Comparison of some runge kutta methods for solving differential algebraic equations

VIETNAM NATIONAL UNIVERSITY, HANOI UNIVERSITY OF SCIENCE - VNUNGUYEN THI HONG THAM COMPARISON OF SOME RUNGE-KUTTA METHODS FOR SOLVING DIFFERENTIAL-ALGEBRAIC EQUATIONS MASTER OF SCIENCE T

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 MathematicsCode: 60460112

MASTER OF SCIENCE THESIS

THESIS SUPERVISOR:

Assoc Prof Dr VU HOANG LINH

Hanoi - 2017

I wish to thank all the other lectures and professors at Faculty of matics, Mechanics and Informatics of University of Science for their teaching,continuous support, tremendous research and study environment they havecreated I also thank to my classmates for their friendship and support Iwill never forget their care and kindness.

Mathe-Finally, I express my deep appreciation to my family for all the 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 nothave been able to finish my study and to complete this thesis

wonder-Hanoi, April 28th 2017

Student

Nguyen Thi Hong Tham

1.1 Differential-algebraic equations 1

1.1.1 Definition of DAEs 1

1.1.2 Index of a DAE 3

1.1.3 Classification of DAEs 4

1.1.4 Special DAE Forms 7

1.2 Runge-Kutta methods 8

1.2.1 Formulation of Runge-Kutta methods 8

1.2.2 Classes of Runge-Kutta methods 10

1.2.3 Simplifying assumptions 12

2 Implicit RK methods and half-explicit RK methods for semi-explicit DAEs of index 2 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

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

3 Partitioned HERK methods for semi-explicit DAEs of index 2 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 39

3.3 Construction of partitioned half-explicit Runge-Kutta methods 42 3.3.1 Methods of order up to 4 43

3.3.2 Methods of order 5 and 6 47

3.4 Numerical experiment 48

3.5 Discussion 50

In recent years, the use of differential equations in connection with gebraic constraints on the variables, for example due to laws of conserva-tion or position constraints, has become a widely accepted tool for modelingthe dynamical behaviour of physical processes Such combinations of bothdifferential and algebraic equations are called differential-algebraic equations(DAEs) Differential-algebraic equations arise in a variety of applications such

al-as modeling constrained multibody systems, electrical networks, aerospaceengineering, chemical processes, computational fluid dynamics, gas transportnetworks Therefore, their analysis and numerical treatment plays an impor-tant role in modern applied mathematics Fast and efficient numerical solversfor DAEs are highly desirable for finding solutions Many numerical meth-ods have been developed for DAEs Most numerical methods for differentialalgebraic equations based on standard methods from the theory of ordinarydifferential equations It is well known that the robust and numerically stableapplication of these ODE methods to higher index DAEs has to be based onthe 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-explicitDAEs of index 2

Here, we are concerned with one-step methods for index 2 DAEs in senberg form These methods combine efficient integrators for ODE theorywith a method to handle algebraic part We aim to present three classes ofRung-Kutta methods and give a comparison

Hes-We introduce primarily about implicit Rung-Kutta methods Then, we alsointroduce half-explicit Runge-Kutta methods (HERK) that allows to solve

more efficiently certain problems of the semi-explicit DAEs of index 2 formarising 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-explicitRung-Kutta methods (PHERK) A detailed analysis of these methods is alsopresented in this thesis We examine the existence and uniqueness of theproposed numerical solutions, the influence of perturbations, the local errorand global convergence and order conditions of the methods Furthermore,

we use MATLAB for numerical experiments on the Radau IIA, HERK andPHERK methods for DAEs of index 2 are presented

The thesis is organized as follows Chapter 1 provides some backgroundmaterial 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 arepresented in chapter 2 Chapter 3 is the main part of the thesis, in which wepay 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

Chapter 1

Introduction

Differential-algebraic equations (DAEs) arise in a variety of applicationssuch as chemical process, physical process and electrical networks and mod-eling constrained multi-body system Therefore, their analysis and numericaltreatment 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

variable), x(t) ∈ Rn is the unknown function, and ˙x(t) = dxdt(t) The function

F : R × Rn×Rn →Rn

is assumed to be differentiable

The system (1.1) is a very general form of DAEs We consider in this thesisonly initial value problem, i.e., 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

Remark 1.1.1

• In general, if the Jacobian matrix ∂F∂ ˙x is non-singular (invertible) thenthe system F (t, x, ˙x) = 0 can be transformed into an ordinary differ-ential equation (ODE) of the form ˙x = f (t, x) Numerical methods forODE models have been already well discussed Therefore, the mostinteresting case is when ∂F∂ ˙x is singular

• The method for solving of a DAE will depend on its structure Aspecial but important class of DAEs of the form (1.1) is the semi-explicit DAE or ordinary differential equation (ODE) with constraints

