DESCRIPTION AND USE OF LSODE, THE LIVERMORE SOLVER …

NASA

Reference

Publication

1327

Lawrence

Livermore Nationai

Laboratory Report

UCRL-ID-113855

1993

Description and Use of LSODE,

the Livermore Solver for

Ordinary Differential

Equations

,

Krishnan Radhakrishnan

Sverdrup Technology, Inc.

U w i s Research Center Group

Alan C. Hindmarsh

Lawrence Livermore National Laboratory

Livermore, CA

National Aermauhcs and

SpaceAdministration

Oftice of Management

Scienwic and Technical

Infmatim Rogram

1993

Preface

This document provides a comprehensive description of LSODE, a solver for

initial value problems in ordinary differential equation systems. It is intended to

bring together numerous materials documenting various aspects of LSODE,

including technical reports on the methods used, published papers on LSODE,

usage documentation contained within the LSODE source, and unpublished notes

on algorithmic details.

The three central chapters-n

methods, code description, and code usage-are

largely independent. Thus, for example, we intend that readers who are familiar

with the solution methods and interested in how they are implemented in LSODE

can read the Introduction and then chapter 3, Description of Code, without

reading chapter 2, Description and Implementation of Methods. Similarly, those

interested solely in how to use the code need read only the Introduction and then

chapter 4, Description of Code Usage. In this case chapter 5, Example Problem,

which illustrates code usage by means of a simple, stiffchemical kinetics problem,

supplements chapter 4 and may be of further assistance.

Although this document is intended mainly for users of LSODE, it can be used

as supplementary reading material for graduate and advanced undergraduate

courses on numerical methods. Engineers and scientists who use numerical

solution methods for ordinary differential equations may also benefit from this

document.

...

Ill

Contents

ListofFigures

.................................................

ix

,

ListofTables

...................................................

Chapter1 Introduction

..........................................

1

Chapter 2 Description and Implementation of Methods ...............7

7

2.1 Linear Multistep Methods ....................................

2.2 Corrector Iteration Methods ...................................

9

2.2.1 Functional Iteration ..................................

11

2.2.2 Newton-Raphson Iteration ............................

13

15

2.2.3 Jacobi-Newton Iteration ..............................

2.2.4 Unified Formulation .................................

15

2.3 MatrixFormulation ........................................

17

2.4 Nordsieck's History Matrix .................................

-20

28

2.5 Local Truncation Error Estimate and Control ....................

2.6 Corrector Convergence Test and Control ........................

32

2.7 Step Size and Method Order Selection and Change ...............33

2.8 Interpolation at Output Stations ...............................

36

2.9 Starting Procedure .........................................

39

Chapter 3 Description of Code ...................................

41

3.1 Integration and Corrector Iteration Methods .....................

41

3.2 Codestructure ............................................

42

46

3.3 Internal Communication ....................................

3.4 Special Features ........................................... 46

3.4.1 -Initial Step Size Calculation ...........................

57

3.4.2 Switching Methods .................................. 64

3.4.3 Excessive Accuracy Specification Test ................... 64

3.4.4 Calculation of Method Coefficients ....................

-65

3.4.5 Numerical Jacobians .................................

66

3.4.6 Solution of Linear System of Equations ..................69

70

3.4.7 Jacobian Matrix Update ..............................

V

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