Analysis and Implementation of Adaptive Explicit TwoStep Methods
(2015) In Master's Theses in Mathematical Sciences NUMM11 20141Mathematics (Faculty of Engineering)
 Abstract
 Recently a new way of constructing variable stepsize multistep methods has been proposed, that parametrizes the entire domain of multistep methods. In the presented work the case of explicit twostep methods is looked at, analyzed and related to the already known theory of multistep methods. The error coefficient is derived as a function of the stepsize ratio and an upper limit to the method domain due to zero stability is found. The theory is used to introduce an implementation of a variable stepsize methods from a pair of explicit twostep methods and optimal parameters then chosen empirically. The chosen method is tested on benchmark problems.
 Popular Abstract
 Differential equations are a common type of mathematical problems to be encountered in many fields of study, such as a change of stock price in economics, a chemical reaction in chemistry or movement of an object in a gravitational field in physics. A differential equation is a mathematical model that describes the rate of change of a system. Knowing some starting value for the system, referred to as the initial value, the differential equation describes how the value will change.
The predatorprey equations, also known as the LotkaVolterra equations, are a famous example that describes the change in the populations of two distinct species of animals where one is the predator and the other is the prey. The differential equation would... (More)  Differential equations are a common type of mathematical problems to be encountered in many fields of study, such as a change of stock price in economics, a chemical reaction in chemistry or movement of an object in a gravitational field in physics. A differential equation is a mathematical model that describes the rate of change of a system. Knowing some starting value for the system, referred to as the initial value, the differential equation describes how the value will change.
The predatorprey equations, also known as the LotkaVolterra equations, are a famous example that describes the change in the populations of two distinct species of animals where one is the predator and the other is the prey. The differential equation would then be the rate of change over time of the two populations as they depend on each other. If there is plenty of prey, the predator population will grow, but as it gets larger, the prey will be eaten up faster and the population will shrink. As the prey population gets smaller, the predator population will follow due to lack of food. When the predator population gets smaller again the prey population will grow again and the system is back to the original populations.
When we talk about solving a differential equation, we wish to find a new mathematical model that allows us to find the future value of a system, knowing the initial value. While such a solution would be optimal, a mathematical model of this type cannot always be found. For the cases when finding such solution is not possible, we rely on computers to estimate the solution using numerical methods. A numerical method is a computer algorithm that with an approximation forwards the solution in small iterative steps. A numerical solution however contains an error from the computation and the task of getting the best solution is to minimize the error, but at the same time keep the computational time to the minimum. For this reason there exists broad range of numerical methods as different problems have different requirements. One way of getting less error is to make the stepsize of the method smaller. That however would mean more steps to reach the final destination and therefore longer computational time. As the same accuracy is not always needed throughout the entire solution, most modern methods allow the stepsize to change between steps. Those methods are referred to as variable stepsize methods.
One type of methods is known as multistep methods as they use the information from multiple previous steps to further the solution to the next step. A few such methods have been well known for quite some time but a recent development in the field has made it possible to construct all possible variable stepsize multistep methods. In this thesis the multistep methods that use the information from the previous two steps to compute the new step, known as twostep methods, were considered. As multistep methods have already been studied in detail, the work presented here was on analyzing and generalizing the already known theory for the new way of constructing methods. Furthermore the theoretical results were used to construct and implement a variable stepsize method. The successful implementation was applied to test problems for demonstration. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/studentpapers/record/8230514
 author
 Gardarsson Myrdal, Kjartan Kari ^{LU}
 supervisor

 Carmen ArĂ©valo ^{LU}
 organization
 course
 NUMM11 20141
 year
 2015
 type
 H2  Master's Degree (Two Years)
 subject
 keywords
 numerical methods, twostep, variable, multistep methods, ODE, adaptive, differential equations, parametric, explicit
 publication/series
 Master's Theses in Mathematical Sciences
 report number
 LUNFNA30202015
 ISSN
 14046342
 other publication id
 2015:E51
 language
 English
 id
 8230514
 date added to LUP
 20160330 16:22:12
 date last changed
 20160330 16:22:12
@misc{8230514, abstract = {Recently a new way of constructing variable stepsize multistep methods has been proposed, that parametrizes the entire domain of multistep methods. In the presented work the case of explicit twostep methods is looked at, analyzed and related to the already known theory of multistep methods. The error coefficient is derived as a function of the stepsize ratio and an upper limit to the method domain due to zero stability is found. The theory is used to introduce an implementation of a variable stepsize methods from a pair of explicit twostep methods and optimal parameters then chosen empirically. The chosen method is tested on benchmark problems.}, author = {Gardarsson Myrdal, Kjartan Kari}, issn = {14046342}, keyword = {numerical methods,twostep,variable,multistep methods,ODE,adaptive,differential equations,parametric,explicit}, language = {eng}, note = {Student Paper}, series = {Master's Theses in Mathematical Sciences}, title = {Analysis and Implementation of Adaptive Explicit TwoStep Methods}, year = {2015}, }