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\"{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.