Fourth-order Adaptive Mesh Refinement both in space and in time for incompressible Navier-Stokes equations with Dirichlet boundary conditions

Abstract: We present a fourth-order projection method with adaptive mesh refinement (AMR) for numerically solving the incompressible Navier-Stokes equations (INSE) with subcycling in time. Our method features (i) a reformulation of INSE so that the velocity divergence decays exponentially on the coarsest level, (ii) a derivation of coarse-fine interface conditions that preserves the decay of velocity divergence on any refinement level of the AMR hierarchy, (iii) an approximation of the coarse-fine interface conditions via spatiotemporal interpolations to facilitate subcycling in time, (iv) enforcing to machine precision solvability conditions of elliptic equations over each connected component of the subdomain covered by any refinement level, (v) a composite projection for synchronizing multiple levels, and (vi) geometric multigrid for solving linear systems with optimal complexity. Different from current block-structured AMR algorithms, our method never adopts refluxing at the coarse-fine interface, nor is fine-to-coarse averaging applied to projected velocities. Results of numerical tests demonstrate the high accuracy and efficiency of the proposed method.