A Low‐Cost, Second‐Order Artificial Compression Method for Incompressible Navier–Stokes Equations

In this article, we propose a straightforward and efficient second‐order artificial compression (AC) method for solving the incompressible Navier–Stokes equations. The core strategy of this method involves incorporating a time filter at each time step following the original backward Euler AC method. This approach not only decouples the velocity and pressure but also enhances the temporal accuracy from first‐order to second‐order without introducing additional computational complexity. We rigorously establish the unconditional stability of the method and derive detailed error estimates. Furthermore, by integrating the time filter technique with adaptive algorithm design principles, we develop a simplified adaptive stepsize AC algorithm and a simplified variable‐stepsize, variable‐order AC algorithm. Extensive numerical experiments are conducted to demonstrate the efficiency and applicability of the proposed algorithms. The results resolve a well‐known open problem in doubly‐adaptive artificial compression methods for incompressible flow.