A global time integration approach for realistic unsteady flow computations

A novel time integration approach is explored for unsteady flow computations. It is a multi-block formulation in time where one solves for all time levels within a block simultaneously. The time discretization within a block is based on the summation-by-parts (SBP) technique in time combined with the simultaneous-approximation-term (SAT) technique for imposing the initial condition. The approach is implicit, unconditionally stable and can be made high order accurate in time. The implicit system is solved by a dual time stepping technique. The technique has been implemented in a flow solver for unstructured grids and applied to an unsteady flow problem with vortex shedding over a cylinder. Four time integration approaches being 2 to 5 order accurate in time are evaluated and compared to the conventional 2 order backward difference (BDF2) method and a 4 order diagonally implicit Runge-Kutta scheme (ESDIRK64). The obtained orders of accuracy are higher than expected and correspond to the accuracy in the interior of the blocks, up to 8 order accuracy is obtained. The influence on the accuracy from the size of the time blocks is small. Smaller blocks are computationally more efficient though, and the efficiency increases with increased accuracy of the SBP operator and reduced size of time steps. The most accurate scheme, with a small time step and block size, is approximately as efficient as the ESDIRK64 scheme. There is a significant potential for improvements ranging from convergence acceleration techniques in dual time, alternative initialization of time blocks, and by introducing smaller time blocks based on alternative SBP operators.