DOI
10.17077/etd.133rx6r5
Document Type
Dissertation
Date of Degree
Summer 2018
Degree Name
PhD (Doctor of Philosophy)
Degree In
Applied Mathematical and Computational Sciences
First Advisor
Jay, Laurent
First Committee Member
Curtu, Rodica
Second Committee Member
Zhu, Xueyu
Third Committee Member
Xiao, Shaoping
Fourth Committee Member
Sugiyama, Hiroyuki
Abstract
In many applications, ordinary differential equations can be additively partitioned
\[y'=f(y)=\sum_{m=1}^{N}\f{}{m}(y).]
It can be advantageous to discriminate between the different parts of the right-hand side according to stiffness, nonlinearity, evaluation cost, etc. In 2015, Sandu and G\"{u}nther \cite{sandu2015gark} introduced Generalized Additive Runge-Kutta (GARK) methods which are given by
\begin{eqnarray*}
Y_{i}^{\{q\}} & = & y_{n}+h\sum_{m=1}^{N}\sum_{j=1}^{s^{\{m\}}}a_{i,j}^{\{q,m\}}f^{\{m\}}\left(Y_{j}^{\{m\}}\right)\\
& & \text{for } i=1,\dots,s^{\{q\}},\,q=1,\dots,N\\
y_{n+1} & = & y_{n}+h\sum_{m=1}^{N}\sum_{j=1}^{s^{\{m\}}}b_{j}^{\{m\}}f^{\{m\}}\left(Y_{j}^{\{m\}}\right)\end{eqnarray*}
with the corresponding generalized Butcher tableau
\[\begin{array}{c|ccc} \c{}{1} & \A{1,1} & \cdots & \A{1,N}\\\vdots & \vdots & \ddots & \vdots\\
\c{}{N} & \A{N,1} & \cdots & \A{N,N}\\\hline & \b{}{1} & \cdots & \b{}{N}\end{array}\]
The diagonal blocks $\left(\A{q,q},\b{}{q},\c{}{q}\right)$ can be chosen for example from standard Runge-Kutta methods, and the off-diagonal blocks $\A{q,m},\:q\neq m,$ act as coupling coefficients between the underlying methods. The case when $N=2$ and both diagonal blocks are implicit methods (IMIM) is examined.
This thesis presents order conditions and simplifying assumptions that can be used to choose the off-diagonal coupling blocks for IMIM methods. Error analysis is performed for stiff problems of the form
\begin{eqnarray*}\dot{y} & = & f(y,z)\\
\epsilon\dot{z} & = & g(y,z)\end{eqnarray*}
with small stiffness parameter $\epsilon.$ As $\epsilon\to 0,$ the problem reduces to an index 1 differential algebraic equation provided
$g_{z}(y,z)$ is invertible in a neighborhood of the solution. A tree theory is developed for IMIM methods applied to the reduced problem. Numerical results will be presented for several IMIM methods applied to the Van der Pol equation.
Keywords
numerical differential equations, Runge-Kutta methods, stiff differential equations, time integration
Pages
xi, 112 pages
Bibliography
Includes bibliographical references (page 112).
Copyright
Copyright © 2018 Gregory Mark Tanner
Recommended Citation
Tanner, Gregory Mark. "Generalized additive Runge-Kutta methods for stiff odes." PhD (Doctor of Philosophy) thesis, University of Iowa, 2018.
https://doi.org/10.17077/etd.133rx6r5