The Variable Time-stepping DLN-Ensemble Algorithms for Incompressible Navier-Stokes Equations

In the report, we propose a family of variable time-stepping ensemble algorithms for solving multiple incompressible Navier-Stokes equations (NSE) at one pass. The one-leg, two-step methods designed by Dahlquist, Liniger, and Nevanlinna (henceforth the DLN method) are non-linearly stable and second-order accurate under arbitrary time grids. We design the family of variable time-stepping DLN-Ensemble algorithms for multiple systems of NSE and prove that its numerical solutions are stable and second-order accurate in velocity under moderate time-step restrictions. Meanwhile, the family of algorithms can be equivalently implemented by a simple refactorization process: adding time filters on the backward Euler ensemble algorithm. In practice, we raise one time adaptive mechanism (based on the local truncation error criterion) for the family of DLN-Ensemble algorithms to balance accuracy and computational costs. Several numerical tests are to support the main conclusions of the report. The constant step test confirms the second-order convergence and time efficiency. The variable step test verifies the stability of the numerical solutions and the time efficiency of the adaptive mechanism.