Fourth-order Adaptive Mesh Refinement both in space and in time for incompressible Navier-Stokes equations with Dirichlet boundary conditions
Journal:
arXiv
Published Date:
Jun 3, 2025
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.
