Partitioned Krylov subspace iteration in implicit Runge–Kutta methods

Application of the method of lines to partial differential equation leads to very large, sparse systems of ordinary differential equations, which are usually stiff. We consider high-order implicit Runge–Kutta methods for the time-integration of these systems, and propose a partitioned Krylov method for the solution of the resulting linear systems. It is shown that this method allows for parallel computation in the construction of Krylov subspaces and in a few examples it converges much faster than GMRES.