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Construction of Adaptive Multistep Methods for Problems with Discontinuities, Invariants, and Constraints

Mohammadi, Fatemeh LU (2018)
Abstract (Swedish)
The wide variety of physical phenomena, such as motion of objects, reaction among chemical substances, electricity flow in a circuit can be described by equations with quantities that vary along time.
The rate at which a quantity is changing with respect to its independent variable (time), is represented by its derivative. Thus, these phenomena are modeled by equations with differential variables that are called
differential equations.

It is often impossible or cumbersome to find the exact solution of a differential equation since either there is no analytical solution for the model or the system is huge. Thus numerical methods are developed
to approximate the solution of differential equations. Numerical methods... (More)
The wide variety of physical phenomena, such as motion of objects, reaction among chemical substances, electricity flow in a circuit can be described by equations with quantities that vary along time.
The rate at which a quantity is changing with respect to its independent variable (time), is represented by its derivative. Thus, these phenomena are modeled by equations with differential variables that are called
differential equations.

It is often impossible or cumbersome to find the exact solution of a differential equation since either there is no analytical solution for the model or the system is huge. Thus numerical methods are developed
to approximate the solution of differential equations. Numerical methods made it possible for human beings to fulfill their dream to travel to other planets by computing the trajectory of spaceship with
the help of computers. Indeed the development in numerical methods is parallel to the growth in computer technology. On one hand the accuracy of the numerical solution is of high importance and
on the other hand how fast the solution is calculated.

To uniquely determine the solution of an ordinary differential equation, some outside condition is needed, typically an initial value or a boundary value.
Some numerical methods, in particular \textit{multistep methods}, demand several initial values to start the calculation of the solution. We suggest some techniques to provide adequate number of high accurate initial values
with least effort.

Often the numerical methods calculate the solution of differential equations at discrete time points. If these time points are equally spaced we have a \textit{fixed step-size} numerical solution and a
\textit{variable step-size} one otherwise.
In differential equations, variable step-size methods also called \textit{adaptive methods} are of significant importance. The location of the time points has to be selected such that an accurate numerical solution is obtained while keeping
the number of points small.
Adaptive methods take smaller step-sizes when needed while they allow for larger step-sizes when the accuracy is not affected by it.

We present an adaptive form of two classes of multistep methods.
The first class is called \textit{$\beta-$blocked multistep methods}. These are used for the solution of systems that contain both differential and non-differential equations. The second class is called \textit{strong stability preserving methods} and Those are applied to the solution of models such as those of sea-waves that experience crashes.
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Abstract
Adaptive multistep methods have been widely used to solve initial value problems. These ordinary differential equations (ODEs) may arise from semi-discretization of time-dependent partial differential equations
(PDEs) or may combine with some algebraic equations to represent a differential algebraic equations (DAEs).

In this thesis we study the initialization of multistep methods and parametrize some well-known classes
of multistep methods to obtain an adaptive formulation of those methods. The thesis is divided into three main parts; (re-)starting a multistep method, a polynomial formulation of strong stability preserving (SSP)
multistep methods and parametric formulation of $\beta-$blocked multistep... (More)
Adaptive multistep methods have been widely used to solve initial value problems. These ordinary differential equations (ODEs) may arise from semi-discretization of time-dependent partial differential equations
(PDEs) or may combine with some algebraic equations to represent a differential algebraic equations (DAEs).

In this thesis we study the initialization of multistep methods and parametrize some well-known classes
of multistep methods to obtain an adaptive formulation of those methods. The thesis is divided into three main parts; (re-)starting a multistep method, a polynomial formulation of strong stability preserving (SSP)
multistep methods and parametric formulation of $\beta-$blocked multistep methods.

Depending on the number of steps, a multistep method requires adequate number of initial values to
start the integration. In the view of first part, we look at the available initialization schemes and introduce two family of Runge--Kutta methods derived to start multistep methods with low computational cost and accurate initial values.
The proposed starters estimate the error by embedded methods.

The second part concerns the variable step-size $\beta-$blocked multistep methods. We use the polynomial formulation of multistep methods applied on ODEs to parametrize $\beta-$blocked multistep methods for
the solution of index-2 Euler-Lagrange DAEs. The performance of the adaptive formulation is verified by some numerical experiments.

For the last part, we apply a polynomial formulation of multistep methods to formulate SSP multistep methods that are applied for the solution of semi-discretized hyperbolic PDEs. This formulation
allows time adaptivity by construction.
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Please use this url to cite or link to this publication:
author
supervisor
opponent
  • Professor Weinmüller, Ewa B., Vienna University of Technology, Austria
organization
publishing date
type
Thesis
publication status
published
subject
keywords
multistep methods, Initialization, \beta-blocking, Discontinuity, Time adaptivity, Strong stability preserving, Differential algebraic equations
pages
162 pages
publisher
Lund University, Faculty of Science, Centre for Mathematical Sciences
defense location
Lecture hall MH:Gårding, Center for Mathematical Sciences, Sölvegatan 18, Lund
defense date
2018-09-28 10:00
ISBN
978-91-7753-787-8
978-91-7753-786-1
language
English
LU publication?
yes
id
3bfc6d50-827f-4ab3-bb8c-8382a66ca793
date added to LUP
2018-09-03 16:45:22
date last changed
2018-11-21 21:41:25
@phdthesis{3bfc6d50-827f-4ab3-bb8c-8382a66ca793,
  abstract     = {Adaptive multistep methods have been widely used to solve initial value problems. These ordinary differential equations (ODEs) may arise from semi-discretization of time-dependent partial differential equations<br/>(PDEs) or may combine with some algebraic equations to represent a differential algebraic equations (DAEs).<br/><br/>In this thesis we study the initialization of multistep methods and parametrize some well-known classes<br/>of multistep methods to obtain an adaptive formulation of those methods. The thesis is divided into three main parts; (re-)starting a multistep method, a polynomial formulation of strong stability preserving (SSP)<br/>multistep methods and parametric formulation of $\beta-$blocked multistep methods.<br/><br/>Depending on the number of steps, a multistep method requires adequate number of initial values to<br/>start the integration. In the view of first part, we look at the available initialization schemes and introduce two family of Runge--Kutta methods derived to start multistep methods with low computational cost and accurate initial values.<br/>The proposed starters estimate the error by embedded methods.<br/><br/>The second part concerns the variable step-size $\beta-$blocked multistep methods. We use the polynomial formulation of multistep methods applied on ODEs to parametrize $\beta-$blocked multistep methods for<br/>the solution of index-2 Euler-Lagrange DAEs. The performance of the adaptive formulation is verified by some numerical experiments. <br/><br/>For the last part, we apply a polynomial formulation of multistep methods to formulate SSP multistep methods that are applied for the solution of semi-discretized hyperbolic PDEs. This formulation<br/>allows time adaptivity by construction. <br/>},
  author       = {Mohammadi, Fatemeh},
  isbn         = {978-91-7753-787-8},
  keyword      = {multistep methods,Initialization,\beta-blocking,Discontinuity,Time adaptivity,Strong stability preserving,Differential algebraic equations},
  language     = {eng},
  pages        = {162},
  publisher    = {Lund University, Faculty of Science, Centre for Mathematical Sciences},
  school       = {Lund University},
  title        = {Construction of Adaptive Multistep Methods for Problems with Discontinuities, Invariants, and Constraints},
  year         = {2018},
}