˙

y = f (t, y, z),

0 = g(t, y, z),

which appear frequently in applications

Example 1.1.1 The system

x1− ˙x1+ 2 = 0,

˙

x1x2+ 2 = 0

is a DAE To see this, determine the Jacobian ∂F∂ ˙x of

1.1.2 Index of a DAE

Generally, the idea of all these index concepts is to classify DAEs withrespect to their difficulty in the analytical as well as the numerical solution.There are different index definitions: Kronecker index (for linear constantcoefficient DAEs), differentiation index (Brenan et al 1996), perturbationindex (Hairer et al 1996), tractability index (Griepentrog et al 1986), geo-metric 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 fore, DAEs are commonly solved by using numerical methods

There-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 achievedthrough repeated derivations of the system with respect to time t

Definition 1.1.1 The minimum number of differentiation steps required totransform a DAE into an ODE is known as the (differentiation) index of theDAE.

Remark 1.1.2

• Index measures the distance from a DAE to its related ODE It revealsthe mathematical structure and potential complications in the analysisand the numerical solution of the DAE

• The higher the index of a DAE is, the more difficulties for its numericalsolutions appear

Example 1.1.2 Let q(t) be a given smooth function in Rn

The scalar equation

˙

y2 = ¨y1 = ¨q(t) This is an index-2 DAE (constraint differentiated twice to getODE for y2)

1.1.3 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

• 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 adifferential variable

• The equation 0 = g(t, x, z) called algebraic equation or a straint

con-Example 1.1.3 (Simple pendulum) Consider a simple pendulum scribed in the figure leads to equation

de-m¨x = −F

l x,m¨y = mgF

l y.

Conservation of mechanical energy: x2+ y2 = l2.

We have a semi-explicit DAE system:

is called fully implicit

Example 1.1.4 The system

1.1.4 Special DAE Forms

The general DAE F (t, y, ˙y) = 0 can include problems which are notwell-defined in mathematical sense, as well as problems which will result

in failure for any direct discretization method (i.e., without reformulation).Fortunately, many of higher-index problems encountered in practice can beexpressed as a combination of more restrictive structures of ODEs coupledwith constraints Differential and algebraic variables can be identified andtreated appropriately Algebraic variables can be eliminated with the samenumber of differentiations These are called Hessenberg forms

Hessenberg index-1 DAEs have the form

x0 = f (t, x, z),

0 = g(t, x, z), ∂g

∂z non-singular for all t.

Also called semi-explicit index-1 DAE This is very closely related to implicitODEs because we can solve (in principle) for z in terms of x, t (implicitfunction theorem)

Hessenberg index-2 DAEs have the form

x0 = f (t, x, z),

0 = g(t, x), ∂g

∂x

∂f

∂z nonsingular for all t.

Note that g is independent of z It is also called pure index-2 DAE (algebraicvariables are index-2 only, not a mixture of index 1 and 2)

It is also possible to take a semi-explicit index-2 DAE

˙

x = f (t, x, z),

0 = g(t, x)

to a fully implicit, index-1 form: Let ˙w = z Then,

one-RK methods have also been extended to DAEs In this section we describesome RK terminologies that will be useful later

1.2.1 Formulation of Runge-Kutta methods

In carrying out a step we evaluate s stage values

Y1, Y2, , Ysand s stage derivatives

k1, k2, , ks,using the formula ki = f (Yi)

Each Yi is defined as a linear combination of the kj added on to y0

Definition 1.2.1 Let b1, , bs, aij(i, j = 1, , s) be real numbers Let

is called an s − stage Runge-Kutta method

There are many different Runge-Kutta methods; individual ones are ally described by presenting their Butcher Array

cs as1 · · · ass

b1 · · · bswhere the ci are the row-sums of the matrix

Example 1.2.1

Runge-Kutta methods can be divided into two main types according tothe style of the matrix A If the A matrix is strictly lower triangular, thenthe method is called ” explicit”; otherwise, the method is called ” implicit”

1.2.2 Classes of Runge-Kutta methods

1.2.2.1 Explicit Runge-Kutta methods

For explicit methods, the Butcher array is strictly lower triangular Thus

it is common to omit the upper-diagonal terms when writing the Butcherarray:

c2 a21 0

. . .

cs as1 as,s−1 0

b1 bs−1 bswhere c2 = a21, c3 = a31+ a32, , cs = as1+ · · · + as,s−1

1 2

1-stage Explicit Euler method 2-stage Runge-Kutta method

2 3

1 2

2 6

2 6

1 6

1.2.2.2 Implicit Runge-Kutta methods

Implicit Runge-Kutta (IRK) methods are more complicated The diagonaland supra-diagonal elements of the Butcher array may be nonzero

c1 a11 a1s

cs as1 ass

b1 bsExamples:

