ODE recursions and iterative solvers for linear equations

Timestepping to a steady-state solution is increasingly applied in engineering and scientific applications as a means for solving equilibrium problems. In the present work we examine the relation between the recursion in timestepping algorithms for semidiscrete systems of ODEs and certain types of iterative methods for solving discretized systems of equilibrium PDEs. We consider, in particular, the possibility of accelerating the ODE approach using recursions that are not time accurate together with parameter selection based on the theory of iterative methods. As one example, we take the parameters arising from the Chebyshev-type iterative methods and use them in a two-stage Runge-Kutta scheme. A comparison study for a representative steady-state diffusion problem indicates a dramatic improvement in convergence and efficiency. We remark that this approach can be trivially incorporated into existing time-integration codes to significant advantage. This yields a hybrid adaptive approach in a single code.