Almost Runge-Kutta Methods for Stiff and Non-Stiff Problems

170 Pages·2006·4.06 MB·English

Almost Runge-Kutta Methods
for Stiff and Non-Stiff Problems
Nicolette Rattenbury
A thesis submitted in fulfilment of the requirements for the degree of
Doctor of Philosophy, The University of Auckland, 2005.

Abstract
Ordinary differential equations arise frequently in the study of the physical world. Un-
fortunately many cannot be solved exactly. This is why the ability to solve these equations
numerically is important.
Traditionallymathematicians haveusedoneoftwoclassesofmethodsfornumericallysolving
ordinary differential equations. These are linear multistep methods and Runge–Kutta methods.
General linear methods were introduced as a unifying framework for these traditional methods.
They have both the multi-stage nature of Runge–Kutta methods as well as the multi-value
nature of linear multistep methods. This extremely broad class of methods, besides containing
Runge–Kuttaandlinearmultistepmethodsasspecialcases, alsocontains hybridmethods,cyclic
composite linear multistep methods and pseudo Runge–Kutta methods.
In this thesis we present a class of methods known as Almost Runge–Kutta methods. This
is a special class of general linear methods which retains many of the properties of traditional
Runge–Kutta methods, but with some advantages.
Mostofthisthesisconcentratesonexplicitmethodsfornon-stiffdifferentialequations,paying
particular attention to a special fourth order method which, when implemented in the correct
way, behaves like order five. We will also introduce low order diagonally implicit methods for
solving stiff differential equations.

Acknowledgements
Duringthecourseof my PhDI have beenvery fortunateto receive the guidanceand support
of many wonderful people.
My supervisor Prof. John Butcher is an inspiration. His enthusiasm is contagious. It is hard
not to be excited about an idea when discussing it with him. Nobody could ask for a more
patient, caring and supportive supervisor and friend.
Dr Robert Chan, my co-supervisor, has also been very supportive. He was always available
when I wished to discuss my work.
My office mate, Dr Allison Heard, has been a wonderful mentor and friend. The many hours
she has spent proof-reading my work has been invaluable. As have the many hours spent poring
over the odd cryptic crossword!
Ourweekly numerical analysis meetings have also been a great source of support. They have
given me the chance to present my work informally and receive feedback. Apart from those I
have already mentioned, I would particularly like to thank Dr Will Wright, Dr Shirley Huang,
Angela Tsai andDr Helmut Podhaisky. They have all become good friendsas well as supportive
colleagues.
FinallyIwouldliketothankmyhusband,DrNicholasRattenbury. Thereisagreatquotation
from the famous Winnie the Pooh that sums up how I feel about him “If you live to be 100,
I want to live to be 100 minus one day, so I never have to live without you”. I am extremely
lucky to have found someone who believes in me as much as he does.

Contents
Abstract iii
Acknowledgements v
Contents vii
List of Tables xi
List of Figures xiii
1 Introduction 1
1.1 Ordinary differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Existence and uniqueness of solutions . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Stiff differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Delay differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 A brief history of numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 General linear methods 9
2.1 Consistency and stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 Stability regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Tree theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.1 Algebraic analysis of order . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Expansion of the exact solution . . . . . . . . . . . . . . . . . . . . . . . 19
Elementary weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Expansion of the numerical solution . . . . . . . . . . . . . . . . . . . . 23
viii Contents
2.4 Examples of general linear methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4.1 Runge–Kutta methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4.2 Linear multistep methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Adams methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
BDF methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.3 DIMSIMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.4 IRKS methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 Almost Runge–Kutta methods 35
3.1 General form of explicit ARK methods . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2 Order and related conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4 Methods with s = p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4.1 RK stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4.2 Third order methods with three stages . . . . . . . . . . . . . . . . . . . . . 47
Order conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Derivation of methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Some example methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.4.3 Fourth order methods with four stages . . . . . . . . . . . . . . . . . . . . . 53
Order conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Derivation of methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Classification of the methods . . . . . . . . . . . . . . . . . . . . . . . . 57
Some example methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.5 Methods with s = p+1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.5.1 RK-stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.5.2 Third order methods with four stages . . . . . . . . . . . . . . . . . . . . . . 64
Order conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Derivation of methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Some example methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.5.3 Fourth order method with five stages . . . . . . . . . . . . . . . . . . . . . . 71
Contents ix
Order conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Derivation of methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Some example methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4 A special ‘fifth’ order method 79
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2 Obtaining order 5 performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.3 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.4 Error estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.5 Optimising these methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.5.1 Fifth order error coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.5.2 Sixth order error coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5 Stiff ARK methods 99
5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.2 Order 3 stiff ARK methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Order conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Derivation of methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Some example methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.3 Order 4 stiff ARK methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Order conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Derivation of methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Some example methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.4 Starting the method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6 Numerical Experiments 115
6.1 Non-stiff methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.1.1 Fixed stepsize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.1.2 Fixed variable stepsize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.1.3 Variable stepsize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.1.4 DDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.2 Stiff methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7 Conclusions 135
x Contents
A Test Problems 139
A.1 DETest problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
A.2 Stiff problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
A.2.1 Oregonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
A.2.2 HIRES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
A.2.3 Prothero-Robinson problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
A.3 Delay differential equation problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
A.3.1 Equation 1.1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
A.3.2 Equation 1.1.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
A.3.3 Equation 1.1.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
A.3.4 Equation 1.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
A.3.5 Equation 1.4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
A.3.6 Equation 1.4.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
References 149
Index 155