Generalized additive Runge-Kutta methods for stiff odes
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\&qu...
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| Format: | Others |
| Language: | English |
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University of Iowa
2018
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| Online Access: | https://ir.uiowa.edu/etd/6507 https://ir.uiowa.edu/cgi/viewcontent.cgi?article=8007&context=etd |
| Summary: | 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. |
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