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 to thank all the people who have helped me make this thesis possible It is not possible to list all here but I will name just 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 other lectures and professors at Faculty of Mathematics, Mechanics and 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 Finally, I express my deep appreciation to my family for all the wonderful, never-ending, unlimited support and encouragement I thank my parents, who have sacrificed so much for my education and have encouraged me toward 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 Introduction 1.1 1.2 Differential-algebraic equations 1.1.1 Definition of DAEs 1.1.2 Index of a DAE 1.1.3 Classification of DAEs 1.1.4 Special DAE Forms Runge-Kutta methods 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 13 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 Half-explicit Rung-Kutta methods 22 2.3 2.4 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 Discussion 30 Partitioned HERK methods for semi-explicit DAEs of index 31 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 of partitioned half-explicit Runge-Kutta methods 3.3 39 Construction of partitioned half-explicit Runge-Kutta methods 42 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 Bibliography 53 Abstract In recent years, the use of differential equations in connection with algebraic constraints on the variables, for example due to laws of conservation 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 important role in modern applied mathematics Fast and efficient numerical solvers for DAEs are highly desirable for finding solutions Many numerical methods 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 equations of index-1 have already discussed in my undergraduate thesis Therefore, this thesis concentrates on numerical methods for semi-explicit DAEs of index Here, we are concerned with one-step methods for index DAEs in Hessenberg form These methods combine efficient 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 Then, we also introduce half-explicit Runge-Kutta methods (HERK) that allows to solve more efficiently certain problems of the semi-explicit DAEs of index 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 superconvergence, we also pay a particular attention to partitioned half-explicit Rung-Kutta 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 is organized as follows Chapter provides some background material on differential-algebraic equations and Runge-Kutta methods Implicit Runge-Kutta and half-explicit Runge-Kutta methods applied to semiexplicit 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) arise in a variety of applications such as chemical process, physical process and electrical networks and modeling constrained multi-body system Therefore, their analysis and numerical treatment play an important role in modern applied mathematics This chapter gives an introduction to the theory of DAEs Some background material on DAEs and Runge-Kutta methods will be provided 1.1 1.1.1 Differential-algebraic equations Definition of DAEs A differential-algebraic equation (DAE) is an equation involving an unknown function and its derivatives A first order DAE is a system of equations of the form F (t, x, x) ˙ = 0, (1.1) where t ∈ ❘ is the independent variable (generally referred to as the ” time” variable), x(t) ∈ ❘n is the unknown function, and x(t) ˙ = dx dt (t) The function F : ❘ × ❘n × ❘n → ❘n is assumed to be differentiable The system (1.1) is a very general form of DAEs We consider in this thesis only initial value problem, i.e., system of the form (1.1) subject to the additional initial condition x(t0 ) = x0 for some initial time t0 ∈ ❘ and value x0 ∈ ❘n 