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

Laurent Jay

First Committee Member

Rodica Curtu

Second Committee Member

Xueyu Zhu

Third Committee Member

Shaoping Xiao

Fourth Committee Member

Hiroyuki Sugiyama

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

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