Analysis and Implementation of Adaptive Explicit Two-Step Methods

Recently a new way of constructing variable step-size multistep methods has been proposed, that parametrizes the entire domain of multistep methods. In the presented work the case of explicit two-step 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 step-size 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 step-size methods from a pair of explicit two-step methods and optimal parameters then chosen empirically. The chosen method is tested on benchmark problems.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 predator-prey equations, also known as the Lotka-Volterra 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 step-size 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 step-size to change between steps. Those methods are referred to as variable step-size 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 step-size multistep methods. In this thesis the multistep methods that use the information from the previous two steps to compute the new step, known as two-step 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 step-size method. The successful implementation was applied to test problems for demonstration