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 when ∂F ∂ x˙ is singular • The method for solving of a DAE will depend on its structure A special but important class of DAEs of the form (1.1) is the semiexplicit DAE or ordinary differential equation (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 + = is a DAE To see this, determine the Jacobian ∂F ∂ x˙ of   x1 − x˙ +  F (t, x, x) ˙ = x˙ x2 +   x˙ with x˙ =  , so that x˙  ∂F = ∂ x˙ ∂F1  ∂ x˙ ∂F2 ∂ x˙  ∂F1 ∂ x˙  ∂F2 ∂ x˙  −1 = x2   , ( see that, det ∂F ∂ x˙ = 0) Hence, the Jacobian is a singular matrix irrespective of the values of x2 Furthermore, we observe that in this example the derivative x˙ does not appear 1.1.2 Index of a DAE Generally, the idea of all these index concepts is to classify DAEs with respect to their difficulty in the analytical as well as the numerical solution There are different index definitions: Kronecker index (for linear constant coefficient DAEs), differentiation index (Brenan et al 1996), perturbation index (Hairer et al 1996), tractability index (Griepentrog et al 1986), geometric index (Rabier et al 2002), and strangeness index (Kunkel et al 2006) In this thesis, the focus is set on the differentiation index DAEs are usually very complex and hard to be solved analytically Therefore, DAEs are commonly solved by using numerical methods Question: Is it possible to use numerical methods of ODEs for the solution of DAEs? Idea: Attempt to transform the DAE into an ODE This can be achieved through repeated derivations of the system with respect to time t method (3.8) is defined by δ y (y0 , z0 , h) = y(t0 + h) − y0 − hΦ(y0 , z0 , h), δ z (y0 , z0 , h) = z(t0 + h) − z0 − ϕ(y0 , z0 , h), (3.9) (3.10) where (y(t), z(t)) is the exact solution of (2.1) with initial values y(t0 ) = y0 , z(t0 ) = z0 Theorem 3.2.1 Let us consider the system (2.1), and the application of a PHERK method (3.2) rewritten as (3.8), with initial values (y0 , z0 ) satisfying (2.3) and gy (y0 )f (y0 , z0 ) = O(hl ) Let us denote h = max hi If in addition to |ws1 | < 1, (3.7) and δ y (y, z, h) = O(hp+1 ), δ z (y, z, h) = O(hm ) (3.11) are satisfied in a neighborhood of the solution, then, for nh < Constant, yn − y(tn ) = O(hmin(p,m+k,2m,m+l,l+k+1,2l+1) ), (3.12) zn − z(tn ) = O(hmin(p,m,l) ) (3.13) The following result, generalization of Lemma 3.9 of [7], will be used in the proof of next theorem: Lemma 3.2.4 Let un and be two sequences of non-negative numbers such that un+1 ≤ (1 + O(h))un + O( )vn + hM, vn+1 ≤ O(1)un + (α + O(h))vn + hN, with α, M, N ≥ 0, then, the following estimates hold for sufficiently small h, ≤ ch, nh ≤ Const : un ≤ C(u0 + v0 + M + N ), ≤ C(u0 + ( + (α∗ )n )v0 + M + hN ), if α∗ = |α + O(h)| < 1, ≤ C(u0 + v0 + M + N ), if |α| = 40 Proof It can be proven in a very similar way to Lemma 3.9 of [7], and it is based  on the decomposition  + O(h) O( )  = O(1) α + O(h)  −1   O( ) + O(h)     α + O(h) O(1) O(1) O( )   Theorem 3.2.2 Let us consider the system (2.1), and the application of a PHERK method given by (3.2), with initial values (y0 , z0 ) satisfying (2.3) and gy (y0 )f (y0 , z0 ) = O(hl ) Let us denote h = max hi If |α| < 1, and (3.11) and (3.7) are satisfied in a neighborhood of the solution, then, for nh ≤ Constant, yn − y(tn ) = O(hmin(p,m+k,2m,m+l,l+k+1,2l+1) ), zn − z(tn ) = O(hmin(p,m,l) ) (3.14) Proof The statement of this theorem can be proven comparing the numerical solution (yn , zn ) of (3.2) with the exact solution (y(tn ), z(tn )) of (2.1) with initial values (y0 , z00 ), where z00 is the solution of gy (y0 )f (y0 , z00 ) = which is closest to z0 Let us denote ∆yn = yn − y(tn ) and ∆zn = zn − z(tn ) The hypothesis of the theorem together with Lemma 3.2.3 implies that        k+1 y + h||∆zn ||) ∆y δ (y(tn ), z(tn ), hn ) ∆y I + O(h) O(h  n+1  =    n −  z ∆zn+1 