A component framework for the parallel solution of the incompressible Navier-Stokes equations with Radau-IIA methods

An efficient solution strategy for the simulation of incompressible fluids needs adequate and accurate space and time discretisation schemes. In this paper for the space discretisation we use an inf--sup stable finite element method and for the time discretisation Radau-IIA methods of higher order, which have the advantage that the pressure component has convergence order $s$ in time, where $s$ is the number of internal stages. The disadvantage of this approach is that we have a high computational amount of work, since large nonlinear systems of equations have to solved. In this paper we use a transformation of the coefficient matrix and the simplified Newton method. This approach has the effect that our large nonlinear systems split into smaller ones, which can now also be solved in parallel. For the parallelisation of the code we use the software component technology and the Component emplate Library (CTL). Numerical examples show that high order in the pressure component can be achieved and that the proposed solution technique is very effective.