1

2

1 2

1

1 12 12

1 2

1 2

1-stage Implicit Midpoint 2-stage Implicit Trapezoidal Rule

1

3

5 12

−1 12

1 34 14

3 4

1 4

0 12 −12

1 12 12

1 2

1 2

Remarks :

Implicit Runge-Kutta methods are suitable for stiff problems because theirregions of absolute stability are large In contrast, explicit Runge-Kuttamethods are not acceptable for stiff problems because they have an boundedregion of absolute stability

1.2.3 Simplifying assumptions

The following are properties that an RK method usually has For historicreasons these properties are known as the simplifying assumptions For themethod description we have to introduce:

q −1 are integrated exactly on the interval [0, ci], for each i, by the quadratureformula with weights ai1, , ais

An RK method is said to have stage order q if C(q) is satisfied

Chapter 2

Implicit RK methods and half-explicit RK methods for semi-explicit DAEs of index 2

Most numerical methods for differential-algebraic equations (DAEs) arebased on standard methods from the theory of ordinary differential equation(ODEs) 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 (e.g index, semi-explicit form, Hessenberg form ) In the presentchapter we focus on implicit Runge-Kutta (IRK) methods and half-explicitRunge-Kutta (HERK) methods for index 2 DAEs in Hessenberg form Weconsider the numerical solution of systems of semi-explicit index 2 differential-algebraic equations (DAEs) by methods based on Runge-Kutta coefficients

These methods combine efficient s-stage Runge-Kutta method from theory for differential part with a method to handle the algebraic part [2].

vari-gy(y)fz(y, z) is invertible (2.2)The initial values y0, z0 at t0are supposed to be given and to be consistent,i.e., to satisfy

The system of DAEs (2.1) is thus of index 2 [3, 7]

The system (2.1) can be integrated considering the system obtained from(2.1) differentiating once the constraint, i.e., the index 1 DAE

˙

y = f (y, z), 0 = gy(y)f (y, z) (2.4)Here, we are concerned with one-step methods that directly integratethe problem (2.1), without making use of the algebraic constraints of (2.3)

(the so called hidden constraints) The first one step methods for integratingdirectly the system (2.1) studied in the literature are implicit Runge-Kuttamethods [3, 7] (see also [5]) Half-explicit Runge- Kutta methods, proposed

in [3] and developed in [4] allows to solve more efficiently certain problems ofthe form (2.1) arising in the 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 definition of an s-stage Runge-Kutta method applied tosemi-explicit index 2 DAEs is described as follows [3, 7] We consider onestep with stepsize h starting from yn at tn The numerical solution yn+1,

zn+1 approximating the exact solution y(t), z(t) at tn+1 = tn+ h is given by

1 From the above standard application of implicit RK methods it is portant to notice that from (2.7b) all internal stages Yni(i = 1, , s)satisfy the constraint g(y) = 0, whereas the numerical solution yn+1generally does not However, for stiffly accurate RK methods, i.e.,for methods satisfying asi = bi for i = 1, , s we have yn+1 = Yns.Therefore g(yn+1) = 0 is automatically satisfied for such methods sincefrom (2.7b) for i = s we have g(Yns) = 0 Superconvergence of stifflyaccurate methods has been demonstrated [3, 5, 7].

im-2 Y1, Y2, , Ys and Z1, Z2, , Zs are obtained by solving (m +n) × s nonlinear equations at the same time Therefore, it can be verycomputationally expensive

2.2.2 Convergence of implicit Rung-Kutta methods

Existence and uniqueness of the system of nonlinear equations arisingwith implicit Runge-Kutta methods are shown in the following theorem

Theorem 2.2.1 Let η, ξ satisfy

g(η) = O(h2), gy(η)f (η, ξ) = O(h), (2.8)and suppose that (2.2) holds in an h-independent neighbourhood of (η, ξ) Ifthe Runge-Kutta matrix A = (aij) is invertible, then the nonlinear system

Definition 2.2.1 The difference between the numerical solution (y1, z1) andthe exact solution at x + h:

δyh(x) = y1− y(x + h), δzh(x) = y1− z(x + h)

is called the local error

Lemma 2.2.1 Suppose that the Runge-Kutta method satisfies with p ≥ q + 1and q ≥ 1 the conditions

k for k = 1, , q and all i.

