A second-order energy-stable low-regularity integrator for the Navier--Stokes equations

A second-order exponential low-regularity integrator is proposed for the incompressible Navier–Stokes equations. The method requires only linear solves and satisfies a discrete energy-decay property. Second-order temporal convergence is established under a uniform $H^4$ solution bound, with an error constant that remains bounded as the viscosity tends to zero. For the Fourier–Galerkin discretization, the optimal spatial convergence rate is established for fixed viscosity. The method can also be extended to conforming finite element methods while retaining the discrete energy-decay structure. Numerical results confirm the convergence orders and robustness with respect to the viscosity.