O(1) αI + O(h + ||∆zn ||) ∆zn δ (y(tn ), z(tn ), hn ) First, assuming that the errors ||∆zn || are sufficiently small so that α∗ = |α + O(h) + max ||∆zn ||| < 1, Lemma 3.2.4 can be applied with un = ||∆yn ||, = ||∆zn ||, = h, α replaced by α∗ , M = O(hp ), and N = O(hm−1 ), which implies that ∆zn = O(hmin(p,m,l) ) 41 Second, we again apply Lemma 3.2.4 with un = ||∆yn ||, = ||∆zn ||, = hmin(p,m,l,k)+1 , M = O(hp ), and M = O(hm−1 ), which gives the estimate for ||∆yn || Remark 3.2.3 Lemma 3.2.2 guarantees that (3.7) is fulfilled at least for k = In Subsection 5.4 of [8], we develop a procedure to obtain in a systematic way sufficient conditions on the coefficients of the methods for (3.7) to be satisfied for k > The application with consistent initial values of a PHERK method such that |α| < is of order p for the differential variables if (3.11) and (3.7) are satisfied with m ≥ k, p − k Moreover, if m > p − k, the leading term of yn − y(tn ) is not influenced by the local error of the algebraic component In the case of |α| = 1, using the similar arguments to those of the proof of the Theorem, it can be proven that yn − y(tn ) = O(hmin(p,m+k,l+k+1,2m−1,2l+1,m+l) ), zn − z(tn ) (3.15) = O(hmin(p,m−1,l) ) 3.3 Construction of partitioned half-explicit Runge-Kutta methods If we want to construct PHERK methods of a given order, according to Theorem 3.2.1, it is desirable to express condition (3.11) and (3.7) in terms of the coefficients of the methods A systematic way of obtaining conditions on the parameters of a general partitioned Runge-Kutta method for (3.11) 42 and (3.7) to be satisfied (for given p, m, and k) is developed in Section of [10] Sufficient condition for 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 i−1 a ¯ij , ≤ i ≤ s + 1, aij , c¯i = ci = j=1 j=1 bi = as+1,i = a ¯si , ≤ i ≤ s, ≤ i ≤ s 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 s bj aji = bi (1 − ci ), (1 ≤ i ≤ s) j=i+1 Since Ys = Y¯s−1 = y1 , cs and c¯s−1 must be equal to Let us consider the following simplifying assumptions i ¯ C(q) : a ¯ij cl−1 j j=1 c¯li = , l ≤ i ≤ s, ≤ l ≤ q Remark 3.3.1 ¯ The condition C(1) is just the definition of the coefficient c¯i As the PHERK methods satisfy asi = bi for all i = 1, 2, , s, (3.7) holds at least for k = 1, and therefore, if only convergence of order is required for algebraic variable, according to Theorem of [10], the ¯ condition C(2) need to be satisfied We have the following results 43 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 wsi c¯3i = 3, i=1 then (3.11) is fulfilled with m = 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 method, s ¯ determining the remaining parameters so that C(2) and i=1 wsi c¯3i = hold, and choosing the free parameters w41 = and c¯2 , c¯4 in such a way that the local error coefficients and |w41 | are reasonably small Construction of the 3th order PHERK method divide in two steps: First, choose a standard order 3-stage IRK as follows 0 1 4 6 Hence, we determines the coefficients aij = for j ≥ i, a21 = 1, a31 = 14 , a32 = , c1 = 0, c2 = 1, c3 = 13 , b1 = b2 = 16 , b3 = 23 It is then straightforward to see that a 4-stage PHERK method have the coefficients satisfy a ¯31 = a ¯32 = 16 (= b1 = b2 ), a ¯33 = b3 = 32 ¯ Secondly, we will determine the remaining parameters so that C(2) and i=1 w3i a ¯3i = hold, and choose free parameters w41 = and w41 ∈ (−1, 1); c¯2 , c¯4 in such way that local error coefficients are reasonably small 44 So, we choose w41 = 12 , c¯2 = and c¯4 = 23 Then we need to find a ¯21 , a ¯22 , a ¯41 , a ¯42 , a ¯43 , a ¯44 satisfy the system of equations    a ¯21 + a ¯22 = 12        ¯41 + a ¯42 + a ¯43 + a ¯44 = a i c¯  a ¯ij cj = 2i    j=1       w c¯3 = i=1 ≤ i ≤ 4, 4i i Solving this systems (with c¯1 = 0, c¯2 = 21 , c¯3 = 1, c¯4 = 32 ) we get a ¯21 = 