 = ) we get 14 Its coefficients are 3.1 a¯ij ci displayed in Table c j ¯i Table 3.1: PHERK of order 3method 8 1 Note that: Construction of the pth order PHERK method divide in two steps: 1 1 2 4 6 1 First, choose a standard the pth coefficients order explicit RungeKutta method This determines aij of the PHERK method 184 1 2 296 − 459 14 And second, determine the coefficients in order thatwhen the corresponding PHERK method is of the required order applied to systems of the form (2.1) It is well of known thathigher the construction of explicit Runge- ifKutta methods or- der than is best accomplished the standard simplifying assumption D(1) is made s−1 C¯ (2) and D(1) Σ D(1) : bjaji = bi(1 − ci), ≤ i ≤ s − j=1 Let us now consider s-stage PHERK methods such that are the satisfied, and the underlying explicit method is of order We have following result Theorem 3.3.2.and Given s-stage PHERK method (3.2) such that is D(1) is satisfied theaunderlying explicit Runge-Kutta method of order for ¯ ODEs, C in (3) is satisfied, then (3.11) holds with p = and m = If if addition, Σ condition w a c = jl l Σ si i i c¯ i,j, = l si c¯i (3.16) 4, w a¯ij holds, then (3.11) is fulfilled with p = and m = Using this stages) result, we have constructed a 5-stage (4 effective of order (both y and z) basedPHERK on themethod classical 3/8-rule, C¯ (3) and (3.16) hold, and determining the remaining parameters so that choosing the free parameters (ws1 ƒ= and c¯3 ) in such a way that the local error coefficients displayed in Tableand 3.2.|ws1| are reasonably small Its coefficients are Table 3.2: PHERK method of order c aij c¯i a¯ i ij 0 3 - 31 1 -1 1 8 8 8 161 102 147 512 441 102 1 8 8 693 500 1701 5000 243 625 81 1250 10 81 − 2500 3.3.2 Methods of order and construction of higher order explicit Runge-Kutta methods for The ODEs usually relies the simplifying assumption D(1) and l C ∗ (q) : b = 0,j=i Σ i j l− j c = l i , ≤ i ≤ s, ≤ l ≤ q a When constructing PHERK methods of higher order, it seem interesting to consider methods satisfying (3.7) for on k the ≤ 2, so that influence of the error for the z-components global error the for the y-components is reduced, and high order can be achieved for the differential variables with more modest order of convergence for the algebraic variables We now interested in PHERK methods such that (3.7) is satisfied for k = Given a s-stage PHERK method, we assume that there exist real numbers ¯b1 , , ¯bs−1 such that s−1 D¯ (1) : ¯bj a¯ji = bi (1 − ci ), ≤ i ≤ s − 1, ¯b1 = 0, l j=1 i=Σ ¯b c¯l−1 = , ≤ l ≤ p B¯ (p) Σ : the (ss−− 1)-stage D(1)underlying and Theorem 3.3.3 explicit Let usRunge-Kutta consider a method s-stagesatisfies PHERK method (3.2) such that i i ∗ C (2) and it is of order for ODEs If C¯ (3), D¯ (1), and B¯ (5) are fulfilled, and s−1 (3.17) Σ¯bi c¯i a¯i2 = 0, i=2 holds, then if(3.11) and 1, (3.7) satisfied p =5 5, = 3, k = 2, so that, |w | < the are method is ofwith order formdifferential variables and ofs1order for the algebraic variables If in addition (3.16) is satisfied, the method is of order for the algebraic variables, and the local error for the zcomponent does not affect the leading term of the global error for the y-component 3.4 Numerical experiment To illustrate superconvergence results, we have applied the 4-stage PHERK the method and 5-stage PHERK method with constant stepsize h to the semi-explicit system of index DAEs (2.18): To test bethe tween convergence integrated this problem on t of = method and (3.2), t = 1weusing constant stepsize h = 1/n (for various values of n) Let erry and errz be the error made on the y-component, z-component respectively pafter n steps of length h = 1/n As erry ≈ Ch p, and errz ≈ Ch for h1 ≥ h2 the value of numerical order: erry (h1) log err (hh ) log( y py = ) log errz (h2 ) z1 h1 is taken as an approximation for p Similarly err log( (h ) h2 h1) is an approx- pz = imation for p The results are displayed in Figure 3.1 and Figure 3.2 Remark 3.4.1 The numerical results clearly show the order of convergence Figure 3.1: The error for 4-stage PHERK in the test problem on interval [0, 1] withresults h = 0.1 h1/h2 py1 py2 pz1 0.1/0.05 2.980 2.939 2.980 0.05/0.0 2.986 2.970 2.987 2.992 2.993 2.996 2.984 3.017 2.993 25 0.025/0 01 0.01/0.0 05 Table The approximation of order of convergence for 4-stage PHERK3.1: method p = coeffi3 which confirm results Theorem 3.3.1 and the cients thatthe wetheoretical have just found it in above Remark 3.4.2 The numerical results clearly show the order of convergence p = which confirm the theoretical results in Theorem 3.3.2 and Table 3.2 Figure 3.2: The error for 5-stage PHERK in the test problem on interval [0, 1] withresults h = 0.1 h1/h2 py1 py2 pz1 0.1/0.05 4.080 4.080 4.080 0.05/0.0 3.959 3.959 3.894 3.999 3.999 3.003 25 0.025/0 01 Table The approximation of order of convergence for 5-stage PHERK3.2: method 3.5 Discussion We presented a systems new class of 2half-explicit methods for differential-algebraic of index that is based on the halfexplicit Runge-Kutta methods of Hairer at al [4] They differ from these half-explicit methods in the sub- stitution of the first stage by an explicit Runge-Kutta stages In