A fourth-order approximate projection method for the incompressible Navier-Stokes equations on locally-refined periodic domains

In this follow-up of our previous work [Zhang et. al., A fourth-order accurate finitevolume method with structured adaptive mesh refinement for solving the advectiondiffusion equation, SIAM J. Sci. Comput. 34 (2012) B179-B201], the author proposes a high-order semi-implicit method for numerically solving the incompressible NavierStokes equations on locally-refined periodic domains. Fourth-order finite-volume stencils are employed for spatially discretizing various operators in the context of structured adaptive mesh refinement (AMR). Time integration adopts a fourthorder, semi-implicit, additive Runge-Kutta method to treat the non-stiff convection term explicitly and the stiff diffusion term implicitly. The divergence-free condition is fulfilled by an approximate projection operator. Altogether, these components yield a simple algorithm for simulating incompressible viscous flows on periodic domains with fourth-order accuracies both in time and in space. Results of numerical tests show that the proposed method is superior to previous second-order methods in terms of accuracy and efficiency. A major contribution of this work is the analysis of a fourth-order approximate projection operator.