38 , a ¯22 = 18 , a ¯41 = 296 459 , a42 = 27 , a43 184 = − 459 , a44 = 14 51 Its coefficients are displayed in Table 3.1 ci aij 0 1 4 1 6 c¯i a ¯ij 8 1 6 3 296 459 27 − 184 459 14 51 Table 3.1: PHERK method of order Note that: Construction of the pth order PHERK method divide in two steps: First, choose a standard pth order explicit Runge- Kutta method This determines the coefficients aij of the PHERK method And second, determine the coefficients in order that the corresponding PHERK method is of the required order when applied to systems of the form (2.1) 45 It is well known that the construction of explicit Runge- Kutta methods of order higher than is best accomplished if the standard simplifying assumption D(1) is made s−1 bj aji = bi (1 − ci ), ≤ i ≤ s − D(1) : j=1 ¯ Let us now consider s-stage PHERK methods such that C(2) and D(1) are satisfied, and the underlying explicit method is of order We have the following result Theorem 3.3.2 Given a s-stage PHERK method (3.2) such that D(1) is satisfied and the underlying explicit Runge-Kutta method is of order for ¯ ODEs, if C(3) is satisfied, then (3.11) holds with p = and m = If in addition, condition wsi c¯4i = 4, i wsi c¯i a ¯ij ajl cl = i,j,l (3.16) holds, then (3.11) is fulfilled with p = and m = Using this result, we have constructed a 5-stage PHERK method (4 effective stages) of order (both y and z) based on the classical 3/8-rule, ¯ determining the remaining parameters so that C(3) and (3.16) hold, and choosing the free parameters (ws1 = and c¯3 ) in such a way that the local error coefficients and |ws1 | are reasonably small Its coefficients are displayed in Table 3.2 Table 3.2: PHERK method of order 46 ci 3.3.2 aij 0 3 - 31 1 -1 1 8 c¯i a ¯ij 8 161 1024 147 512 441 1024 1 8 8 10 693 5000 1701 5000 243 625 81 1250 81 − 2500 Methods of order and The construction of higher order explicit Runge-Kutta methods for ODEs usually relies the simplifying assumption D(1) and i ∗ aij cl−1 j C (q) : b2 = 0, j=1 cli = , ≤ i ≤ s, ≤ l ≤ q l When constructing PHERK methods of higher order, it seem interesting to consider methods satisfying (3.7) for k ≤ 2, so that the influence of the error for the z-components on the global error 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 ¯bj a ¯ji = bi (1 − ci ), ¯ D(1) : ≤ i ≤ s − 1, ¯b1 = 0, j=1 s−1 ¯ B(p) : ¯bi c¯l−1 = , i l i=2 ≤ l ≤ p Theorem 3.3.3 Let us consider a s-stage PHERK method (3.2) such that the underlying (s − 1)-stage explicit Runge-Kutta method satisfies D(1) and 47 ¯ ¯ ¯ C ∗ (2) and it is of order for ODEs If C(3), D(1), and B(5) are fulfilled, and s−1 ¯bi c¯i a ¯i2 = 0, (3.17) i=2 holds, then (3.11) and (3.7) are satisfied with p = 5, m = 3, k = 2, so that, if |ws1 | < 1, the method is of order for differential variables and of order 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 z-component does not affect the leading term of the global error for the y-component 3.4 Numerical experiment To illustrate the superconvergence results, we have applied the 4-stage PHERK method and 5-stage PHERK method with constant stepsize h to the semi-explicit system of index DAEs (2.18): To test the convergence of method (3.2), we integrated this problem between on t = and t = using 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 after n steps of length h = 1/n As erry ≈ Chp , and errz ≈ Chp for h1 ≥ h2 the value of numerical order: log py = erry (h2 ) erry (h1 ) log( hh21 ) log is taken as an approximation for p Similarly pz = errz (h2 ) errz (h1 ) h log( h2 ) is an approx- 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 48 Figure 3.1: The error results for 4-stage PHERK in the test problem on interval [0, 1] with h = 0.1 h1/h2 py1 py2 pz1 0.1/0.05 2.980 2.939 2.980 0.05/0.025 2.986 2.970 2.987 0.025/0.01 2.992 2.993 