Then the local error is of magnitude

δyh(x) = O(hq+1), P (x)δyh(x) = O(hq+2)

δzh(x) = O(hq)

(2.11)

Here P (x) is a projection given by

P (x) = I − Q(x), Q(x) = (fz(gyfz)−1gy)(y(x), z(x)) (2.12)The limit of the stability function at infinity plays a decisive role fordifferential algebraic equations For invertible A it is given by

R(∞) = 1 − bTA−11, where 1 = (1, 1, , 1)T (2.13)Convergence for the y-component

Theorem 2.2.2 Suppose that the estimate (2.2) holds in a neighbourhood

of the solution (y(x), z(x)) of (2.1) and that the initial values are consistent

If the Runge-Kutta matrix A is invertible, |R(∞)| < 1 and the local errorsatisfies

δyh(x) = O(hr), P (x)δyh(x) = O(hr+1) (2.14)

with P (x) given in (2.12), then the method (2.5) is convergent of order r,i.e.,

yn− y(xn) = O(hr) for xn = nh ≤ Const

If in addition δyh(x) = O(hr+1), then we have g(yn) = O(hr+1)

The proof of this theorem is given in [4, Theorem VII.4.5] and [10, rem 4.4]; it is therefore omitted

Theo-Convergence for the z-component

The following theorem shows that the global error for the z-component isessentially equal to its local error, if |R(∞)| < 1

Theorem 2.2.3 Suppose that the estimate (2.2) holds in a neighbourhood

of the solution (y(x), z(x)) of (2.1) and that the initial values are consistent.Assume that the Runge-Kutta matrix A is invertible and |R(∞)| < 1 If theglobal error of the y-component is O(hk), g(yn) = O(hk+1) and the local error

of the z-component is O(hk), then we have for the global error

zn− z(xn) = O(hk) for xn = nh ≤ Const

2.2.3 Order conditions

The goal now is to obtain a local error estimate for differential variables

y1 compared to the exact solution y(t) at t0 + h passing through consistentinitial values (y0, z0) at t0 In this thesis we will not define once again thewhole tree theory for semi-explicit index 2 DAEs which is found for example

in [3, 7] Definitions of tree t ∈ Ty and related quantities p(t), γ(t),etc., are

as in [3, Section 5] and [7, Section VII.5] We only mention the main idea,algorithm and results

We will derive the conditions on the method coefficients which ensure thatthe local error is of a given order This is obtained by comparing the Taylorexpansions of the exact and numerical solution

Taylor expansions of the exact solution

Theorem 2.2.4 For the exact solution of (2.1) we have:

where α(t), α(u) are integer coefficients

Taylor expansions of the numerical solution

Theorem 2.2.5 For the numerical solution of (2.7) we have:

α(u)γ(u)Φi(u)F (u)(y0, z0),

where α(t), α(u) are the same integer coefficients as in Theorem 2.2.4.Again (wij) denotes the inverse of the Runge-Kutta matrix (aij) Theorder conditions are now given as follows:

Theorem 2.2.6 The local error yh(x0) = y1− y(x0 + h) satisfies

yh(x0) = O(hr), P (x0)δyh(x0) = O(hr+1) (2.15)with P (x0) = I − (fz(gyfz)−1gy)(y0, z0), if the conditions

The order conditions can be read directly from the trees as follows: Weattach one letter i, j, k, l, to each vertex of the tree in Ty The left-handside of (2.16) is obtained as the sum over i, j, k, l, where the summand isthe product of

bi , if the letter i is attached to the root,

alm , if the vertex l is followed by a meagre vertex m,

wnp , if the vertex n is followed by a fat vertex p

Figure 2.1: Order conditions for order r

2.2.4 Numerical experiment

To illustrate the superconvergence results, we have applied the stiffly rate Radau II method with constant stepsize h to the following semi-explicitsystem of index 2 DAEs:

For the initial conditions y1(0) = 1, y2(0) = 1 at t0 = 0 the exact solution

to this test problem is given by

Table 2.1: The approximation of order of convergence for Radau IIA methods

Remark 2.2.3 The numerical results clearly show the order of convergence

p = 3 which confirm the theoretical results in Theorem 2.2.2 and Theorem2.2.3

Figure 2.2: The error results for Radau IIA in the test problem on interval[0, 1] with h = 0.1.

2.3 Half-explicit Rung-Kutta methods

This part introduces half-explicit Runge-Kutta methods for the solution

of differential-algebraic systems of index 2 Half-explicit methods only treatthe algebraic variables implicitly, hence they have computational advantagescompared to fully implicit schemes Definition of half-explicit Runge-Kuttamethod, convergence results and order conditions are presented A numericalexperiment, illustrating the theoretical results, is included

2.3.1 Formula of half-explicit Runge-Kutta methods

For the numerical solution of the differential-algebraic system (2.1), weconsider the following method (half-explicit Runge-Kutta method):

ob-2.3.2 Discussion of the convergence

The convergence analysis given here is based on the results for generalimplicit Runge-Kutta methods