Comparing Multistep Methods Within Parametric Classes to Determine Viability in Solver Applications
(2021) In Master's Theses in Mathematical Sciences NUMM11 20201Mathematics (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 prestudy 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 prestudy 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)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/studentpapers/record/9044000
 author
 Kraut, Michael Andrew ^{LU}
 supervisor

 Claus Führer ^{LU}
 organization
 alternative title
 En jämförande studie av parametriska flerstegs metoder
 course
 NUMM11 20201
 year
 2021
 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
 LUNFNA30342021
 ISSN
 14046342
 other publication id
 2021:E17
 language
 English
 id
 9044000
 date added to LUP
 20210611 16:57:14
 date last changed
 20210611 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 prestudy 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 = {{14046342}}, 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}}, }