Numerical analysis of a 4th-order time parallel algorithm for the time-dependent Navier-Stokes equations

This work studies the stability and convergency properties for a time parallel algorithm: RIDC4 with reduced stencils, namely reduced 4th revisionist integral deferred correction method. For the spatial approximation of the velocity and the pressure, a finite element space discretization is applied. We proved that the scheme is almost unconditionally stable and is of 4th-order accuracy in time. Finally, numerical experiments are carried out to verify its time and space convergence orders and the formidable parallel efficiency demonstrated by the time speed-up compared with the classical Backward Euler (BE) methods. In conclusion, within almost same time to BE, the (reduced) RIDC4 scheme could achieve 4th order accuracy in time direction, provided four cores are available.