Immersed Boundary Smooth Extension (IBSE): A high-order method for solving incompressible flows in arbitrary smooth domains

摘要

The Immersed Boundary method is a simple, efficient, and robust numericalscheme for solving PDE in general domains, yet for fluid problems it onlyachieves first-order spatial accuracy near embedded boundaries for the velocityfield and fails to converge pointwise for elements of the stress tensor. In aprevious work we introduced the Immersed Boundary Smooth Extension (IBSE)method, a variation of the IB method that achieves high-order accuracy forelliptic PDE by smoothly extending the unknown solution of the PDE from a givensmooth domain to a larger computational domain, enabling the use of simpleCartesian-grid discretizations. In this work, we extend the IBSE method toallow for the imposition of a divergence constraint, and demonstrate high-orderconvergence for the Stokes and incompressible Navier-Stokes equations: up tothird-order pointwise convergence for the velocity field, and second-orderpointwise convergence for all elements of the stress tensor. The method isflexible to the underlying discretization: we demonstrate solutions producedusing both a Fourier spectral discretization and a standard second-orderfinite-difference discretization.