2.996 0.01/0.005 2.984 3.017 2.993 Table 3.1: The approximation of order of convergence for 4-stage PHERK method p = which confirm the theoretical results in Theorem 3.3.1 and the coefficients that we have just found it 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 49 Figure 3.2: The error results for 5-stage PHERK in the test problem on interval [0, 1] with h = 0.1 h1/h2 py1 py2 pz1 0.1/0.05 4.080 4.080 4.080 0.05/0.025 3.959 3.959 3.894 0.025/0.01 3.999 3.999 3.003 Table 3.2: The approximation of order of convergence for 5-stage PHERK method 3.5 Discussion We presented a new class of half-explicit methods for differential-algebraic systems of index that is based on the half-explicit Runge-Kutta methods of Hairer at al [4] They differ from these half-explicit methods in the substitution 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- 50 der explicit Runge-Kutta methods for ODEs can be extended to half-explicit for differential-algebraic systems 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 to the half-explicit Runge-Kutta methods the stage vector Yi in (3.2) is computed explicitly at the ith stage, and Zi is obtained by solving the equation that results when replacing 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 differential-algebraic equations (DAEs) can be solved efficiently by partitioned half-explicit Runge-Kutta methods and half-explicit Runge-Kutta methods However, the implicit methods is more suitable than explicit methods for solving stiff problems because their regions of absolute stability is large and have a higher-order accuracy In view of the recent development of half-explicit methods the PHERK methods (3.2) has special advantage in the application to constrained mechanical systems These methods are convergent with the same order as the underlying 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 for index DAEs of the type (2.1), the s extra internal stages Y¯i are required, thus finding more coefficient a ¯ij 51 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 discussed their advantages and disadvantages As the main result of the thesis, we have introduced the partitioned halfexplicit Runge-Kutta methods which have some advantages in comparison with implicit Runge-Kutta and half-explicit Runge-Kutta methods We discuss 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 have tried to discuss feature advantages and disadvantages of some classes of Runge-Kutta methods and some numerical experiments are carried out to illustrate the theoretical 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Differential Equation II: Stiff and Differential-Algebraic Problems, Springer-Verlag, 2nd edition, 1996 53 [8] L Jay, Convergence of a Class of Runge-Kutta Methods for DifferentialAlgebraic Systems of Index 2, BIT, 137-150 (1993) [9] A Murua, Partitioned Half-Explicit Runge-Kutta Methods for Differential-Algebraic Systems of Index 2, Computing 59, 4361 (1997) [10] A Murua, Partitioned Runge-Kutta 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 Thompson, Solving ODEs with MATLAB, Cambridge University Press, 2003 [12] N T H Tham, Solving Differential-algebraic Equations by Half-explicit Runge-Kutta methods, Undergraduate Thesis, Univ of Science (2012) 54 ... 1.1.4 Special DAE Forms Runge- Kutta methods 1.2.1 Formulation of Runge- Kutta methods 1.2.2 Classes of Runge- Kutta methods 10... provides some background material on differential- algebraic equations and Runge- Kutta methods Implicit Runge- Kutta and half-explicit Runge- Kutta methods applied to semiexplicit DAEs of index... 0 1 0 4 6 2-stage Runge- Kutta method 3-stage Runge- Kutta 0 0 2 0 2 0 0 1 6 6 4-stage Runge- Kutta Formula 10 1.2.2.2 Implicit Runge- Kutta methods Implicit Runge- Kutta (IRK) methods are more complicated