# LUP Student Papers

## LUND UNIVERSITY LIBRARIES

### Comparing Multistep Methods Within Parametric Classes to Determine Viability in Solver Applications

(2021) In Master's Theses in Mathematical Sciences NUMM11 20201
Mathematics (Faculty of Engineering)
Mathematics (Faculty of Sciences)
Centre for Mathematical Sciences
Abstract
In the 2017 paper by Arévalo and Söderlind, a framework was established for creating linear multistep methods of various classes based on a polynomial formulation which includes variable step size adaptivity. These classes are: k step explicit and implicit methods of order k and k+1 respectively for nonstiff problems (such as Adams methods), and k step implicit methods of order k for stiff problems (such as BDF methods). For each method class and order, all multistep methods of maximal order, including those which lack zero stability, are given by a parametrization depending on the method class and order. In this paper we conduct a pre-study on low order methods, comparing the properties of methods of the same class and order, and present... (More)
In the 2017 paper by Arévalo and Söderlind, a framework was established for creating linear multistep methods of various classes based on a polynomial formulation which includes variable step size adaptivity. These classes are: k step explicit and implicit methods of order k and k+1 respectively for nonstiff problems (such as Adams methods), and k step implicit methods of order k for stiff problems (such as BDF methods). For each method class and order, all multistep methods of maximal order, including those which lack zero stability, are given by a parametrization depending on the method class and order. In this paper we conduct a pre-study on low order methods, comparing the properties of methods of the same class and order, and present experimental results when these methods are applied to simple test problems. We are motivated by the possibility of using method changes as a primary means of error control in solvers alongside traditional error control tools such as step size variability and order control. This paper also discusses some of the difficulties encountered during the research and concludes with questions for future study. (Less)
Popular Abstract
An ordinary differential equation is an equation which says that the way a system changes depends on its current state. For example, if money is invested in a savings account, then the money will grow at a rate depending on the amount of money in the account. The goal is to determine the exact behavior of the system through time, given a starting value and knowing that it behaves according to a differential equation. Many problems in science and engineering can be stated this way, although for most differential equations it is difficult or impossible to determine an exact solution. Therefore, it is common to use numerical methods to approximate the solution. Numerical methods are not exact and there are many different types of numerical... (More)
An ordinary differential equation is an equation which says that the way a system changes depends on its current state. For example, if money is invested in a savings account, then the money will grow at a rate depending on the amount of money in the account. The goal is to determine the exact behavior of the system through time, given a starting value and knowing that it behaves according to a differential equation. Many problems in science and engineering can be stated this way, although for most differential equations it is difficult or impossible to determine an exact solution. Therefore, it is common to use numerical methods to approximate the solution. Numerical methods are not exact and there are many different types of numerical methods, each with its own strengths and weaknesses. In this paper we examine some new classes of numerical methods and compare methods within each class, first determining the analytic properties of each method, and then using those methods to solve test problems to see how these properties may affect performance. (Less)
author
supervisor
organization
alternative title
En jämförande studie av parametriska flerstegs metoder
course
NUMM11 20201
year
type
H2 - Master's Degree (Two Years)
subject
keywords
Ordinary Differential Equations, Initial Value Problems, Linear Multistep Methods, Numerical Analysis, Numerical Solvers, Parametrized, Algorithms, Control Theory
publication/series
Master's Theses in Mathematical Sciences
report number
LUNFNA-3034-2021
ISSN
1404-6342
other publication id
2021:E17
language
English
id
9044000
2021-06-11 16:57:14
date last changed
2021-06-11 16:57:14
```@misc{9044000,
abstract     = {{In the 2017 paper by Arévalo and Söderlind, a framework was established for creating linear multistep methods of various classes based on a polynomial formulation which includes variable step size adaptivity. These classes are: k step explicit and implicit methods of order k and k+1 respectively for nonstiff problems (such as Adams methods), and k step implicit methods of order k for stiff problems (such as BDF methods). For each method class and order, all multistep methods of maximal order, including those which lack zero stability, are given by a parametrization depending on the method class and order. In this paper we conduct a pre-study on low order methods, comparing the properties of methods of the same class and order, and present experimental results when these methods are applied to simple test problems. We are motivated by the possibility of using method changes as a primary means of error control in solvers alongside traditional error control tools such as step size variability and order control. This paper also discusses some of the difficulties encountered during the research and concludes with questions for future study.}},
author       = {{Kraut, Michael Andrew}},
issn         = {{1404-6342}},
language     = {{eng}},
note         = {{Student Paper}},
series       = {{Master's Theses in Mathematical Sciences}},
title        = {{Comparing Multistep Methods Within Parametric Classes to Determine Viability in Solver Applications}},
year         = {{2021}},
}

```