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 TIEU LUAN MOI download : skknchat@gmail.com 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. VU HOANG LINH Hanoi - 2017 TIEU LUAN MOI download : skknchat@gmail.com 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.
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 Mathe- matics, 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 wonder- ful, never-ending, unlimited support and 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 1 TIEU LUAN MOI download : skknchat@gmail.com Contents 1 Introduction 1 1.1 Differential-algebraic equations .1 Definition of DAEs .2 Index of a DAE .3 Classification of DAEs .4 Special DAE Forms .2 Runge-Kutta methods .1 Formulation of Runge-Kutta methods .2 Classes of Runge-Kutta methods. 12 2 Implicit RK methods and half-explicit RK methods for semi- explicit DAEs of index 2 13 2.2 Implicit Runge-Kutta methods .1 Formula of implicit Runge-Kutta methods .2 Convergence of implicit Rung-Kutta methods .3 Half-explicit Rung-Kutta methods. 22 2 TIEU LUAN MOI download : skknchat@gmail.1 Formula of half-explicit Runge-Kutta methods .2 Discussion of the convergence.
30 3 Partitioned HERK methods for semi-explicit DAEs of index 2 31 3.2 Partitioned half-explicit Runge-Kutta methods .1 Definition of partitioned half-explicit RK method .2 Existence and influence of perturbations .3 Convergence of partitioned half-explicit Runge-Kutta methods .3 Construction of partitioned half-explicit Runge-Kutta methods 42 3.1 Methods of order up to 4 .2 Methods of order 5 and 6. 50 Bibliography 53 3 TIEU LUAN MOI download : skknchat@gmail.com Abstract In recent years, the use of differential equations in connection with al- gebraic constraints on 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, this thesis concentrates on numerical methods for semi-explicit DAEs of index 2. Here, we are concerned with one-step methods for index 2 DAEs in Hes- senberg 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 TIEU LUAN MOI download : skknchat@gmail.com more efficiently certain problems of the semi-explicit DAEs of index 2 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 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 2 are presented. The thesis is organized as follows. Chapter 1 provides some background material on differential-algebraic equations and Runge-Kutta methods. Im- plicit Runge-Kutta and half-explicit Runge-Kutta methods applied to semi- explicit DAEs of index 2 and the characteristic properties of each method are presented in chapter 2.
Chapter 3 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 2 and their numerical experiments. Finally, we discuss the pros and cons of each family of the methods. 2 TIEU LUAN MOI download : skknchat@gmail.com Chapter 1 Introduction Differential-algebraic equations (DAEs) arise in a variety of applications such as chemical process, physical process and 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 Differential-algebraic equations 1.1 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, ẋ) = 0, (1.1) where t ∈ R is the independent variable (generally referred to as the ” time” 1 TIEU LUAN MOI download : skknchat@gmail. The function F : R × Rn × Rn → Rn is assumed to be differentiable.1) is a very general form of DAEs.
We consider in this thesis only initial value problem, i., system of the form (1.1) subject to the ad- ditional initial condition x(t0 ) = x0 for some initial time t0 ∈ R and value x0 ∈ Rn. • In general, if the Jacobian matrix ∂F ∂ ẋ is non-singular (invertible) then the system F (t, x, ẋ) = 0 can be transformed into an ordinary differ- ential equation (ODE) of the form ẋ = f (t, x). Numerical methods for ODE models have been already well discussed. Therefore, the most interesting case is when ∂F ∂ ẋ 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 semi- explicit DAE or ordinary differential equation (ODE) with constraints ẏ = f (t, y, z), 0 = g(t, y, z), which appear frequently in applications. The system x1 − ẋ1 + 2 = 0, ẋ1 x2 + 2 = 0 2 TIEU LUAN MOI download : skknchat@gmail.com is a DAE. To see this, determine the Jacobian ∂F ∂ ẋ of x1 − ẋ1 + 1 F (t, x, ẋ) = ẋ1 x2 + 2 ẋ1 with ẋ = , so that ẋ2 ∂F1 ∂F1 ∂F −1 0 , ( see that, det ∂F = 0).
= ∂ ẋ1 ∂ ẋ2 = ∂ ẋ ∂F2 ∂F2 x2 0 ∂ ẋ ∂ ẋ1 ∂ ẋ2 Hence, the Jacobian is a singular matrix irrespective of the values of x2. Furthermore, we observe that in this example the derivative ẋ2 does not appear.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), geo- metric index (Rabier et al. 2002), and strangeness index (Kunkel et al. In this thesis, the focus is set on the differentiation index.
DAEs are usually very complex and hard to be solved analytically. There- fore, 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.
3 TIEU LUAN MOI download : skknchat@gmail. The minimum number of differentiation steps required to transform a DAE into an ODE is known as the (differentiation) index of the DAE. • Index measures the distance from a DAE to its related ODE. It reveals the mathematical structure and potential complications in the analysis and the numerical solution of the DAE.
• The higher the index of a DAE is, the more difficulties for its numerical solutions appear. Let q(t) be a given smooth function in Rn. The scalar equation y = q(t) is a (trivial) index-1 DAE ( with a differentiation, you obtain an ODE for y). One differentiates the first equation to get y2 = ẏ1 = q̇(t) = y2 and then ẏ2 = ÿ1 = q̈(t) This is an index-2 DAE (constraint differentiated twice to get ODE for y2 ).3 Classification of DAEs Frequently, DAEs posses mathematical structure that are specific to a given application area.
As a result we have nonlinear DAEs, linear DAEs, etc. 4 TIEU LUAN MOI download : skknchat@gmail.com Nonlinear DAEs: In the DAE F (t, x, ẋ) = 0 if the function F is nonlinear with respect to any one of t, x or ẋ, then it is said to be a nonlinear DAE. Linear DAEs: A DAE of the form A(t)ẋ(t) + B(t)x(t) = c(t), where A(t) and B(t) are n × n matrices, is linear. If A(t) ≡ A and B(t) ≡ B, then we have time-invariant linear DAE.
Semi-explicit DAEs A DAE system given in the form x0 = f (t, x, z), (1.3) is called semi-explicit. • Note that the derivative of the variable z does not appear in the DAE. • Such a variable z is called an algebraic variable; while x is called a differential variable. • The equation 0 = g(t, x, z) called algebraic equation or a con- straint.
Consider a simple pendulum de- scribed in the figure leads to equation F mẍ = − x, l F mÿ = mg y. l 5 TIEU LUAN MOI download : skknchat@gmail.com Conservation of mechanical energy: x2 + y 2 = l2. We have a semi-explicit DAE system: ẋ1 = x3 , ẋ2 = x4 , F ẋ3 = − x1 , ml F ẋ4 = g x2 , l 0 = x2 + y 2 − l 2. Fully-implicit DAEs A DAE system of the form: F (t, x, ẋ) = 0 is called fully implicit.5) is a fully-implicit DAE.
• Any fully-implicit DAE can be always transformed into semi-explicit DAEs with index increased by one as follows: For F (t, y, ẏ) = 0, let ẏ = z. Then, we have ẏ = z, 0 = F (t, y, z).