addition, the PHERK methods have additional parameters a¯ij that may be chosen to get a high order of convergence With this modification well known high or- der explicit Runge-Kutta methods for systems ODEs can be extended to half-explicit for differential-algebraic without any order reduction Furthermore, the construction of high order method is simplified since most of the order conditions coincide with classical order condition for the underlying explicit Runge-Kutta method Similar methods the and stage vector Yito in the (3.2)half-explicit is computedRunge-Kutta explicitly at the ith stage, Z is obtained by solving the equation that results when replacingi in g(Yi) = the expression for Yi Their effective number stages only is s − Because of these two reasons, PHERK methods only require to solve s − nonlinear systems separately whose size m×m Thus, they are relatively simple, in comparison with implicit methods that need to solve (m + n) × s nonlinear equations at the same time Non-stiff equations (DAEs) can be solved efficientlydifferential-algebraic by partitioned half-explicit Runge-Kutta methods and half-explicit Runge-Kutta methods However, the implicit methods is more suitable than explicit meth- ods for solving stiff problems because their regions of absolute stability is large and have a higher-order accuracy In viewmethof the recent development of half-explicit methods the PHERK ods (3.2) has special advantage in the application to constrained mechanical systems These methods are convergent with the same order as the under- lying explicit Runge-Kutta methods if the local discretization error in the algebraic components is sufficiently small and stability conditions are satisfied For the approximation of the algebraic components the methods can be extended by additional stages However, a drawback of this approach is the fact that forrequired, index DAEs of the type the s extra stages are thus finding more(2.1), coefficient a¯ internal ij Y¯i Conclusion This thesis is devoted to the numerical solution of semi-explicit differentialalgebraic equations of index by Runge-Kutta methods We have presented implicit Runge-Kutta and half-explicit Runge-Kutta methods and also dis- cussed their advantages and disadvantages As the main of Runge-Kutta the thesis, methods we havewhich introduced the partitioned half-result explicit have some advantages in comparison with implicit Runge-Kutta and halfexplicit Runge-Kutta methods We dis- cuss the convergence, order conditions and implementation of PHERK in detail A numerical experiment is given to illustrate the superconvergence results of PHERK methods The main contributions of this thesis are that we to discuss feature advantages and disadvantages of have some tried classes of Runge-Kutta methods and some numerical experiments are carried out to illustrate the theoretical results Bibliography [1]M Arnold, Methods of withIndex Explicit Stages for Half-Explicit Differential-Runge-Kutta Algebraic Systems 2, submitted to BIT (1995) [2]M Arnold,sysA tems Murua, Non-stiff integrators for differentialalgebraic of index 2, Numerical Algorithms, 1998 Springer [3]M K Strehmel andSystems R Weiner, NumericalMethods, Solution of Arnold, Differential-Algebraic by The Runge-Kutta Lecture-Notes in Mathematics, Vol 1409, Springer- Verlag (1989) [4]V Brasey and E Hairer,Equations Half-Explicit Semi-Explicit Differential-Algebraic of Methods Index 2, for Siam J Numer Anal., Vol 30, No 2, 538-552 (1993) [5]K E Brenan L R Petzold, The Numerical of higher in- dexand Differential/ Algebraic Equations Solution by Implicit Runge-Kutta Methods, SIAM J Numer Anal., 26, 976-996 (1989) [6]E Hairer, S Equation P N ∅restt, and G Problems., Wanner Solving Ordinary Differential I: Nonstiff Springer-Varlag, 2nd edition, 1993 [7]E HairerII:and G Differential-Algebraic Wanner, Solving Ordinary Differential Equation Stiff and Problems, SpringerVerlag, 2nd edition, 1996 [8]L Jay, Convergence of Systems a Class ofofRunge-Kutta DifferentialAlgebraic Index 2, BIT,Methods 137-150for (1993) [9]A for Murua, Partitioned Half-Explicit Runge-Kutta Methods Differential-Algebraic Systems of Index 2, Computing 59, 4361 (1997) [10]A Murua, Partitioned RungeKutta methods for semi-explicit differential-algebraic systems of index 2,, Tech Report EHU-KZAAIKT-196, Univ of the Basque country, 1996 [11]L F Shampine, I Gladwell, and S Press, Thompson, with MATLAB, Cambridge University 2003.Solving ODEs [12]N T H Tham, Solvingmethods, Differential-algebraic Equations Halfexplicit Runge-Kutta Undergraduate Thesis, by Univ of Science (2012) ... 1.2.1 Formulation of Runge- Kutta methods 1.2.2 Classes of Runge- Kutta methods 10 1.2.3 Simplifying assumptions 12 Implicit RK methods and half-explicit RK methods for semiexplicit DAEs of. .. simulation of multi-body systems in (index 2) descriptor form 2.2 Implicit Runge- Kutta methods 2.2.1 Formula of implicit Runge- Kutta methods The standard of index an s-stage Runge- Kutta method... equations provides some background on as differential- algebraic and Runge- Kutta methods Im- plicit Runge- Kutta and half-explicit Runge- Kutta methods applied to semi- explicit DAEs of index and the